Dependencies

For \(\beta \ge 0\), let \(k_\beta (t)=|t|^\beta \) on \(\mathbb {R}\), a continuous real function, with \(0^0=1\).

For arbitrary \(a,b\) define \(A\); for \(a\ne b\) define its inverse coordinate \(B\):

References: Chebfun interval scaling [DHT14]; [DLMF, 3.11.12].

Let \(\mathcal H_q=\bigoplus _{s}^{\ell ^2}H_{t,m(s)}\), indexed by all real cosine and nonzero sine sectors \(s\). A physical polynomial \(BL_{m(s)}p\) embeds into sector \(s\) as \(\sqrt{\kappa _{m(s)}}\, p\).

Let \(0{\lt}q{\lt}1\), \(r^2=x^2+y^2\), and let \(p\) be a real Cartesian polynomial. With the smooth boundary factor \(F_a\) of Theorem 5.4.62, set

For \(t\in \mathbb R\setminus \{ 0\} \) and \(\gamma \in \mathbb R\), set \(Ep=(X(1-X)p)'\) and \(A_{t,\gamma }p=(4/t)((t-X)(Ep)'-(\gamma +1)Ep)\). These are linear maps on real polynomials.

For \(t\in \mathbb R\setminus \{ 0\} \), define the polynomial \(W_t(X)=t^{-2}X(1-X)\).

For fixed \(t{\gt}1\) and real \(\gamma \), equip radial polynomials with the inner product \((p,z)_E=-\langle Ap,z\rangle _{1,1,\gamma }\) of Theorem 5.4.22, and let \(H_{t,\gamma }\) be its Hilbert completion, with the canonical isometric polynomial embedding \(\iota \). For an annular angular sector, set \(\gamma =m\).

For \(0{\lt}q{\lt}1\), the open annular domain is \(\Omega _q=\{ (x,y):q{\lt}x^2+y^2{\lt}1\} \). A test \(\phi \in C_c^\infty (\Omega _q)\) determines its almost-everywhere class in the physical \(L^2(dA_q)\) space.

For \(q\ne 1\), write \(B_q=(1-x^2-y^2)(x^2+y^2-q)\). Define the boundary-weighted lift by \(p\mapsto B_qL_mp\).

For \(q\ne 1\) and an angular mode, set \(\kappa _m=c_m/(2t(q)^{m+3})\), using the natural-number power. Here \(c_m\) is the squared angular norm, with \(c_m=0\) for the zero sine mode.

For \(q\ne 1\) and a real angular harmonic \(H_m\), the linear lift sends \(p\) to \(L_mp=p(t(q)(1-x^2-y^2))H_m(x,y)\).

For \(q\in \mathbb R\setminus \{ 1\} \), define \(t(q)=(1-q)^{-1}\).

For continuous kernels \(K_n\) on the closed Volterra triangle, define \(B_n=I-P_nA_{K_n}\) on normalized coefficient space.

A barycentric evaluation configuration consists of a nonempty finite set of distinct nodes and an evaluation point \(z\) different from every node.

For \(f\in C([-1,1],\mathbb C)\), define

the infimum over complex polynomials \(p\) of degree at most \(N\), restricted to \([-1,1]\). Here \(\| \cdot \| _\infty \) is the norm of \(C([-1,1],\mathbb C)\). The interpolant \(I_N f\) is the cardinal sum \(\sum _{j=0}^N f(x_j)\ell _j\), evaluating \(f\) only at its extrema nodes in \([-1,1]\).

Reference: [ATAP, Chapter 16].

For source and target bases \(b,c\), a linear residual map \(L\) and a linear map \(B\) of \(q\) boundary constraints, the bordered coefficient map sends finite coefficients \(v\) to \((Bp,\operatorname {repr}_c(Lp))\), where \(p=\operatorname {repr}_b^{-1}v\).

Reference: [OT13, Sections 2.4 and 3.2].

A continuously differentiable convolution kernel consists of continuous real functions \(g\) and \(s\) on \(\mathbb {R}\), with \(g'(t)=s(t)\) at every \(t\in \mathbb {R}\).

For \(\beta {\gt}1\), equip \(k_\beta \) with its continuous slope \(s_\beta (t)=\beta |t|^{\beta -1}\operatorname {sign}(t)\).

Let \(\mu \) be a measure on \([-1,1]\) and \(\nu \) a real measure. Write \(\bar\mu =\iota _*\mu \) for the real-coordinate measure. The \(k\)th outer measure is the shifted disk measure of Definition 5.3.37, using the transport of \(\bar\mu \) to \(D_k\). Its geometric Cartesian measure is the pushforward in Definition 5.3.42. A real source supported on \([-1,1]\) is transported to the interval without loss. A Cartesian disk system for \((\mu ,\nu )\) consists of an inner polynomial sequence \((q_k)\) and, for each \(k\), an outer polynomial sequence \((p_r^{[k]})\) such that \((q_k)\) is orthogonal for \(\nu \) and \((p_r^{[k]})\) for the \(k\)th outer measure, each \(q_k\) has the parity of \(k\), all leading coefficients are positive, and all squared norms are one. Its polynomials are

with \(L_k\) the parity lift of Definition 5.3.30; on the disk with \(|x|{\lt}1\) this is \(p_{n-k}^{[k]}(x)(1-x^2)^{k/2}q_k(y/\sqrt{1-x^2})\). At the tips \((\pm 1,0)\), the values are given by the polynomial lift \(L_kq_k\).

For a commutative ring \(R\) and every integer \(n\), define \(T_n,U_n\in R[X]\) by \(T_0=U_0=1\), \(T_1=X\), \(U_1=2X\), and the common recurrence \(p_{n+2}=2Xp_{n+1}-p_n\), run forwards for nonnegative and backwards for negative indices.

Reference: [DLMF, Table 18.9.1].

Let the Chebyshev coordinate basis of \(\mathbb {R}[X]\) be \((T_n)_{n\in \mathbb {N}}\), in the polynomial normalisation \(T_n(1)=1\). It differs from the normalised weighted Hilbert basis.

Reference: [OT13, Sections 2–3].

For coefficient sequences \(a,b\) such that \((a_mb_n)_{|m-n|=k}\) is summable, define

Both sums contribute when their output degrees coincide. This retains the full-constant convention, including multiplication by \(T_0\).

References: [DLMF, 18.18.21]; [OT13, Section 2.2, equation (2.7)].

At output index \(k\), consider pairs of coefficient sequences \(a,b\) for which the family \((a_mb_n)_{|m-n|=k}\) is summable. This condition concerns only the chosen output index.

For \(\rho {\gt}1\), \(|a_n|\le C\rho ^{-n}\) and \(k\ge 0\), regard \(S_a|_{E_\rho }\) as an element of \(\mathcal D^k(E_\rho )\). Its order-\(k\) derivative value is \(D_{E_\rho }^kS_a\).

For \(c\in \mathbb C\) and \(r\in \mathbb R\), let \(S=\{ z:|z-c|=|r|\} \) and \(\gamma (\theta )=c+r e^{i\theta }\). Let \(\mathcal C_{c,r}\) consist of functions \(f:S\to \mathbb C\) for which \(\theta \mapsto i r e^{i\theta }f(\gamma (\theta ))\) is integrable on \([0,2\pi ]\). For \(f\in \mathcal C_{c,r}\), define

Addition in \(\mathcal C_{c,r}\) is pointwise addition of functions. Reference: [ATAP, proof of Theorem 8.1].

For a positive natural order \(N\) and a natural angular index \(j\), define \(x_{N,j}=\cos (j\pi /N)\). The extrema grid consists of \(0\le j\le N\). Reference: [ATAP, Chapter 4].

On the open interval \(I=(-1,1)\), define the continuous nonnegative weights \(w_U(x)=\sqrt{1-x^2}\) and \(w_T(x)=1/\sqrt{1-x^2}\).

For a nonempty set \(S\subseteq \mathbb {R}\) that is bounded above, define \(\sup S\) to be its least upper bound in \(\mathbb {R}\).

Let \(R\) be a ring, not necessarily commutative, and let \(a,b,c:\mathbb {N}\to R\). For \(n\) coefficients starting at index \(k\), define \(S_{0,k}=(0,0)\). If \(S_{n,k+1}=(u,v)\), set

If \(S_{n,0}=(u,v)\), the evaluator returns \(u p_0+v p_1\). Here \(n\) is the number of coefficients, not the largest degree. The initial values \(p_0,p_1\) are supplied separately.

References: [Cle55]; [DLMF, 3.11(ii), 18.40(i)].

For \(N\ge 1\), define \(\mathcal Q_Nf=\sum _{k=0}^Nc_k(f)m_k\), where \(m_k=\int _{-1}^1T_k\). Separately, set \(\mathcal Q_0f=2f(1)\) for the singleton interpolant.

Reference: [ATAP, Chapter 19].

For \(N\ge 1\) and \(0\le j\le N\), write \(x_j=\cos (j\pi /N)\), \(d_j=1/2\) at \(j=0,N\) and \(d_j=1\) otherwise, and let \(m_k\) be the classical moment from Definition 2.7.10. Define

The nodal endpoints and highest DCT mode have separate half factors.

Reference: [ATAP, Chapters 4 and 19].

For \(n\in \mathbb {N}\), define

Functions in this domain retain their pointwise values. Addition and real scalar multiplication are pointwise. The defining condition concerns \(fT_n\), not \(f\) alone. Reference: [ATAP, Chapter 3].

At each index \(n\), for \(f\in \mathcal I_n\), set

The expansion convention is \(a_0+\sum _{n\geq 1}a_nT_n\).

Reference: [ATAP, Chapters 3 and 8], with the constant-term convention made explicit here.

Let \(\mu \) have all moments finite and let \(f\in \mathcal L^1_T(\mathbb {C})\). The degree-\(N\) Chebyshev truncation in \(L^2(\mu ;\mathbb {C})\) is

where \([T_n]\) denotes the polynomial’s \(L^2(\mu ;\mathbb {C})\) class. Reference: [ATAP, Theorem 8.2].

For \(a:\mathbb N\to \mathbb C\), let \(D_a\) consist of the complex points \(z\) where the natural-order partial sums \(\sum _{n{\lt}N}a_nT_n(z)\) converge to a finite complex limit. Define \(S_a:D_a\to \mathbb C\) to be that limit. This includes conditionally convergent series; no value is assigned elsewhere.

Reference: [ATAP, Chapters 3 and 8].

For \(n\in \mathbb {N}\), define

retaining pointwise function values. Real functions are regarded as complex functions by scalar inclusion; real and imaginary parts are taken pointwise. Reference: [ATAP, Chapter 3].

For \(f\in \mathcal I_n^{\mathbb {C}}\), define \(a_n(f)\) by the weighted integrals of Definition 2.2.3, with the same full constant coefficient convention: \(a_0(f)=\pi ^{-1}\int f\, d\mu _T\) and \(a_n(f)=2\pi ^{-1}\int fT_n\, d\mu _T\) for \(n{\gt}0\).

Reference: [ATAP, Chapters 3 and 8].

Let \(\mathcal L^1_T(\mathbb {C})\) be the set of functions \(f:\mathbb {R}\to \mathbb {C}\) that are \(\mu _T\)-integrable, retaining their pointwise representatives. Reference: [ATAP, Chapter 3].

For monic \(g\) of degree \(n\), give \(K[x]/(g)\) the basis whose coordinates are the sequence coordinates \([r]_0,\ldots ,[r]_{n-1}\) of the remainder \(r\) modulo \(g\); by Theorem 2.9.3 its vectors are the images of \(P_0,\ldots ,P_{n-1}\). Define \(C_g\) to be the matrix of multiplication by \(x\) in this basis, with basis images as columns.

References: [Bar75]; [ATAP, Chapter 18].

Let \(\beta \in \mathbb R\) and \(0\le \delta \le 1\). Form the Cartesian disk measure \(\sigma _{\beta ,\delta }\) from the symmetrically deformed outer Jacobi measure with parameters \((\beta +1/2,\beta +1/2)\) and inner Jacobi measure with parameters \((\beta ,\beta )\). For \(r(x,y)=A+Bx^2+Cy^2\), require \(r\ge 0\) almost everywhere for \(\sigma _{\beta ,\delta }\). Transport the source to \(\{ r\ge 0\} \) and take its polynomial density measure; this is the deformed quadratic disk measure.

For \(s\in \mathbb N\), let \(W_s\) consist of complex sequences \(v\) for which \(((n+1)^s v_n)_{n\ge 0}\in \ell ^2\), with norm \(\| v\| _s=\| ((n+1)^s v_n)_n\| _{\ell ^2}\). Write \(U_sv=((n+1)^sv_n)_n\).

On \(W_s\), define the diagonal \(D_dv=(d_nv_n)_n\) for a bounded sequence \(d\), the shifts \((Lv)_n=v_{n+1}\) and \((Rv)_0=0\), \((Rv)_{n+1}=v_n\), and the truncation \((P_mv)_n=v_n\) for \(n{\lt}m\) and \(0\) otherwise. Let \(\iota _s:W_s\to \ell ^2\) be the coordinate inclusion. For a bounded linear operator \(A\) on \(\ell ^2\), its weighted transport is \(U_s^{-1}AU_s\).

For an open set \(\Omega \subseteq \mathbb R^n\), let \(L=L^2(\Omega )\) and \(\mathcal G=\bigoplus _{i=1}^n{}^{\ell ^2}L\). Write \(D_i\phi \) for the Cartesian derivative of \(\phi \in C_c^\infty (\Omega )\), viewed globally with support compactly contained in \(\Omega \). The zero-boundary Sobolev graph is

The closure uses the product maximum norm. Elements are represented by function/gradient pairs.

For a measure \(\mu \) on \(D_\delta \), define

For \(0\le \delta \le 1\), a real source measure is identified with its measure on \(D_\delta =\mathbb R\). In the Cartesian disk application, the scale remains \(\sqrt{1-x^2}\) and the inner factor measure is unchanged.

For \(\delta \in \mathbb R\), let \(D_\delta =\{ t\in \mathbb R:1-\delta +\delta t^2\ge 0\} \). The positive branch \(\phi _\delta :D_\delta \to \mathbb R\) is \(\phi _\delta (t)=\sqrt{1-\delta +\delta t^2}\).

The Cartesian chart \((x,z)\mapsto (x,\sqrt{1-x^2}z)\) has domain \([-1,1]\times \mathbb {R}\). For a measure \(\mu \) on \([-1,1]\) and a real inner measure \(\nu \) with the product-kernel measurability condition of Definition 5.3.33, its pushforward of \(\mu \otimes \nu \) is the Cartesian measure. Real outer measures supported on \([-1,1]\) are transported to this domain. The disk measure uses the Jacobi factors \(\mu _{\beta +1/2,\beta +1/2}\) and \(\mu _{\beta ,\beta }\) in this construction. The disk orthogonality range is \(\beta {\gt}-1\).

For \(k\in \mathbb N\), let \(D_k=\{ x\in \mathbb R:(1-x^2)^k\ge 0\} \). For a measure \(\eta \) on \(D_k\), the shifted disk outer measure is the real-coordinate pushforward of \((1-x^2)^k\, d\eta (x)\). The domain is all of \(\mathbb R\) for even \(k\) and \([-1,1]\) for odd \(k\).

For \(h\in \mathbb R[X]\) and \(l\in \mathbb N\), put \(w(x)=(1-x^2)^l h(x)\). A measure \(\eta \) on \(\{ x:w(x)\ge 0\} \) has modified outer measure given by the real-coordinate pushforward of \(w\, d\eta \). All source points lie in the nonnegative-density domain.

For \(p(x)=\sum _j c_jx^j\) define the bivariate polynomial \(L_np(x,y)=\sum _{j=0}^n c_j y^j(1-x^2)^{\lfloor (n-j)/2\rfloor }\).

For \(\beta {\gt}-1\), continuous \(f:\mathbb {R}^2\to \mathbb {R}\), and \(x\in (-1,1)\), put \(r_x=\sqrt{1-x^2}\). The function

is denoted by \(F_{\beta ,f,x}\). Its domain is the open slice \((-r_x,r_x)\).

For \(\beta {\gt}-1\) and continuous \(f:\mathbb {R}^2\to \mathbb {R}\), set \(r_x=\sqrt{1-x^2}\) for \(x\in (-1,1)\) and define

The integrand is the slice function of Definition 5.3.46.

For a positive measurable weight \(h\), define \(D_h:L^2(\mu ;\mathbb {R})\to L^2(h^2\mu ;\mathbb {R})\) as the linear isometry characterized by \((D_hu)h=u\) \(h^2\mu \)-almost everywhere for every \(u\in L^2(\mu ;\mathbb {R})\). This defines an \(L^2\) equivalence class and assigns no pointwise quotient at a zero of \(h\).

For every parameter and index choice for which \((D_{n,k}^{\beta })^2\) is integrable with respect to \(\mu _\beta \), define

The domain consists of all such square-integrable polynomials; no additional parameter restrictions are imposed.

Set \(D_{n,k}^{\beta }(x,y)=P_{n-k}^{(\beta +k+1/2,\beta +k+1/2)}(x) L_kP_k^{(\beta ,\beta )}(x,y)\), using \(0\le k\le n\) in the degree-\(n\) block. This is the Cartesian Jacobi normalization, with increasing \(k\) within a block; it is not yet orthonormally scaled.

Reference: [DX14, Section 2.3].

For \(\rho {\gt}1\), define the open filled ellipse by

Reference: [ATAP, Chapter 8].

A real continuous error \(e\) on \([a,b]\) equioscillates for degree \(N\) if there are \(a\le x_0{\lt}\cdots {\lt}x_{N+1}\le b\) and \(\sigma \in \{ -1,1\} \) such that \(e(x_i)=\sigma (-1)^i\| e\| _\infty \). The zero-error case is included.

Reference: [ATAP, Theorem 10.1].

For \(a,b\in \mathbb {C}\) with \(\Re a,\Re b{\gt}0\), define

The powers are principal complex powers of positive real bases; the integration variable belongs to the open interval. Reference: [DLMF, 5.12.1].

For \(\delta \in \mathbb R\setminus \{ 0\} \) and a real polynomial \(p\), let \(E_p(s)=\sum _j[X^{2j}]p\, s^j\). Define \(\mathcal T_\delta p=E_p((X^2-1+\delta )/\delta )\).

For \(N\ge 1\), fold a frequency \(k\) by setting \(r=k\bmod 2N\) and \(A_N(k)=\min (r,2N-r)\). For \(N=0\), set \(A_0(k)=0\).

For \(N\ge 1\) and \(0\le j\le N\), write \(x_j=\cos (j\pi /N)\) for the extrema nodes. Set \(F_N=(X^2-1)T_N'\), and let \(e_j=2\) for \(j=0\) or \(j=N\), and \(e_j=1\) otherwise. The normalised barycentric weights are \(w_j=(-1)^j/e_j\).

Reference: [ATAP, Theorem 5.2].

For a finite-grid index \(j\), let \(\ell _{N,j}\) be its classical Lagrange cardinal polynomial. At order zero it is the constant polynomial \(1\). Reference: [ATAP, Chapter 4].

For \(N\ge 1\), put \(x_j=\cos (j\pi /N)\) and \(S_N(g)=\tfrac 12g(0)+\sum _{j=1}^{N-1}g(j)+\tfrac 12g(N)\). For real-valued data and \(0\le k\le N\), define

For \(N=0\), define \(P_0f\) to be the constant polynomial \(f(1)\).

Reference: [ATAP, Chapters 4 and 6].

For \(N\ge 0\), index the grid by \(0\le j\le N\). When \(N{\gt}0\) its points are \(x_{N,j}=\cos (j\pi /N)\). For \(N=0\) there is one node, \(x_{0,0}=1\).

For \(N\ge 1\), let \(I_Nf\in \mathbb {C}[X]\) be the Lagrange interpolant of \(f:\mathbb {R}\to \mathbb {C}\) at the \(N+1\) distinct Chebyshev extrema \(x_j=\cos (j\pi /N)\), \(0\le j\le N\). For \(N=0\), define \(I_0f\) to be the constant polynomial \(f(1)\).

Reference: [ATAP, Chapters 4 and 8].

Define \(\lambda _N(x)=\sum _{j=0}^N|\ell _{N,j}(x)|\) using the classical finite-grid cardinal polynomials, for every real \(x\) and \(N\ge 0\). Reference: [ATAP, Chapter 15].

For \(N\ge 0\), define \(\Lambda _N\) as the classical real supremum of \(\{ \lambda _N(x):x\in [-1,1]\} \). Reference: [ATAP, Chapter 15].

For a real error function \(e\) on \([a,b]\), set \(v_N(x)=e(x)(1,x,\ldots ,x^N)\in \mathbb R^{N+1}\).

Reference: [ATAP, Theorem 10.1], convex-separation formulation.

Over a ring, take \(n\) coefficients \(c_0,\ldots ,c_{n-1}\) and recurrence coefficients \(a_k,b_k\) only for \(0\le k{\lt}n-2\). Define the states for zero, one and two coefficients by \((0,0)\), \((c_0,0)\) and \((c_0,c_1)\), respectively. For \(n\ge 3\), shift all three finite arrays by one index, compute the suffix state \((u,v)\) recursively, and return \((c_0+b_0v,u+a_0v)\). The evaluator applied to the resulting state \((u,v)\) returns \(up_0+vp_1\). Thus no recurrence coefficient is evaluated for an expansion with at most two coefficients.

References: [Cle55]; [DLMF, 18.40(i)].

For a finite family \(p_0,\ldots ,p_{N-1}\), let \(A_{ij}=[X^i]p_j\). For \(q\) let \(D_j(q)\) be the determinant obtained by replacing column \(j\) of \(A\) with the first \(N\) coefficients of \(q\). Over a field, on the domain \(\det A\ne 0\), put \(c_j(q)=D_j(q)/\det A\). The Jacobi versions use target parameters \((\alpha ,\beta )\), source parameters \((\gamma ,\delta )\), and target degrees \(0,\ldots ,n\) for a source of degree \(n\).

For a degree-graded sequence, define \(A_n\) by \((A_n)_{ij}=[xP_j]_i\) for \(0\le i,j{\lt}n\). For an orthogonal sequence we call it the finite Jacobi matrix in the original polynomial normalization.

Reference: [DLMF, 3.5(vi)].

For a polynomial sequence \(P\), a measure \(\mu \), and a cutoff \(n\), the finite-kernel domain requires integrability of \(P_j^2\) and \(h_j\ne 0\) for every \(0\le j\le n\). No conditions are imposed on higher indices.

For endpoint value \(c\in \mathbb C\) and derivative coordinates \(f\in \ell ^2\), define \(v=(I+K_B)^{-1}(c,f_0,f_1,\ldots )\), using the inverse of the invertible bordered derivative. The reconstructed solution has coefficients \(Rv\).

Let \(\ell ^2_1\) be the real sequences \(q\) for which \(((n+1)q_n)_{n\ge 0}\) belongs to \(\ell ^2\), equipped with norm

Reference: [GO20, Definition 5.3]; [Gut22, Section 2.7].

Let \(U\subseteq \mathbb C\) be open and \(f:U\to \mathbb C\). For complex polynomial coefficients \(A_0,\ldots ,A_m\) and a point \(z\in U\) at which all derivatives of orders \(0,\ldots ,m\) exist in the sense of Definition 1.1.19, define \(\mathcal Lf(z)=\sum _{k=0}^m A_k(z)J_kf(z)\). The highest derivative need only exist at \(z\); no value is assigned at points without the required jet.

Reference: [OT13, Section 3].

Let \(D=\mathbb C\setminus \{ 0,-1,-2,\ldots \} \). For \(z\in D\), let \(N=\lfloor \max (1-\Re z,0)\rfloor \) and define

where \(t\) ranges over the strictly positive reals and the power uses the real logarithm of \(t\). The real Gamma function is its restriction to \(\mathbb R\setminus \{ 0,-1,-2,\ldots \} \). Gamma has no assigned value at a pole.

Reference: [DLMF, 5.2.1, 5.5.1].

Over a field of characteristic zero, call \((a,b,c)\) admissible when \((c)_n=0\) implies \((a)_n(b)_n=0\) for every \(n\ge 0\). Its classical Gauss coefficient is zero after numerator termination; otherwise it is

In a terminating case, termination is taken before any later denominator pole; this includes coincident terminating parameters.

Reference: [DLMF, 15.2.1, 15.2.4–15.2.6].

On the domain of Definition 3.6.18, define \(u_i(x)=\sqrt{w_x}P_i(x)/\sqrt{h_i}\), \(i{\lt}n\). The square roots have nonnegative arguments and their denominators are positive. Reference: [DLMF, 3.5(vi)].

For a point \(x\) and each coordinate \(i{\lt}n\), the Gauss-vector domain requires integrability of the Gaussian weight’s normalized nodal product \(\ell _x\), nonnegativity of \(w_x=\int \ell _x\, d\mu \), and an integrable polynomial square with strictly positive squared norm \(h_i\). No conditions are required when \(n=0\). A point need not be a Gaussian node if these local conditions hold.

Gegenbauer polynomials are also called ultraspherical polynomials; see [DLMF, §18.1(ii)]. Let \(R\) be a commutative \(\mathbb {Q}\)-algebra, let \(\lambda \in R\), and let \(n\in \mathbb {N}\). Define

Reference: [DLMF, 18.5.10].

For a field \(K\), parameters \(\lambda ,x\in K\), \(n\) coefficients \(c_0,\ldots ,c_{n-1}\), and the denominator conditions of Definition 4.2.8, use finite Clenshaw evaluation with

References: [Cle55]; [DLMF, Table 18.9.1].

For a field \(K\) and a coefficient count \(n\), the Gegenbauer Clenshaw domain requires \((k+2)\ne 0\) in \(K\) for every \(0\le k{\lt}n-2\). There is no condition when \(n\le 2\).

For \(f\in \mathcal A_{\mu _\lambda ,C_n^{(\lambda )}}^{\mathbb {R}}\), define \(a_n(f)=h_n^{-1}\int fC_n^{(\lambda )}\, d\mu _\lambda \), where \(h_n=\int (C_n^{(\lambda )})^2\, d\mu _\lambda \).

Reference: [DLMF, 18.2(viii)].

For real \(\lambda \), let \(w_\lambda =w_{\lambda -1/2,\lambda -1/2}:(-1,1)\to \mathbb {R}_{\ge 0}\) be the Jacobi weight of Definition 3.2.9, and let \(\mu _\lambda =\mu _{w_\lambda }\) be the measure of Definition 1.1.2.

Reference: [DLMF, Table 18.3.1].

For \(\lambda {\gt}-1/2\), \(\lambda \ne 0\), let \(e_n=h_n^{-1/2}J_{\mu _\lambda }C_n^{(\lambda )}\), using the positive square root of the squared norm.

Reference: [DLMF, Table 18.3.1].

For real \(\lambda {\gt}-1/2\), \(\lambda \ne 0\), let the Gegenbauer basis of \(\mathbb {R}[X]\) be the basis built from the sequence \(C_n^{(\lambda )}\) in its original normalisation. Theorem 4.1.9 identifies the vectors of all three bases.

Reference: [DLMF, Table 18.3.1].

For a commutative \(\mathbb {Q}\)-algebra \(R\) and \(\lambda \in R\), normalize \(R_0^{(\lambda )}=1\). For \(n\ge 1\), define

Reference: [DLMF, 18.5.10], [DLMF, §18.7(iii)].

For a cutoff \(n\) in the domain of Definition 3.1.21 and arbitrary real \(x,y\), set \(K_n(x,y)=\sum _{j=0}^n P_j(x)P_j(y)/h_j\).

Reference: [DLMF, 18.2.12].

For a degree-indexed sequence \(P\) and a measure with finite polynomial moments, a normalization index is a natural number \(n\) together with \(h_n{\gt}0\); define \(e_n=h_n^{-1/2}J_\mu P_n\) on this domain. For \(f\in \mathcal A_{\mu ,P_n}^{\mathbb {R}}\), define the original-normalisation coefficient by \(a_n(f)=h_n^{-1}\int fP_n\, d\mu \).

Reference: [DLMF, 18.2(viii)].

For a real measure \(\mu \) with finite polynomial moments, let \(J_\mu :\mathbb {R}[X]\to L^2(\mu )\) send \(p\) to the class of \(x\mapsto p(x)\).

Write \(k_n=\operatorname {lc}(P_n)\). For \(B_n\) and \(C_n\), respectively, use the domains of Definition 3.1.18. Define

References: [DLMF, 18.2(iv)]; [Sze75, Chapter 3].

For a degree-indexed polynomial sequence \(P\), a measure \(\mu \), an index \(n\), with \(P_n\in \mathcal Q_\mu \), define \(h_n=\int P_n(x)^2\, d\mu (x)\). Zero squared norms are permitted.

For the preceding polynomial differential system, let \(T_n=I+P_nK\). At a cutoff \(n\) for which \(T_n\) is invertible, define \(v_n=T_n^{-1}P_nf\).

Let \(N=k+1\), \(a_0,\ldots ,a_k\in \mathbb R[x]\), and \(B:W_s\to W_s\) be bounded. Set

For \(k\in \mathbb N\) and admissible \((a,-k,c)\), define

Admissibility allows earlier termination by the other numerator and termination before a later denominator pole. Reference: [DLMF, 15.2.4–15.2.5].

For \(F:\mathbb R\to \mathbb R\) and a measure \(\mu \) on \(\mathbb R\), let \(\mathcal D_\mu (F)\) consist of the integrable functions \(h\in \mathcal I(\mu )\) for which \(F\) has derivative \(h(x)\) at \(\mu \)-almost every \(x\). This is an almost-everywhere derivative representation; no derivative is asserted at the exceptional points.

For a measure \(\mu \) on a measurable space \(X\), let \(\mathcal I(\mu )\) be the set of functions \(f:X\to \mathbb {R}\) that are \(\mu \)-integrable.

For a measure \(\mu \) on \(\mathbb {R}\), define the square-integrability domain

No condition is imposed on other polynomial moments.

For a polynomial family \(P\), a cutoff \(N\), and \(\mathbb K=\mathbb {R}\) or \(\mathbb {C}\), define the simultaneous coefficient domain

Thus a member is one pointwise function \(f:\mathbb {R}\to \mathbb K\) for which each \(P_n^2\) and \(fP_n\) is integrable and \(\int P_n^2\, d\mu {\gt}0\), for \(0\le n\le N\). No condition is imposed at higher indices.

For \(\mathbb K=\mathbb {R}\) or \(\mathbb {C}\), a real polynomial \(p\) and a measure \(\mu \), define

Functions retain their pointwise representatives. These conditions concern only \(p\), not other polynomial moments. Reference: [DLMF, 18.2(viii)].

For a natural index \(n\), define \(m_1=0\) and, for \(n\ne 1\),

The exceptional odd mode is defined separately, without division by zero.

For \(g\in C(\mathbb {R};\mathbb {R})\), an integrable real function \(f\) on \((-1,1)\), and \(x\in \mathbb {R}\), consider the integrable function \(y\mapsto g(x-y)f(y)\). Define \(V_gf(x)=\int _{-1}^1 g(x-y)f(y)\, dy\) with this integrable product as integrand.

For real endpoints \(a,b\), consider real functions \(f\) with finite Lebesgue integral of \(|f|\) between the endpoints. Either orientation is allowed. At equal endpoints every function is admitted.

For \(t\in D_K\), define \(R_K(t)\) as the inverse of \(I+tK\).

For a family \(A:I\to R\) in a monoid, let \(I_A=\{ i\in I:A_i\text{ is invertible}\} \). For \(i\in I_A\), define \(A_i^{-1}\) by inversion in the group of units of \(R\). There is no inverse value at an index outside \(I_A\).

For a filter \(\mathcal F\) on \(I\), the induced filter on \(I_A\) is its pullback along the inclusion \(I_A\hookrightarrow I\). For \(I=\mathbb N\) and \(\mathcal F\) the filter of tails, this means tending to infinity through invertible indices.

For a finite real square matrix \(K\), let \(D_K=\{ t\in \mathbb R: I+tK\text{ is invertible}\} \). An invertible pencil parameter is an element of \(D_K\).

For a commutative \(\mathbb {Q}\)-algebra \(R\), \(\alpha ,\beta \in R\), and \(n\in \mathbb {N}\), define

Reference: [DLMF, 18.5.7].

For a field \(K\), parameters \(\alpha ,\beta ,x\in K\), \(n\) coefficients \(c_0,\ldots ,c_{n-1}\), and the denominator conditions of Definition 4.2.12, return \(0\) if \(n=0\) and \(c_0\) if \(n=1\). For \(n\ge 2\), use finite Clenshaw evaluation with

where \(d_k=2(k+2)(k+\alpha +\beta +2)(2k+\alpha +\beta +2)\), and \(p_0=1\), \(p_1=\alpha +1+\tfrac 12(\alpha +\beta +2)(x-1)\). References: [Cle55]; [DLMF, 18.9.1–2].

For a field \(K\), parameters \(\alpha ,\beta \in K\) and a coefficient count \(n\), the Jacobi Clenshaw domain requires \(2\ne 0\) when \(n\ge 2\), and

Empty and constant expansions impose no denominator conditions.

For \(f\in \mathcal A_{\mu _{\alpha ,\beta },P_n^{(\alpha ,\beta )}}^{\mathbb {R}}\), define \(a_n(f)=h_n^{-1}\int fP_n\, d\mu _{\alpha ,\beta }\), where \(h_n=\int P_n^2\, d\mu _{\alpha ,\beta }\).

Reference: [DLMF, 18.2(viii)].

For \(0\le k\le n\) define polynomial expressions

These are defined over every commutative ring.

Reference: [DLMF, 18.18.14].

Use the roots of \(P_n^{(\alpha ,\beta )}\) as nodes and integrate their Lagrange basis polynomials against the original Jacobi measure, on the domain where the normalized nodal product is integrable, as in Definition 3.6.1. These weights do not renormalise the measure to have total mass one.

Reference: [DLMF, 3.5(v)].

For \(a+1\notin \{ 0,-1,-2,\ldots \} \), \(\alpha ,\beta {\gt}-1\), and an integer \(k\ge 0\), define

where \(h_n=\int (P_n^{(\alpha ,\beta )})^2\, d\mu _{\alpha ,\beta }\). The constants depend on \(k\) and the parameters, not on \(n\).

For real \(\alpha ,\beta \), define the continuous map \(w_{\alpha ,\beta }:(-1,1)\to \mathbb {R}_{\ge 0}\) by \(w_{\alpha ,\beta }(x)=(1-x)^\alpha (1+x)^\beta \). Let \(\mu _{\alpha ,\beta }=\mu _{w_{\alpha ,\beta }}\) be the measure of Definition 1.1.2 for this weight on \((-1,1)\).

Reference: [DLMF, Table 18.3.1].

For \(\alpha ,\beta {\gt}-1\), set \(e_n=h_n^{-1/2}J_{\mu _{\alpha ,\beta }}P_n^{(\alpha ,\beta )}\), where \(h_n\) is the squared integral norm.

Reference: [DLMF, Table 18.3.1].

For real \(\alpha ,\beta {\gt}-1\), let the Jacobi basis of \(\mathbb {R}[X]\) be the basis built from the Jacobi sequence, in the original normalisation. This is an algebraic polynomial basis, distinct from the weighted L2 Hilbert basis.

Reference: [DLMF, Table 18.3.1].

For \(\alpha ,\beta \ge 0\), define the continuous map \(\overline w_{\alpha ,\beta }:[-1,1]\to \mathbb {R}_{\ge 0}\) by \(\overline w_{\alpha ,\beta }(x)=(1-x)^\alpha (1+x)^\beta \), where \(0^0=1\).

Reference: [DLMF, Table 18.3.1].

Define \(J:\mathbb C\setminus \{ 0\} \to \mathbb C\) by \(J(z)=(z+z^{-1})/2\).

Reference: [ATAP, Chapter 8].

For \(f\in C([-1,1],\mathbb C)\) and \(n\ge 0\), define the integrable function on the unit circle

Its numerator is the Joukowski pullback of the interval data. Reference: [ATAP, Theorems 3.1 and 8.1].

Let \(D\subseteq \mathbb R\) be an outer domain, \(\mu \) a measure on \(D\), \(\nu \) a measure on \(\mathbb R\), \(\rho :D\to \mathbb R\) a scale and \(a\) a polynomial. Write \(\bar\mu =\iota _*\mu \), where \(\iota :D\hookrightarrow \mathbb R\) is the real coordinate map. When \(a\ge 0\) almost everywhere for \(\bar\mu \), transport \(\bar\mu \) to \(D_{a,k}\) and let \(\mu _k\) be its degree-dependent outer measure from Definition 5.6.5. Let \(\omega \) be the pushforward of \(\mu \otimes \nu \) by \((x,z)\mapsto (\iota (x),\rho (x)z)\). The input includes almost-everywhere measurability certificates for both the product kernel \(x\mapsto (z\mapsto (x,z))_*\nu \) and the coordinate map \((x,z)\mapsto (x,\rho (x)z)\) under \(\mu \otimes \nu \). The scale may be discontinuous, and no interval support condition is imposed. Global scales use \(D=\mathbb R\). A real measure supported almost everywhere in a measurable \(D\) is represented by its restriction to that domain; its coordinate pushforward recovers the original measure.

For real polynomials \(b,h,g\), a positive integer \(s\) and \(\ell \in \mathbb {N}\), define

Reference: [DX14, §2.6.1].

For a real polynomial \(a\) and \(k\in \mathbb N\), let \(D_{a,k}=\{ x\in \mathbb R:a(x)^k\ge 0\} \). For a source measure \(\eta \) on this domain, the \(k\)th outer measure is the polynomial density measure \(a^k\, d\eta \), transported to real coordinates. For even \(k\), the source domain is all of \(\mathbb R\).

Require \(a(\iota (x))\ge 0\) almost everywhere for \(\mu \). Choose positive-leading orthonormal sequences \(q_k\) for \(\nu \) and \(p_r^{[k]}\) for \(\mu _k\), together with polynomial lifts \(L_k\) satisfying \(L_k(x,\rho (x)z)=\rho (x)^kq_k(z)\) almost everywhere in \(x\). Set \(P_{n,k}=p_{n-k}^{[k]}L_k\) for \(0\le k\le n\). Two constructions of such systems are defined, under finite moments and positive definite polynomial pairings for \(\bar\mu \) and \(\nu \): one for a nonzero polynomial scale \(c\) restricted to \(D\) (\(a=c^2\)), and one for \(\rho (x)=\sqrt{a(\iota (x))}\) on \(D=\{ x\in \mathbb R:a(x)\ge 0\} \) when \(a\ne 0\) and \(\nu \) is even. The square-root scale has no values outside \(D\). The label \(n\) is the separated index.

For an admissible index \(i\) of Definition 2.4.10, define \(\ell _i(X)=\prod _{j\in s\setminus \{ i\} }(X-v_j)/(v_i-v_j)\). When \(i\notin s\) this is the normalized nodal product, extending the cardinal-polynomial construction. Reference: [DLMF, 3.3(i)].

For a finite index set \(s\) and field-valued nodes \(v\), define the admissible indices to be those \(i\) satisfying \(v_i\ne v_j\) for every \(j\in s\setminus \{ i\} \). An admissible index need not belong to \(s\).

For a finite interpolation configuration \((s,v)\) and values \((r_i)_{i\in s}\), define \(I r=\sum _{i\in s}r_i\ell _i\). For the empty set this is the zero polynomial. Reference: [DLMF, 3.3(i)].

A finite interpolation configuration consists of a finite index set \(s\) and field-valued nodes \((v_i)_{i\in s}\) with \(v_i\ne v_j\) whenever \(i\ne j\). The set \(s\) may be empty.

For an admissible index \(i\) of Definition 2.4.10, define \(\omega _i=\prod _{j\in s\setminus \{ i\} }(v_i-v_j)^{-1}\). This also permits \(i\notin s\) whenever all displayed denominators are nonzero. Reference: [DLMF, 3.3(i)].

Let \(T\in \mathbb R\setminus \{ 0\} \) and let \(\mathbb R/T\mathbb Z\) be the circle with signed period \(T\). For a Laurent polynomial \(p=\sum _{k\in \mathbb Z}p_kz^k\) with finite support, its continuous circle symbol is

This defines a complex-linear map from Laurent polynomials over \(\mathbb C\) to continuous complex-valued functions on the circle.

Let \(R\) be a commutative \(\mathbb {Q}\)-algebra and let \(n\in \mathbb {N}\). Define (Rodrigues’ formula)

Reference: [DLMF, 18.5.5, Table 18.5.1 and Table 18.3.1].

For fixed \(n\), let \(f:\mathbb {R}\to \mathbb {R}\) satisfy \(fP_n\in L^1(\mu _L)\), with pointwise values retained. Define \(a_n(f)=\frac{2n+1}{2}\int f(x)P_n(x)\, d\mu _L(x)\).

Reference: [DLMF, 18.2(viii)].

Let \(\mu _L\) be Lebesgue measure restricted to \((-1,1]\), the Legendre weight one.

Reference: [DLMF, Table 18.3.1].

Define \(e_n=\sqrt{(2n+1)/2}\, J_{\mu _L}P_n\).

Reference: [DLMF, Table 18.3.1].

For a variable set \(\sigma \) and \(N\in \mathbb {N}\), consider the submodule of polynomials of total degree at most \(N\).

For a finite real square matrix \(K\), let \(P_K(X)=I+XK\) be its polynomial matrix pencil. For compatible real matrices \(F,E\), define the polynomial \(q_{ij}(X)=(F\operatorname {adj}(P_K(X))E)_{ij}\).

Let \(\mu _T=\mu _{w_T}\) be the measure of Definition 1.1.2 for the weight \(w_T\) of Definition 2.1.3.

Reference: [DLMF, Table 18.3.1].

Let \(\mu _U=\mu _{w_U}\) be the measure of Definition 1.1.2 for the weight \(w_U\) of Definition 2.1.3.

Reference: [DLMF, Table 18.3.1].

For finite \(\sigma \), assume explicitly that \(\mu \) has finite polynomial moments and positive-definite polynomial integral. Equip the polynomial space with the inner product \(\langle p,q\rangle _\mu =\int pq\, d\mu \). There is no connectedness or product-support restriction.

For a measure \(\mu \) on \(\mathbb {R}^\sigma \), an integrable polynomial pair consists of polynomials \(p,q\) together with \(pq\in L^1(\mu )\). Its pairing is the finite integral \(\langle p,q\rangle _\mu =\int pq\, d\mu \), defined on these pairs only.

A measure \(\mu \) on \(\mathbb {R}^\sigma \) has finite polynomial moments when every polynomial square is \(\mu \)-integrable.

For a real polynomial \(p\) in finitely many variables, put \(D_p=\{ x:p(x)\ge 0\} \). For a measure \(\eta \) on \(D_p\), define its polynomial density measure as the coordinate pushforward of \(p\, d\eta \).

The polynomial integral is positive definite when every polynomial square is integrable and every nonzero polynomial has strictly positive squared integral.

For a real polynomial sequence, a measure and a finite size \(n\), the matrix domain requires integrability of every mixed product \(xP_jP_i\) and every square \(P_i^2\) for \(i,j{\lt}n\), together with \(h_i=\int P_i^2\, d\mu {\gt}0\). On this domain set

Reference: [DLMF, 3.5(vi)].

Let \(\mathbb K\) be a nontrivially normed field (for example \(\mathbb R\) or \(\mathbb C\)), let \(U\subseteq \mathbb K\) be open, and let \(f:U\to \mathbb K\). We call \(f\) analytic on \(U\) if, for every \(x\in U\), there exist a function \(F:\mathbb K\to \mathbb K\), coefficients \(a_n\in \mathbb K\) for \(n\in \mathbb N\), and \(r\in (0,\infty ]\) such that:

1. \(F(y)=f(y)\) for every \(y\in U\);

2. \(r\le R\), where the supremum is taken in \([0,\infty ]\) and

   \[ R=\sup \left\{ s\ge 0:\ \exists C\in \mathbb R,\quad \| a_n\| s^n\le C\text{ for every }n\in \mathbb N\right\} ; \]

3. for every \(h\in \mathbb K\) with \(\| h\| {\lt}r\), the partial sums

   \[ \sum _{n=0}^{N-1}a_nh^n \]

   converge to \(F(x+h)\) as \(N\to \infty \).

Let \(\mathbb K\) be a nontrivially normed field (for example \(\mathbb R\) or \(\mathbb C\)). Let \(U\subseteq \mathbb K\) be open, \(f:U\to \mathbb K\), \(x\in U\) and \(d\in \mathbb K\). Say that \(f\) has derivative \(d\) at \(x\) if there exists \(F:\mathbb K\to \mathbb K\) agreeing with \(f\) on \(U\) such that \(F'(x)\) exists and equals \(d\).

Let \(\mathbb K\) be a nontrivially normed field (for example \(\mathbb R\) or \(\mathbb C\)). For \(f:U\to \mathbb K\) and \(m\ge 0\), a jet point is \(x\in U\) at which every derivative of order \(0,\ldots ,m\) exists in the preceding pointwise sense. At such a point define \(J_kf(x)\) to be its unique order-\(k\) derivative value.

Let \(\mathbb K\) be a nontrivially normed field (for example \(\mathbb R\) or \(\mathbb C\)). For an open set \(U\subseteq \mathbb K\) and \(n\ge 0\), let \(\mathcal D^n(U)\) consist of functions \(f:U\to \mathbb K\) possessing a derivative chain of order \(n\). For \(f\in \mathcal D^n(U)\) define \(D_U^n f\) to be the unique final function of that chain. No value is assigned without such a chain.

Let \(\mathbb K\) be a nontrivially normed field (for example \(\mathbb R\) or \(\mathbb C\)). For open \(U\subseteq \mathbb K\), write \(D_U^n f=g\) when there is a chain \(f_0=f,f_1,\ldots ,f_n=g\) of functions \(U\to \mathbb K\) such that \(f_{k+1}(x)\) is the derivative of \(f_k\) at every \(x\in U\), for every \(k{\lt}n\).

Let \(\mathbb K\) be a nontrivially normed field (for example \(\mathbb R\) or \(\mathbb C\)). Let \(f:U\to \mathbb K\), with \(U\) open, and \(x\in U\). Its order-zero derivative at \(x\) is \(f(x)\). For \(n=k+1\), say that its order-\(n\) derivative at \(x\) is \(d\) if some open neighbourhood \(V\subseteq U\) of \(x\) carries an order-\(k\) derivative chain of \(f|_V\), whose final function has derivative \(d\) at \(x\). The highest derivative is required only at \(x\).

For \(a,b\in \mathbb {R}\) and a continuous \(w:(a,b)\to \mathbb {R}_{\ge 0}\), form \(w\, dx\) on the open interval \((a,b)\), then push it forward to \(\mathbb {R}\) by inclusion. Denote this measure by \(\mu _w\).

For source and target bases \(b,c\) and a linear map \(L\), define the coefficient map \(c.\mathrm{repr}\circ L\circ b.\mathrm{repr}^{-1}\) on finitely supported coefficients. Its matrix is \(A_{ij}=(c.\mathrm{repr}(L(b_j)))_i\). For natural indices, lower bandwidth \(\ell \) and upper bandwidth \(u\) mean \(A_{ij}=0\) whenever \(j+\ell {\lt}i\) or \(i+u{\lt}j\). Parameter conversion uses the identity map with different source and target bases.

Reference: [OT13, §2].

For admissible real or complex parameters, the classical Gauss series is

Its argument domain consists exactly of the points where the partial sums in increasing natural index converge. Its value is their unique limit. The domain contains only arguments where these partial sums converge.

Reference: [DLMF, 15.2.1, 15.2.4–15.2.5].

For \(P:\operatorname {Polynomial.Sequence}\mathbb {R}\) and a measure \(\mu \), require integrability of each product \(P_mP_n\) and vanishing integrals when \(m\ne n\). This predicate alone imposes no normalisation or positivity of squared norms.

References: [DLMF, 18.2(i)], [Sze75, Chapter 3].

For a real polynomial \(p\), write \(D_p=\{ x\in \mathbb R:p(x)\ge 0\} \). For a measure \(\eta \) on \(D_p\), its polynomial density measure is the real-coordinate pushforward of \(p\, d\eta \). The density has values in \([0,\infty )\) on its entire domain.

For polynomial coefficients \(a_k\) over a commutative ring, assemble the endomorphism \(L=\sum _{k=0}^N M[a_k]D^k\) on the polynomial module, where \(D\) is polynomial differentiation.

Reference: [OT13, Section 3].

For a Borel measure \(\mu \) on \(\mathbb {R}\), an integrable polynomial pair consists of \(p,q\in \mathbb {R}[X]\) together with the condition \(pq\in L^1(\mu )\). Its pairing is the finite integral \(\langle p,q\rangle _\mu =\int pq\, d\mu \). This pairing is defined only on integrable pairs; no assumption on other polynomial moments is required.

Let \(\mathcal P:\mathbb {R}[X]\to L^2(\mathbb {R},\mu _T;\mathbb {R})\) send a polynomial \(p\) to the class of \(x\mapsto p(x)\).

Reference: [DLMF, Table 18.3.1].

Consider the linear maps \(p\mapsto Xp\), \(p\mapsto p'\), and \(p\mapsto p\). For a target basis \(c\) and any family \(v\), the basis matrix is \((c.\mathrm{toMatrix}(v))_{ij}=(c.\mathrm{repr}(v_j))_i\), the \(i\)th coordinate of \(v_j\). This construction allows infinite index sets.

Reference: [OT13, §2].

A Borel measure \(\mu \) on \(\mathbb {R}\) has finite polynomial moments when \(x\mapsto p(x)^2\) is \(\mu \)-integrable for every \(p\in \mathbb {R}[X]\). No interval-support or connectedness restriction is imposed.

References: [DLMF, 18.2(i)], [Sze75, Chapter 3].

Require every polynomial square to be integrable and \(\int p(x)^2\, d\mu (x){\gt}0\) for every nonzero \(p\in \mathbb {R}[X]\).

References: [DLMF, 18.2(i)], [Sze75, Chapter 3].

For a measure \(\mu \), a positive measurable weight is a measurable real function \(h\) such that \(h{\gt}0\) \(\mu \)-almost everywhere. No pointwise positivity on a null set is imposed.

For \(k\in \mathbb N\) and \(\lambda =k+1\), define \(d_n=\lambda /(n+\lambda )\), \(u_n=(n+1)/(2(n+\lambda ))\), and \(\ell _n=(n+2\lambda -1)/(2(n+\lambda ))\). On complex \(\ell ^2\) set \(S_\lambda =(I-L^2)D_d\), \(J_\lambda =RD_u+LD_\ell \), and \(M_\lambda [a]=a(J_\lambda )\) for \(a\in \mathbb R[x]\).

Take \(\alpha {\gt}-1\), \(\lambda {\gt}-1/2\) and \(\alpha +\lambda {\gt}-1/2\). For \(x\in [-1,1]\) and \(f\in C([-1,1],\mathbb {R})\), define the functions \(y\mapsto w_\lambda (y)f(y)\) and \(y\mapsto w_\lambda (y)\) on \(D_x\) as elements of the domain of \(u\mapsto Q_\alpha u(x)\), which requires their products with \(|x-y|^\alpha \) to be integrable on \(D_x\). No value of a singular density is assigned on the excluded endpoints or diagonal.

For \(\alpha {\gt}-1\), \(\lambda {\gt}-1/2\), and \(\alpha +\lambda =-1/2\), define the bounded operator on \(L^2(\mu _\lambda ;\mathbb {R})\) by its integrable weak pairing \(\langle g,Tf\rangle =\int g(x)f(y)\, dK_{\alpha ,\lambda }(x,y)\), where \(K_{\alpha ,\lambda }\) is the classical joint power-law kernel measure.

The measures satisfy \(\mu _0=\mu _T\). For \(\lambda {\gt}-1/2\), choose the normalized Gegenbauer Hilbert basis when \(\lambda \ne 0\), and the normalized Chebyshev \(T\) Hilbert basis when \(\lambda =0\), transported along this equality of measures. Denote this basis by \((b_n)\) and its coordinate isometry by \(U\).

For \(\alpha {\gt}-1\), \(\lambda {\gt}-1/2\), and \(\alpha +\lambda \ge -1/2\), select the bounded operator on \(L^2(\mu _\lambda ;\mathbb {R})\) characterized by integrable pairings \(\langle g,Tf\rangle =\int g(x)f(y)\, dK_{\alpha ,\lambda }\) against the classical joint kernel measure.

Write \(w_\lambda (y)=(1-y^2)^{\lambda -1/2}\) on \((-1,1)\). For \(\alpha {\gt}-1\), \(\lambda {\gt}-1/2\), \(x\in (-1,1)\) and \(f\in C([-1,1],\mathbb {R})\), define \(g(y)=w_\lambda (y)f(y)\) on \(D_x\) as a function in the domain of \(u\mapsto Q_\alpha u(x)\). Taking \(f=1\) gives the weighted constant amplitude; taking \(f=C_n^{(\lambda )}\) for \(n\in \mathbb N\) gives the weighted polynomial modes. At \(x=1\), define \(g(y)=w_\lambda (y)\) on \(D_1\) under the endpoint conditions \(\lambda {\gt}-1/2\) and \(\alpha +\lambda {\gt}-1/2\); no separate lower bound on \(\alpha \) is imposed. Membership in these domains requires the product \(|x-y|^\alpha g(y)\) to be integrable on \(D_x\).

Initial data \((\alpha ,\lambda )\) require all three arguments \((\alpha +1)/2\), \(\lambda +1/2\), and \(\lambda +\alpha /2+1\) to avoid the nonpositive integers. For every \(k\in \mathbb {N}\), define

Here \(H_k\) is the terminating polynomial of Definition 6.2.23. No compatibility or positivity is required for this polynomial definition.

For \(\alpha ,x\in \mathbb {R}\), let \(D_x=\{ y\in (-1,1):y\ne x\} \). An input at \((\alpha ,x)\) is a function \(f:D_x\to \mathbb {R}\) for which \(y\mapsto |x-y|^\alpha f(y)\) is integrable. Define

The parameter and input domain allow every existing weighted integral, without assigning a kernel value on the diagonal. Reference: [GCO22, Section 2].

Let \(D=\{ (x,y)\in (-1,1)^2:x\ne y\} \). For arbitrary real \(\alpha ,\tau ,\lambda \), restrict \(\mu _\tau \otimes \mu _\lambda \) to \(D\), multiply by its positive density \(|x-y|^\alpha \), and push forward to \(\mathbb {R}^2\). This defines the joint two-weight kernel measure. The common-weight kernel measure is its specialization \(\tau =\lambda \). These measures need not be finite. No diagonal or exterior kernel value is assigned.

For \(\beta \ge 0\), \(x\in \mathbb {R}\), and an integrable real function \(f\) on \((-1,1)\), regard \(f|_{D_x}\) as an element of the domain of \(u\mapsto Q_\beta u(x)\): its product with \(|x-y|^\beta \) is integrable on \(D_x\).

For \((\alpha ,\lambda ,M)\) satisfying Definition 6.2.35, with \(\lambda \ne 0\), define \(A=UTU^{-1}\), where \(U\) is the isometry furnished by the normalized Gegenbauer Hilbert basis.

Mass-bound data consist of \(\alpha {\gt}-1\), \(\lambda {\gt}-1/2\), a finite nonnegative real bound \(M\), and \(Q_\alpha w_\lambda (x)\le M\) for every \(x\in (-1,1)\). Here \(Q_\alpha w_\lambda (x)=\int _{D_x}|x-y|^\alpha w_\lambda (y)\, dy\), with an integrable integrand on \(D_x\).

For \((\alpha ,\lambda ,M)\) satisfying Definition 6.2.35, define the bounded operator on \(L^2(\mu _\lambda ;\mathbb {R})\) by the Schur construction for the classical joint power-law kernel measure \(K_{\alpha ,\lambda }\).

For \(\alpha \in \mathbb {R}\) and \(x\ge -1\), an input is a real-valued function on \((-1,x)\) with integrable weighted product \((x-y)^\alpha f(y)\). Define

The collapsed interval at \(x=-1\) is empty.

For \(\alpha {\gt}-1\), \(\lambda {\gt}-1/2\), \(x\in (-1,1)\) and \(P\in \mathbb {R}[X]\), let \(f_P(y)=w_\lambda (y)P(y)\) on \((-1,x)\). Define the linear functional

For \(x\in \mathbb {R}\), the domain of \(\phi _x\) is \(\{ t\in \mathbb {R}:1-xt\ne 0\} \), and \(\phi _x(t)=(x-t)/(1-xt)\). The rational map also retains all admissible real arguments outside that interval.

For \(x\in (-1,1)\) and an integrable real function \(f\) on \((-1,1)\), define its Jacobian-weighted pullback as the integrable input

For \(\alpha {\gt}-1\), \(\lambda {\gt}-1/2\) and \(x\in (-1,1)\), define on \(D_0=(-1,1)\setminus \{ 0\} \) the function

as an element of the domain of \(u\mapsto Q_\alpha u(0)\); its product with \(|t|^\alpha \) is integrable. All bases of real powers in this expression are positive on \(D_0\).

For \(\alpha {\gt}-1\), \(\lambda {\gt}-1/2\) and \(x\in (-1,1)\), define the linear functional on real polynomials by \(P\mapsto Q_\alpha (w_\lambda P)(x)\), where \(w_\lambda P\) is restricted to \(D_x=(-1,1)\setminus \{ x\} \).

For whole-family data with \(\lambda {\gt}-1/2\) and \(\lambda \ne 0\), define the polynomial linear operator \(C_n^{(\lambda )}\mapsto q_n\). An individual coefficient column is the Gegenbauer-basis representation of \(q_n\) and requires only admissible data through degree \(n\), together with the same nondegenerate basis conditions. It does not require later recurrence steps.

For a mode \(n\), take initial data and require \(\alpha +2\ne 0\) if \(n\ge 1\), and \(m+2\lambda +\alpha +2\ne 0\) for every \(m+2\le n\). Define the finite recurrence by the preceding \(q_0\),

and, through the requested mode,

Whole-family data require these nonzero-denominator conditions at every step. A finite mode requires no conditions on later unused steps.

For a real exponent \(\alpha \), admissible cutoffs satisfy \(\delta {\gt}0\) or \(\alpha \ge 0\). On \(\mathbb {R}^2\) define

A zero base is allowed only for a nonnegative exponent, with \(0^0=1\). For comparison with the unregularized kernel, the physical off-diagonal domain consists of pairs \(x,y\in [-1,1]\) with \(x\ne y\).

For \(\alpha \in \mathbb {R}\) and \(x\le 1\), an input is a real-valued function on \((x,1)\) with integrable weighted product \((y-x)^\alpha f(y)\). Define

The collapsed interval at \(x=1\) is empty.

For real \(\alpha \) such that neither \((\alpha +1)/2\) nor \(-\alpha /2\) is a nonpositive integer, define \(C_\alpha =\Gamma ((\alpha +1)/2)\Gamma (-\alpha /2)/\Gamma (1/2)\).

For real \(\alpha ,\lambda \), define the joint kernel measure with exponent \(\alpha \) and Gegenbauer parameter \(\lambda +\alpha /2\) in both variables. It is the classical measure obtained by tilting the original kernel by \(h_\alpha (x)h_\alpha (y)\) on its off-diagonal open-square domain.

For every real \(\alpha \), define the continuous positive weight \(h_\alpha (x)=(1-x^2)^{\alpha /2}\) on \(x\in (-1,1)\).

Define \(A=U_\tau T U_\lambda ^{-1}\) using the normalized source and target Hilbert bases, including the Chebyshev \(T\) choice at a zero parameter.

For a common exponent \(\alpha {\gt}-1\), assume the uniform mass bounds of Definition 6.2.35 for input weight \(\lambda {\gt}-1/2\) and output weight \(\tau {\gt}-1/2\), with respective bounds \(M,N\ge 0\). Define \(T:L^2(\mu _\lambda )\to L^2(\mu _\tau )\) by the Schur construction for the classical two-weight joint kernel measure.

For real \(\alpha ,\lambda \), a weak realization is a bounded operator \(T\) on \(L^2(\mu _\lambda ;\mathbb {R})\) such that the function \(g(x)f(y)\) is integrable against the classical joint kernel measure for every \(L^2\) pair \(f,g\), and \(\langle g,Tf\rangle =\int g(x)f(y)\, dK_{\alpha ,\lambda }\). This domain may be empty.

For an outer measure \(\mu \), the admissible inner measures \(\nu \) are those for which the measure-valued map \(x\mapsto (y\mapsto (x,y))_*\nu \) is \(\mu \)-almost-everywhere measurable. This is the measurability condition used to form the product measure \(\mu \otimes \nu \).

For real parameters and natural indices, the multiplication-entry domain requires integrability of \(x_\ell Q_{n,k}Q_{m,j}\) and \(Q_{m,j}^2\) for the Proriol measure, together with a strictly positive target squared norm \(h_{m,j}=\int Q_{m,j}^2\, d\mu \). Define the multiplication entry by \(M^\ell _{(m,j),(n,k)}=(\int x_\ell Q_{n,k}Q_{m,j}\, d\mu )/h_{m,j}\). This local domain does not require positivity for other polynomial pairings.

For every parameter and index choice for which \((P_{n,k}^{\alpha ,\beta ,\gamma })^2\) is integrable with respect to \(\omega _{\alpha ,\beta ,\gamma }\), define

The domain consists of all such square-integrable polynomials; no additional parameter restrictions are imposed.

For real \(\alpha ,\beta ,\gamma \), write \(\widetilde P_j^{(a,b)}(x)=P_j^{(a,b)}(2x-1)\), and for a polynomial \(q(z)=\sum _j c_jz^j\) of degree at most \(k\) let \(x^kq(y/x)\) denote the polynomial \(\sum _j c_jy^jx^{k-j}\). Define

used for \(0\le k\le n\).

References: [DLMF, 18.37.7]; [DX14, Section 2.4].

For a measure \(\mu \), finite nodes \(v\) indexed by \(s\), and an index \(i\), the quadrature-weight domain requires \(v_i\ne v_j\) for \(j\in s\setminus \{ i\} \) and \(\mu \)-integrability of \(\ell _i(X)=\prod _{j\in s\setminus \{ i\} }(X-v_j)/(v_i-v_j)\). Only this individual polynomial is required to be integrable.

For a measure \(\mu \) on \(\mathbb R\) and a finite node set, define \(w_i=\int \ell _i\, d\mu \) on the domain of Definition 3.6.1, where \(\ell _i\) is the Lagrange basis polynomial of the nodes (Theorem 2.4.15). For a polynomial sequence \(P\), its degree-\(n\) Gaussian weights use the finite set \(s_n\) of distinct real roots of \(P_n\), with the identity node map. No ordering of the roots is chosen. The normalized-product integral also defines a weight at an off-root point whenever it is integrable; the identity node map has no vanishing denominator at any real point.

Reference: [DLMF, 3.5(iv)–(v)].

Define \(G:\mathbb C\to \mathbb C\) by \(G(z)=1/\Gamma (z)\) for \(z\in D\) and \(G(-n)=0\) for \(n\ge 0\).

Reference: [DLMF, 5.2(i)].

The domain of the diagonal coefficient at \(n\) requires integrability of \(P_n^2\) and \(xP_n^2\), and \(h_n\ne 0\). The domain of the backward coefficient at \(n\) requires integrability of \(P_n^2\) and \(P_{n+1}^2\), and \(h_n\ne 0\). No nonzero condition is imposed on \(h_{n+1}\), and no moments at other indices are required.

For an index map \(e:I\to \mathbb {N}\) and output index \(k\), consider coefficient families \(c:I\to \mathbb {C}\) whose restriction to \(\{ i:e(i)=k\} \) is summable. Convergence is required only in this fibre.

For a coefficient family \(c:I\to \mathbb C\) summable on \(\{ i:e(i)=k\} \), define \(c^{[e]}_k=\sum _{i:e(i)=k}c_i\).

Reference: elementary regrouping, used for the aliasing formula [ATAP, Theorem 4.2].

The regularized generalized hypergeometric value is the limit of \(\sum _{n=0}^{N-1}v_n(A;B)z^n\) as \(N\to \infty \), on exactly the domain where this limit exists. Its Gauss specialization \(\mathbf F(a,b;c;z)\) takes \(A=\{ a,b\} \) and \(B=\{ c\} \). The domain includes conditional convergence and excludes divergent series. Reference: [DLMF, 15.2.2, 16.2].

For finite multisets \(A,B\) of complex parameters, define

where \(G\) is the reciprocal Gamma function of Definition 1.2.3.

Reference: [DLMF, 15.2.2, 16.2].

For \(\mathbb K=\mathbb {R}\) or \(\mathbb {C}\), embed the existing real polynomial functions and normalized vectors in \(L^2(\mu ;\mathbb K)\) at positive-norm indices. The scalar kernel vector is the real kernel vector embedded into this space, at a cutoff with positive squared norms throughout. For \(f\in \mathcal A_{\mu ,P_n}^{\mathbb K}\) and \(f\in \mathcal A_{\mu ,P,\le N}^{\mathbb K}\), respectively, define

For \(f\) in the respective domains \(\mathcal A_{\mu _{\alpha ,\beta },P_n^{(\alpha ,\beta )}}^{\mathbb {C}}\) and \(\mathcal A_{\mu _\lambda ,C_n^{(\lambda )}}^{\mathbb {C}}\), define the complex Jacobi and Gegenbauer coefficients \(h_n^{-1}\int fP_n^{(\alpha ,\beta )}\, d\mu _{\alpha ,\beta }\) and \(h_n^{-1}\int fC_n^{(\lambda )}\, d\mu _\lambda \), with \(h_n\) the corresponding squared norms, in the original polynomial normalizations.

Reference: [DLMF, 18.18].

For \(b\in R[x]\), a step \(s\ge 1\) and \(n\in \mathbb {N}\), define

Let \(X^{(\gamma )}_{ij}=[p_i^{(\gamma )}](x p_j^{(\gamma )})\). For \(c,N\in \mathbb {N}\), let \(R_{c,N}\) be the leading \(N\times N\) section of the connection from \(p^{(\gamma )}\) to \(p^{(\gamma +c)}\): \((R_{c,N})_{ij}=[p_i^{(\gamma +c)}]p_j^{(\gamma )}\), \(0\le i,j{\lt}N\). Here \([p_i]q\) denotes the coordinate of \(q\) in the indicated polynomial basis.

For \(t{\gt}1\), \(\alpha ,\beta {\gt}-1\) and \(\gamma \in \mathbb {R}\), let \(B\) be the polynomial basis whose \(n\)th member is \(Q_n\).

Reference: [PGSO24, Section 3].

For \(t\in (1,\infty )\) and arbitrary real \(a,b,c\), define the continuous nonnegative weight \(w_{t,a,b,c}(x)=x^a(1-x)^b(t-x)^c\) on \((0,1)\). Define \(\mu _{t,a,b,c}\) by this density on the open interval, pushed forward to \(\mathbb {R}\) by inclusion. The measure need not be finite.

Reference: [PGSO24, Section 3].

For \(t{\gt}1\), \(\alpha ,\beta {\gt}-1\) and arbitrary real \(\gamma \), let \((Q_n)\) be a degree-indexed polynomial sequence chosen to be orthonormal for \(x^\alpha (1-x)^\beta (t-x)^\gamma \, dx\) on \((0,1)\), with positive leading coefficients; such a sequence exists by Theorem 5.2.4.

Reference: [PGSO24, Section 3].

For \(0\le i{\lt}N+1\) and \(0\le j{\lt}N\), define

Let \(b,c\) be polynomial bases over a commutative ring \(R\), let \(L:R[x]\to R[x]\) be \(R\)-linear, and let \(\sigma :R\to \mathbb C\) be a ring map. Write \(M_{ij}=[Lb_j]_{c,i}\). At index \(i\), an admissible input is a complex sequence \(a\) for which \(\sum _j a_j\sigma (M_{ij})\) converges unconditionally. Define \(\beta _i\) only for such an input, as this sum.

Reference: [OT13, Sections 2–3].

Let \(D\subseteq \mathbb R\) and \(p\in \mathbb R[X]\). A measure \(\eta \) on \(\{ x\in D:p(x)\ge 0\} \) defines a measure on \(D\) by weighting by \(p\) and pushing forward along the inclusion into \(D\).

For \(n,k\in \mathbb N\) and \(\beta \in \mathbb R\), when every constituent coefficient is defined, set \(\mathcal A_{n,k}(\beta )=A_k(\beta )A_{\max (n-k,0)}(\beta +k+1/2)\).

For \(n,k\in \mathbb N\) and \(\beta \in \mathbb R\), when every constituent coefficient is defined, set \(\mathcal B_{n,k}(\beta )=A_k(\beta )B_{\max (n-k,0)}(\beta +k+1/2)\).

Let \(n,k\in \mathbb N\). For \(k{\lt}2\) the coefficient is zero for every real \(\beta \). For \(k\ge 2\), when every constituent coefficient is defined, set \(\mathcal C_{n,k}(\beta )=B_k(\beta )E_{\max (n-k,0)}(\beta +1+(k-2)+1/2)\).

Let \(n,k\in \mathbb N\). For \(k{\lt}2\) the coefficient is zero for every real \(\beta \). For \(k\ge 2\), when every constituent coefficient is defined, set \(\mathcal D_{n,k}(\beta )=B_k(\beta )F_{\max (n-k,0)}(\beta +1+(k-2)+1/2)\).

For \(n\in \mathbb N\) and \(\beta \in \mathbb R\), when every constituent coefficient below is defined, set \(\widehat A_n(\beta )=A_n(\beta )A_n(\beta +1)\).

For \(n\in \mathbb N\) and \(\beta \in \mathbb R\), when every constituent coefficient below is defined, set \(\widehat B_n(\beta )=A_n(\beta )B_n(\beta +1)+B_n(\beta )A_{\max (n-2,0)}(\beta +1)\).

For \(n\in \mathbb N\) and \(\beta \in \mathbb R\), when every constituent coefficient below is defined, set \(\widehat C_n(\beta )=B_n(\beta )B_{\max (n-2,0)}(\beta +1)\).

Write \(D_n(\beta )=(2n+2\beta +2)(n+2\beta +2)(2n+2\beta +3)(2n+2\beta +4)\). For \(D_n(\beta )\ne 0\), define

With \(D_n(\beta )=(2n+2\beta +2)(n+2\beta +2)(2n+2\beta +3)(2n+2\beta +4)\), define

on the same domain \(D_n(\beta )\ne 0\).

For \(\beta \in \mathbb R\) and natural degrees, define

The domain at each degree requires exactly the displayed denominator to be nonzero; degree zero admits every real parameter.

Define \(B_0(\beta )=B_1(\beta )=0\) for every real \(\beta \), and

when its displayed denominator is nonzero.

Write \(h_0=\pi \) and \(h_n=\pi /2\) for \(n{\gt}0\), and set \(e_n=h_n^{-1/2}\mathcal P(T_n)\). This is \(T_0/\sqrt\pi \) at index zero and \(T_n\sqrt{2/\pi }\) otherwise.

Reference: [DLMF, Table 18.3.1].

For a field \(K\) and degree \(n\), call \(n\) an admissible primitive degree when the denominators in the adjacent-degree formula are nonzero: there is no condition at \(n=0\), one requires \(4\ne 0\) at \(n=1\), and for \(n=m+2\) one requires \(2(m+3)\ne 0\) and \(2(m+1)\ne 0\) in \(K\).

Over a field \(K\), for an admissible primitive degree \(n\) as in Definition 2.7.7, define \(Q_0=T_1\), \(Q_1=T_2/4\), and

The anchored polynomial is \(Q_n-Q_n(a)\).

Reference: [ATAP, Chapter 19].

For positive measurable weights \(h\) for \(\mu \) and \(g\) for \(\nu \), and a joint measure with marginals bounded by \(C h^2\mu \) and \(D g^2\nu \), where \(C,D\) are finite and nonnegative, define \(T=D_h^* S D_g\). Here \(S\) is the ordinary Schur operator between the two tilted \(L^2\) spaces.

Let \(B_0\) be the Chebyshev coordinate basis and, for integers \(k\ge 1\), let \(B_k\) be the Gegenbauer basis with \(\lambda =k\).

Reference: [OT13, Section 3].

A polynomial density has uniform Cartesian inner bandwidth if there exists one finite bound \(w\) such that, for every admissible pair of factor measures, every Cartesian source system and every complete degree section, the canonical modified Gram factor has zero entries whenever the absolute inner-index gap exceeds \(w\). The factor measures have finite moments, strictly positive polynomial pairings and support in \([-1,1]\); the inner measure is even. Only densities nonnegative almost everywhere for the Cartesian source are tested, using their polynomial density measures on the nonnegative locus. The modified pairing must also have the moments and positivity required to define its canonical factor. The bound is chosen before both the measures and the section.

Let \(\widehat\mu _{\alpha ,\beta }\) be the pushforward of \(\mu _{\alpha ,\beta }\) by \(t\mapsto (t+1)/2\). Set \(\widehat P_n^{(\alpha ,\beta )}(x)=P_n^{(\alpha ,\beta )}(2x-1)\).

On the open interval \((0,1)\), set \(\widehat w_{\alpha ,\beta }(x)=2w_{\alpha ,\beta }(2x-1)\).

For \(p\in R[X]\) over a commutative ring and \(n\in \mathbb {N}\), define

Let \(S\) be synthesis in the signed reflected Proriol basis \(Q_{n,k}\). For finite shifted Jacobi \((1,0)\) coefficients \(c\), set

where \(b\) is a finite triangular coefficient vector and \(D\) selects inner index zero. The \(n\)th column of \(E\) is the coordinate vector of \(P_n^{(1,0)}(2y-1)\). The endpoint factor and output reflection are kept outside \(B_K\). Reference: [GO20, Sections 3.1–3.2].

For a continuous kernel \(K\) on the closed Volterra triangle, define \(R_K=(I-A_K)^{-1}\) on \(\ell ^2\).

For a continuous kernel \(K\) and a natural cutoff \(N\), put \(B_{K,N}=I-P_NA_K\) on normalized coefficient space.

For an integrable row pair \((k,f)\) at \(x\), define its oriented value by

For the row \(k(y)=K(x,y)\) this is denoted \(V_Kf(x)\). The integral exists by the input’s product-integrability condition.

For \(x\in \mathbb R\), put \(J_x=(\min (0,x),\max (0,x))\). A Volterra input at \(x\) consists of two real-valued functions \(k,f\) on \(J_x\) whose product is Lebesgue integrable there. Neither factor is required to be separately integrable, and no endpoint values are part of the input.

For exponent sequences \(i_n,j_n\in \mathbb N\), an operator-series input is a real coefficient sequence \(c_n\) for which the ordered operator family \(A_{c_nx^{i_n}y^{j_n}}\) is unconditionally summable in the Banach space of bounded operators on real \(\ell ^2\).

For polynomial kernels \(p_n\), let \(B_n=I-P_nA_{p_n}\). For an invertible cutoff \(n\) and \(f\in \ell ^2\), define \(v_n\) by restricting \(B_n^{-1}P_nf\) to its first \(n\) coordinates.

For real \(R,S\), put \(B_{R,S}=\{ (x,y)\in \mathbb R^2:|x|\le R,\ |y|\le S\} \). This set is empty if either radius is negative.

For a kernel-series input at \(z=(x,y)\), define \(K(z)=\sum _{n\ge 0}c_nx^{i_n}y^{j_n}\), the sum of its absolutely convergent series.

For nonnegative radii and a summable joint majorant, a kernel-series extension is a continuous function \(F:\mathbb R^2\to \mathbb R\) such that \(F(z)=K(z)\) for every \(z\in B_{R,S}\). Outside the rectangle, \(F\) is an extension and is not asserted to be a series value.

Fix exponent sequences \(i_n,j_n\in \mathbb N\) and a point \(z=(x,y)\in \mathbb R^2\). A kernel-series input at \(z\) is a real coefficient sequence \(c_n\) for which \(\sum _n |c_nx^{i_n}y^{j_n}|{\lt}\infty \). Convergence is required at the selected point; no common rectangle is required to define the series at \(z\).

For a convergent operator-series input, define the unconditional sum \(A=\sum _{n\ge 0}A_{c_nx^{i_n}y^{j_n}}\) in the space of bounded operators on real \(\ell ^2\).

For a Volterra tail input, define the convergent coefficient tail by \(E_N=\sum _{n\ge 0}w_{n+N}\).

For exponent sequences \(i_n,j_n\in \mathbb N\) and real radii \(R,S\), a Volterra tail input is a real coefficient sequence \(c_n\) for which the real family \(w_n=w_{R,S}(i_n,j_n,c_n)\) is absolutely summable. No positivity of the radii is needed to specify this convergence domain.

The physical Volterra triangle is \(\Delta =\{ (x,y)\in \mathbb R^2:0\le y\le x\le 1\} \).

For a real polynomial kernel \(K\), set \(S_N(A_K)=I-P_N+P_NA_KP_N\). At a cutoff in the domain where \(S_N(A_K)\) is invertible, define \(u_{K,N}(q)=S_N(A_K)^{-1}P_ND^{-1}q\) for \(q\in \ell ^2_1\).

For a real polynomial kernel \(K\), let \(A_K\) be its preconditioned coefficient operator and \(D:\ell ^2\to \ell ^2_1\) the isometric integration diagonal. Define \(W_K=DA_K\). For a kernel in the domain where \(A_K\) is invertible and weighted data \(q\in \ell ^2_1\), define \(u_K(q)=A_K^{-1}D^{-1}q\).

For a measure \(\mu \), weak kernel data include almost-everywhere measurability of \(x\mapsto (y\mapsto (x,y))_*\mu \) as a measure-valued map, certifying the product \(\mu \times \mu \). They consist further of a real kernel \(k(x,y)\), a bounded operator \(T\) on \(L^2(\mu ;\mathbb {R})\), and, for every \(L^2\) pair \(f,g\), absolute integrability of \(k(x,y)g(x)f(y)\) and the identity \(\langle g,Tf\rangle =\int k(x,y)g(x)f(y)\, d(\mu \times \mu )\).

For bounded maps \(B,A:W_s\to W_s\), define \(K_{N,B,A}=P_N(B-I)+R^N A\). The first \(N\) coordinates of \(Bv\) specify the boundary data; coordinates of \(Bv\) with index at least \(N\) are unused.

A kernel cutoff is a natural number \(N\) together with \(h_n{\gt}0\) for every \(0\le n\le N\). For \(f\in \mathcal A_{\mu ,P,\le N}^{\mathbb {R}}\), write \(S_Nf=\sum _{n=0}^Na_n(f)P_n\) as a polynomial, and let \(v_{N,x}=\sum _{n=0}^NP_n(x)h_n^{-1/2}q_n\) be the kernel vector in weighted L2, where \(q_n=P_n/\sqrt{h_n}\) and the kernel vector is defined only at a kernel cutoff.

For a real polynomial basis \(b=(b_n)\) and \(p\in \mathbb R[x]\), let \(c_b(p)\in W_s\) be its finite coordinate sequence, regarded as complex coefficients.

For a positive measurable weight \(h\) and a joint measure whose marginals are bounded by \(C h^2\mu \), with \(0\le C{\lt}\infty \), let \(S\) be the ordinary Schur operator on \(L^2(h^2\mu )\). Define \(T=D_h^* S D_h\) on \(L^2(\mu )\).

On \(W_s\), use the same coordinate formulas for \(S_\lambda \), \(J_\lambda \) and \(M_\lambda [a]\) at integer levels \(\lambda \ge 1\). At level zero set \(S_0=\tfrac 12(I-L^2+P_1)\), \(J_0=\tfrac 12(R(I+P_1)+L)\) and \(M_0[a]=a(J_0)\). Write \(S_{j\to N}=S_{N-1}\cdots S_j\) for successive conversion, with \(S_{j\to j}=I\).

Let \(N=k+1\) and \(c_k=2^k k!\). Define \(\mathcal R_N=c_k^{-1}\operatorname {diag}(1,1,1/2,1/3,\ldots )\) on \(W_s\). For \(j\ge 0\), define the preconditioned order-\((j+1)\) derivative by \(E_{N,j+1}=(c_j/c_k)L^{j+1}\).

Let \(\widehat{\mathcal R}_N:W_s\to W_{s+1}\) have the same coordinate formula as \(\mathcal R_N\). For \(x\in [-1,1]\), define \(\mathcal F_N(v)(x)=\sum _{n\ge 0}(\mathcal R_Nv)_nT_n(x)\) using the convergent Chebyshev series.

For a Cartesian polynomial \(p\) and \(s\ge 0\), let \(A_p(s)\) be the function \(\theta \mapsto p(\Phi (s,\theta ))\) on \((-\pi ,\pi )\). Write \(J_p(s)=\int _{-\pi }^{\pi }A_p(s)(\theta )\, d\theta \).

For \(q\ne 1\), define \(\Psi _q(x,\theta )=\Phi (1-x/t(q),\theta )\) on the domain \(1-x/t(q)\ge 0\).

For \(0\le q{\lt}1\) and real \(a,b\), define \(\omega _{q,a,b}=\frac12t(q)^{-(a+b+1)}\omega ^{\rm aux}_{q,a,b}\). The value \(q=0\) includes the disk endpoint.

For \(q\ge 0\), \(q\ne 1\), and real \(a,b\), define \(\omega ^{\rm aux}_{q,a,b}\) as the pushforward by \(\Psi _q\) of the semiclassical measure with parameters \((2,a,b,0)\), regarded as a measure on \([0,1]\), times angular Lebesgue measure on \((-\pi ,\pi )\).

For \(0{\lt}q{\lt}1\), \(a,b{\gt}-1\) and a Cartesian polynomial \(p\), define the function \(B_{q,a,b,p}(s)=(1-s)^a(s-q)^bJ_p(s)\) on \(s\in (q,1)\).

For \(s\in [0,\infty )\) and \(\theta \in \mathbb R\), set \(\Phi (s,\theta )=\sqrt{s}(\cos \theta ,\sin \theta )\).

For real \(a,b\), define \(\omega _{a,b}=2^{-(a+b+2)}\omega ^{\rm aux}_{a,b}\).

For an individual real Zernike mode \(Z\), with \(Z^2\) integrable against \(\omega _{a,b}\), define its norm by \(\sqrt{\int Z^2\, d\omega _{a,b}}\). The normalized polynomial \(Z/\| Z\| \) additionally requires a strictly positive squared integral.

For every parameter and index choice for which \((Z_{n,j}^{a,b})^2\) is integrable with respect to \(\omega _{a,b}\), define

The domain consists of all such square-integrable polynomials; no additional parameter restrictions are imposed.

For \(a,b{\gt}-1\) and a Cartesian polynomial \(p\), define the function \(R_{a,b,p}(r)=r(r^2)^a(1-r^2)^bJ_p(r^2)\) on \(r\in (0,1)\).

For real \(a,b\), let \(\eta _{b,a}\) be the affine Jacobi measure on \([0,1]\), regarded as a measure on \([0,\infty )\). Define \(\omega ^{\rm aux}_{a,b}=\Phi _*(\eta _{b,a}\otimes d\theta |_{(-\pi ,\pi )})\).

Let \(H^c_m\) and \(H^s_m\) be the real polynomials with \(H^c_m+iH^s_m=(x+iy)^m\). For real \(a,b\) and the shifted Jacobi polynomials \(\widetilde P_k^{(a,b)}(s)=P_k^{(a,b)}(2s-1)\), set

In total degree \(n\), the \(j\)th polynomial, \(0\le j\le n\), is the sine mode with frequency \(m=j+1\) when \(n+j\) is odd and the cosine mode with frequency \(m=j\) otherwise, with radial degree \(k=(n-m)/2\).

Reference: [DLMF, 18.37(i)].

For \(a,b{\gt}-1\) and a Cartesian polynomial \(p\), define the function \(S_{a,b,p}(s)=(1-s)^bs^aJ_p(s)\) on \(s\in (0,1)\).

The measure \(\mu _T\) is finite and almost every point belongs to \([-1,1]\).

Reference: [DLMF, Table 18.3.1].

Every real-valued function continuous on \([-1,1]\) is integrable for \(\mu _U\).

Reference: the second-kind weight in [DLMF, Table 18.3.1].

For \(f\in \mathcal I(\mu _U)\),

The function \(x\mapsto f(x)\sqrt{1-x^2}\) is Lebesgue-integrable on \((-1,1)\). Reference: [DLMF, Table 18.3.1].

For integers \(n,m\),

Reference: [DLMF, Table 18.3.1 and equations 18.5.1–18.5.2].

For \(f\in \mathcal I(\mu _U)\), the function \(x\mapsto f(x)(1-x^2)\) is \(\mu _T\)-integrable and

Reference: [DLMF, Table 18.3.1].

For \(\beta \ge 0\), \(k_\beta (t)=|t|^\beta \) on \(\mathbb {R}\), with \(0^0=1\). For \(\beta {\gt}1\) its derivative exists at every real \(t\) and is \(\beta |t|^{\beta -1}\operatorname {sign}(t)\), including value zero at \(t=0\).

If \(\sum _n|a_n|{\lt}\infty \), then \(S_a\) is continuous on \([-1,1]\), \(|S_a(x)|\le \sum _n|a_n|\), and its integral Chebyshev coefficient at index \(k\) is exactly \(a_k\), including the full constant coefficient. More precisely, any \(f\in \mathcal I_k^{\mathbb {C}}\) agreeing with \(S_a\) on \([-1,1]\) has coefficient \(a_k\). All interval points belong to \(D_a\).

Reference: [ATAP, Chapters 3 and 4].

For \(a{\lt}b\), \(A\) and \(B\) are inverse maps between \([-1,1]\) and \([a,b]\), with \(A(-1)=a\) and \(A(1)=b\). Composition \(p\circ B\) is ordinary polynomial composition and preserves degree. Uniform approximation errors transfer unchanged. At points where \(g\) has derivative \(d\) at \(B(x)\), \(g\circ B\) has derivative \(2d/(b-a)\) at \(x\). Moreover,

The integral identity assumes that \(f\) is integrable between \(a\) and \(b\) and also asserts integrability of \(f\circ A\) on \([-1,1]\). It holds for either orientation and for equal endpoints. The approximation and interval-membership assertions assume \(a{\lt}b\); the coordinate \(B\) requires distinct endpoints.

Reference: affine change of variables, as in [DLMF, 3.11.12].

For \(a{\lt}b\) and \(N\ge 1\), \(P_N(f\circ A)\circ B\) interpolates \(f\) at the mapped extrema \(A(x_j)\). Its integral is the scaled rule \(\frac{b-a}{2}\mathcal Q_N(f\circ A)\), which is exact for every physical polynomial of degree at most \(N\). If \(f\circ A\) is interval integrable and its canonical interpolation error is at most \(E\) on \([-1,1]\), the physical quadrature error is at most \((b-a)E\). In particular, let \(\rho {\gt}1\), let \(F\) be analytic on the open ellipse \(E_\rho \), and let \(|F|\le M\) there. For \(f(x)=\Re F(B(x))\) the error is at most

Reference: [ATAP, Chapters 8 and 19], with affine scaling.

Under the hypotheses of Theorem 2.3.3, with \(\nu \ge 1\) and \(N{\gt}\nu \), the degree-\(N\) Chebyshev projection satisfies

The coefficients are absolutely summable.

Reference: [ATAP, Theorem 7.2].

Let \(N\ge 1\) and \(L=\sum _{k=0}^N a_kD^k\), with real polynomial coefficients and \(d_k=\operatorname {natDegree}a_k\). In Chebyshev input and \(B_N\) output, each term has bandwidths \((d_k,d_k+2N-k)\). Hence \(L\) has bandwidths \((\ell ,u)\) whenever \(d_k\le \ell \) and \(d_k+2N-k\le u\) for every \(k\le N\). A common degree bound \(d\) gives \((d,d+2N)\). For input supported on the first \(m\) coefficients, all residual coefficients at indices \(i\ge m+d\) vanish. These are exact polynomial statements.

Reference: [OT13, Section 3].

If a real polynomial changes sign across each of \(n\) consecutive intervals with strictly increasing endpoints, then its natural degree is at least \(n\). Each sign change gives a root in the open interval.

Reference: [ATAP, proof of Theorem 10.1].

Let \(p\) have degree at most \(N\). If \(N+2\) increasing points satisfy \(\sigma (-1)^i(f(x_i)-p(x_i))\ge L\), where \(|\sigma |=1\), then every polynomial \(q\) of degree at most \(N\) satisfies \(L\le \| f-q\| _\infty \). In particular, an equioscillating error certifies a best approximant.

Reference: [ATAP, Theorem 10.1 and Exercise 10.3].

If \(\rho {\gt}1\) and \(|a_n|\le C\rho ^{-n}\) for all \(n\ge 0\), then \(S_a\) is analytic on the open ellipse \(E_\rho \). For \(1{\lt}r{\lt}\rho \) and \(z\in E_r\), the series converges absolutely and \(|S_a(z)|\le C/(1-r/\rho )\). Boundedness on all of \(E_\rho \) is not claimed.

Reference: [ATAP, Theorem 8.3].

For continuous complex data \(f\) on \([-1,1]\) and \(\rho {\gt}1\), an analytic extension to \(E_\rho \) exists if and only if, for every \(1{\lt}r{\lt}\rho \), there is \(C_r\) such that \(|a_n(f)|\le C_r r^{-n}\) for every \(n\ge 0\). The constant may depend on \(r\).

Reference: [ATAP, Theorems 8.1–8.3].

Let \(P\) be orthogonal for a measure \(\mu \) with finite polynomial moments and positive definite polynomial pairing, supported almost everywhere on \([-1,1]\). Let \(\rho {\gt}1\), let \(F\) be analytic on \(E_\rho \) with \(|F|\le M\) there, and let \(f\in L^2(\mu ;\mathbb {R})\) agree almost everywhere with \(\operatorname {Re}F\). Then for every \(x\in [-1,1]\),

Let \(L=\sum _{k=0}^m A_kD^k\) have real polynomial coefficients. On a normally convergent complex Chebyshev expansion, the existing matrix from \(T\) to \(B_m\), under the weighted reconstruction condition of Theorem 4.3.10, represents the analytic differential expression \(\mathcal LS\). For \(m\ge 1\) and \(\operatorname {natDegree} A_k\le d\), each output coefficient uses only indices \(j{\lt}i+d+2m+1\). This is the same matrix as finite polynomial assembly, without introducing a second complex basis definition.

For any continuous real linear load \(\ell \) on \(\mathcal H_q\), there is a unique \(u\) satisfying \((u,z)=\ell (z)\) for all \(z\in \mathcal H_q\). Restrict the load to each normalized sector and solve its existing tridiagonal radial Dirichlet system. The sum \(u_{S,N}\) of the retained sector reconstructions is exactly the orthogonal projection of \(u\) onto \(V_{S,N}\). Hence

In particular, taking every valid angular frequency \(m{\lt}N\) and radial degree below \(N\) gives convergence as \(N\to \infty \). The coordinates use the normalization in Definition 5.4.36.

Applying \(D\) in each normalized real angular sector embeds \(\mathcal H_q\) isometrically into the Hilbert sum of the sector derivative \(L^2\) spaces, with closed range. The error of an actual finite annular solve in this concrete space equals its energy error. Thus the frequency/radial-degree truncations converge strongly as radial derivative classes as well.

Write \(U_p=BL_{m(s)}p\). The embedding of sector \(s\) preserves the planar negative-Laplacian pairing, \((\iota _sp,\iota _sz)=-\int \Delta U_p\, U_z\, d\mu \), where \(\mu \) is the annular measure with \(a=b=0\). Embeddings of distinct sectors are orthogonal. The finite angular and radial trial spaces have dense union in \(\mathcal H_q\).

For \(0{\lt}q{\lt}1\), let \(\Omega _q=\{ (x,y):q{\lt}x^2+y^2{\lt}1\} \). The unweighted annular measure is exactly planar Lebesgue measure restricted to \(\Omega _q\):

This is equality of measures, hence applies to arbitrary measurable functions and their \(L^2\) classes, not only to polynomial integrands.

Every physical datum \(g\in L^2(dA)\) gives a continuous energy load \(\ell _g(x)=\langle g,Tx\rangle _{L^2}\) with \(\| \ell _g\| \leq \| g\| _{L^2}\). Its value on a polynomial trial is \(\int gU_p\, dA\). There is a unique \(x\in \mathcal H_q\) satisfying

The full solution and every assembled finite coefficient solve have energy norm at most \(\| g\| _{L^2}\); the finite physical realizations have the same bound. The physical reconstruction is precisely the finite sum of computed radial reconstructions with their inverse square-root sector scales. For any continuous energy load, these reconstructions converge to \(Tx\) in physical \(L^2\) as both cutoffs increase, also for the common frequency/radial-degree cutoff. Their physical error is bounded by the best energy approximation error in the finite trial space.

For boundary-weighted lifts \(U_p,U_z\) in a real angular sector \((u,m)\),

Distinct angular sectors are orthogonal for this Cartesian gradient pairing.

For every real Cartesian polynomial \(p\), the compactly supported smooth cutoff trials satisfy

For \(0{\lt}q{\lt}1\) and a real Cartesian polynomial \(p\), Every \(u_a\) is smooth; for \(a{\gt}0\) it has compact support contained in \(\Omega _q\). At each point of \(\Omega _q\), the functions \(\phi _n\) eventually agree on a neighbourhood with \(B_qp\), where \(B_q=(1-r^2)(r^2-q)\).

For \(t{\gt}1\) and real \(\gamma \), the map

from radial energy polynomials to \(L^2(\mu _{t,0,0,\gamma +1})\) extends to a linear isometry \(D:H_{t,\gamma }\to L^2(\mu _{t,0,0,\gamma +1})\). Its range is exactly the closed \(L^2\) closure of these polynomial derivative classes; no surjectivity onto all of \(L^2\) is claimed. For every \(g\) in this \(L^2\) space, \(\ell _g(u)=\langle g,Du\rangle \) is a continuous energy load of norm at most \(\| g\| _{L^2}\), and its finite coefficient reconstructions are uniformly bounded by this norm. The existing energy convergence also gives strong \(L^2\) convergence of the realized derivatives.

Let \(t{\gt}1\) and \(\gamma \in \mathbb R\). Set \(S(x)=x(1-x)\), \(Ep=(Sp)'\), and

Then \(A p=L_{t,\gamma }(t^{-2}Sp)\), with the radial Laplacian expression above. In the semiclassical pairings,

Thus this form is symmetric, and \(\langle Ap,p\rangle _{1,1,\gamma }{\lt}0\) for \(p\ne 0\). In particular, \(A\) is injective on polynomials.

In the positive orthonormal semiclassical basis with parameters \((t,1,1,\gamma )\), the actual coefficient matrix of \(A\) is symmetric and tridiagonal: entry \((i,j)\) vanishes if \(j+1{\lt}i\) or \(i+1{\lt}j\). Its coefficient action is exact on finite expansions. Multiplication by the physical boundary factor \(t^{-2}x(1-x)\) has a symmetric pentadiagonal coefficient matrix: entry \((i,j)\) vanishes if \(j+2{\lt}i\) or \(i+2{\lt}j\). The mass pairing is strictly positive on nonzero polynomials.

For \(t\ne 0\), \(Ep=X(1-X)p'+(1-2X)p\), and \(A_{t,\gamma }p=L_{t,\gamma }(W_tp)\), where \(L_{t,\gamma }\) is the radial Laplacian.

For \(t{\gt}1\), real \(\gamma \), and every \(N\ge 0\), the leading \(N\times N\) Dirichlet section \(A_N\) is negative definite, meaning \(-A_N\) is positive definite. The corresponding mass section is positive definite. Both are invertible; in particular, \(A_Nv=g\) has exactly one solution for every finite right-hand side.

For \(t\ne 0\) and every real \(x\), \(W_t(x)=t^{-2}x(1-x)\).

The energy norm satisfies \(\| p\| _E^2=E(p)\). The embedding satisfies \((\iota p,\iota z)=-\langle Ap,z\rangle _{1,1,\gamma }\), and its range is dense in \(H_{t,\gamma }\).

Let \(\ell \) be any continuous linear functional on \(H_{t,\gamma }\), and let \(u\) be the unique energy solution \((u,z)_E=\ell (z)\) for every \(z\in H_{t,\gamma }\). Let \(Q_j\) denote the embedded \((t,1,1,\gamma )\) semiclassical trial polynomials. Solve the actual finite Dirichlet system

Then \(u_N\) is exactly the energy orthogonal projection of \(u\) onto \(V_N\). Consequently

This is radial convergence in a fixed angular sector when \(\gamma =m\).

The completed Cartesian gradient sectors are mutually orthogonal. Their synthesis gives a linear isometry \(G:\mathcal H_q\to \mathcal G_q\) with unconditionally convergent sector series and closed range. On physically scaled polynomial energy vectors it is exactly \(\nabla U_p\). The simultaneous map

is an isometry for the product maximum norm and has closed range. In particular, the physical body-force solution satisfies

The Cartesian-gradient error of every actual finite solve equals its energy error. The common angular/radial cutoff reconstructions converge simultaneously as physical functions and Cartesian gradient classes.

The full annular energy Hilbert sum \(\mathcal H_q\) has an injective continuous linear realization \(T:\mathcal H_q\to L^2(dA)\) given by

The series converges unconditionally in physical \(L^2\). Each physically scaled polynomial energy vector realizes exactly its boundary-weighted polynomial \(U_p\).

Let \(0{\lt}q{\lt}1\), \(t=(1-q)^{-1}\), and \(B=(1-r^2)(r^2-q)\). For a valid real angular harmonic \(H_m\) (excluding the zero sine mode), set \(L_m p=p(t(1-r^2))H_m\) and

With the unweighted annular measure \(\mu _{q,0,0}\),

Pairings between distinct real angular sectors vanish.

For \(0{\lt}q{\lt}1\) and every Cartesian polynomial \(p\), put \(u=B_qp\), where \(B_q=(1-x^2-y^2)(x^2+y^2-q)\). Then

The constant is independent of degree, angular frequency, and \(q\). Writing \(Eu=x\partial _xu+y\partial _yu\), the auxiliary identity is \(\int uEu\, dA=-\int u^2\, dA\).

For \(t{\gt}1\) and real \(\gamma \), write \(E(p)=-\langle Ap,p\rangle _{1,1,\gamma }\); finite moments ensure this pairing exists. If \(u\) is a polynomial and \(u_N\in V_N\) solves the Galerkin system with forcing \(Au\), then

Energy-orthogonal polynomial vectors satisfy the exact Pythagorean identity.

Every \(\phi \in C_c^\infty (\Omega _q)\), viewed globally, has a smooth factorisation \(\phi =B_qg\) with \(g\in C^\infty (\mathbb R^2)\). There is a sequence of real Cartesian polynomials \(p_n\) such that

In particular, the physical function/gradient pair of every such test belongs to the image of \(x\mapsto (Tx,Gx)\) on \(\mathcal H_q\).

Let \(V_N\) be the span of the first \(N\) semiclassical polynomials with parameters \((t,1,1,\gamma )\). For polynomial forcing \(f\), the square-section system is exactly

There is exactly one coefficient vector solving these equations. If the exact radial solution already belongs to \(V_N\), the finite system recovers its coefficients. This is residual orthogonality, not an assertion that the full polynomial residual vanishes for arbitrary forcing.

For every real Cartesian polynomial \(p\), there exists \(x\in \mathcal H_q\) with

Thus the completed energy image contains all boundary-weighted Cartesian polynomial graph pairs.

Let \(0{\lt}q{\lt}1\), let \(dA\) be the unweighted annular measure, and put \(B_q=(1-x^2-y^2)(x^2+y^2-q)\). For arbitrary Cartesian polynomials \(p,z\),

The Laplacian pairing on this boundary-weighted polynomial space is symmetric. More explicitly, the integrated Laplacian of a radial lift with radial polynomial \(F(s)\) is zero in every nonconstant angular sector, and is \(4\pi (F'(1)-qF'(q))\) in the constant sector.

Fix a valid real angular sector \(s=(u,m)\) and let \(\kappa _s\) be its physical energy scale. Polynomial lifting extends to a continuous linear map \(T_s:H_{t,m}\to L^2(dA)\) with

Every \(g\in L^2(dA)\) defines a body load \(b_g(v)=\langle g,T_sv\rangle _{L^2}\), of norm at most \(\sqrt{\kappa _s}\| g\| _{L^2}\). On polynomial trials this is the physical integral \(\int gU_p\, dA\). Feeding the unscaled radial solver the load \(\kappa _s^{-1}b_g\) gives a solution satisfying

The existing radial coefficient reconstructions converge after applying \(T_s\), for any continuous radial load.

For a valid sector \(s=(u,m)\), the completed map \(T_s:H_{t,m}\to L^2(dA)\) is injective. More precisely, for every completed energy vector \(x\) and radial polynomial \(p\),

Let \(\mathcal G_q=L^2(dA)\oplus ^{\ell ^2}L^2(dA)\) be the physical space of two Cartesian gradient components. Polynomial differentiation defines \(\nabla p=(\partial _xp,\partial _yp)\) in this space. For each valid angular sector \(s\), the map

extends to a linear isometry \(G_s:H_{t,m}\to \mathcal G_q\). Its two components on polynomial inputs are exactly the corresponding partial-derivative \(L^2\) classes, with this scale.

For \(g\in L^2(\Omega _q)\), define the continuous load \(L_g(v)=\langle g,v_f\rangle _{L^2(\Omega _q)}\). Its weak solution satisfies

For a physical force \(g_p\in L^2(dA_q)\), put \(g=(A_q(g_p,0))_f\). Then the pulled-back load is precisely

For \(0{\lt}q{\lt}1\), the map

is a surjective linear isometry, where the Sobolev graph carries the function/ gradient product maximum norm. Subscripts \(f\) and \(\nabla \) denote the function and gradient components. Moreover,

Thus the completed polynomial energy is exactly the zero-boundary Sobolev space, with its actual function and Cartesian gradient components.

The equality of polar and area measures induces an isometric identification \(A_q\) of their function/gradient product spaces. For a smooth test \(\phi \), its physical gradient has coordinates \([D_i\phi ]\), and \(A_q\) carries its physical function/gradient pair to its ordinary area-measure graph pair. The cutoff gradients converge in the finite Hilbert sum. Consequently,

for every Cartesian polynomial \(p\), and every completed energy vector satisfies

Let \(L\) be a continuous real linear functional on \(\mathcal H_0^1(\Omega _q)\), and put \(\ell =L\circ E_q\). There is exactly one \(u_L\in \mathcal H_0^1(\Omega _q)\) such that

It is the image under \(E_q\) of the energy Riesz solution for \(\ell \). The finite annular solves with all valid angular frequencies \(m{\lt}N\) and radial degree below \(N\), transported by \(E_q\), converge to \(u_L\) in the Sobolev graph norm as \(N\to \infty \).

For every completed annular energy vector \(x\in \mathcal H_q\), its physical function \(Tx\) and Cartesian gradient \(Gx\) satisfy

The physical test class agrees almost everywhere with \(\phi \), and pairing it with a polynomial is the ordinary integral of their product.

Every lifted polynomial \(B_qL_mp\) vanishes at each real point satisfying \(x^2+y^2=q\) or \(x^2+y^2=1\).

For \(0{\lt}q{\lt}1\), the scale \(\kappa _m\) is positive for every nonzero angular mode.

Let \(0{\lt}q{\lt}1\), \(a,b{\gt}-1\), and let \(p_n\) be the semiclassical basis polynomial with parameters \((t(q),a,b,m)\). For either the cosine or sine angular harmonic \(H_m\), its lift is

which is the annular mode with radial index \(n\) and angular harmonic \(H_m\).

For \(q\ne 1\), \(t(q)\ne 0\) and \(t(q)(1-q)=1\). If \(0{\lt}q{\lt}1\), then \(t(q){\gt}1\).

For the initial coordinate projections on real or complex \(\ell ^2\), write \(S_N(A)=I-P_N+P_NAP_N\). Then \(\| S_N(A)-S_N(B)\| \le \| A-B\| \). If \(B_N\to A\) in operator norm, then \(S_N(B_N)u\to Au\) for every \(u\). An eventual inverse bound \(C\) for \(S_N(A)\) transfers to an eventual bound \(2C\) for \(S_N(B_N)\). Under that stability hypothesis and \(g_N\to Au\), \(S_N(B_N)^{-1}P_Ng_N\to u\). At each invertible approximate section,

Let \(P_n\) be contractions on a real or complex Banach space, converging strongly to \(I\), let \(K\) be compact, and suppose \(L_n\to K\) in operator norm. Then \(B_n=I+P_nL_n\to I+K\) in operator norm. If \(I+K\) is invertible, then eventually \(B_n\) is invertible and \(\| B_n^{-1}\| \le \| (I+K)^{-1}\| +1\). For any contraction \(P\) and bounded \(K,L\), whenever \(B=I+PL\) is invertible,

The fixed-section inverse \((I+PL)^{-1}\) is defined for operators \(L\) such that \(I+PL\) is invertible. The indices \(n\) for which \(B_n\) is invertible include every sufficiently large integer. The error identity and estimate do not require compactness.

For every \(u\in \ell ^2\), \(B_nu=u-P_nA_{K_n}u\).

If \(M\) has bandwidths \((\ell ,u)\), each row sum \(\sum _j a_j\sigma (M_{ij})\) is finite for every sequence \(a\), and \(\beta _i=\sum _{j{\lt}i+u+1}a_j\sigma (M_{ij})\) for every sequence \(a\). On finitely supported sequences embedded from \(R\), this agrees exactly with the existing finite coefficient map. Neither coordinate identity needs an analytic convergence hypothesis.

If a matrix has bandwidths \((\ell ,u)\), \(\ell \le \ell '\) and \(u\le u'\), then it has bandwidths \((\ell ',u')\).

For each node, \(\omega _i\ne 0\) and \(z-v_i\ne 0\).

Let \(\rho {\gt}1\) and suppose complex polynomials \(p_n\) of degree at most \(n\) satisfy \(|f(x)-p_n(x)|\le C\rho ^{-n}\) for every \(n\ge 0\) and \(x\in [-1,1]\). Then \(f\) has an analytic extension to the open ellipse \(E_\rho \). Continuity of \(f\) need not be assumed.

Reference: [ATAP, Theorem 8.3].

The complex best-error infimum \(E_N(f)\) on \([-1,1]\) is attained by a polynomial of degree at most \(N\). For continuous real data on any compact interval \([a,b]\), a real degree-at-most-\(N\) minimizer exists as well. Existence alone does not require \(a{\lt}b\).

Reference: [ATAP, Theorem 10.1].

For \(f\in C([-1,1],\mathbb C)\) and \(\rho {\gt}1\), if \(E_n(f)\le C\rho ^{-n}\) for every \(n\ge 0\), then \(f\) has an analytic extension to \(E_\rho \).

Reference: [ATAP, Theorem 8.3].

The competitor-error set is nonempty and bounded below by zero. Thus \(0\le E_N(f)\le \| f-p\| _\infty \) for every polynomial of degree at most \(N\).

Reference: [ATAP, Chapter 16].

Both real and complex analytic Jacobi and Gegenbauer expansions satisfy weighted L2 error bounds and converge uniformly on \([-1,1]\). The complex bounds have exactly the constants of the real-valued theorems above, with \(|F-S_Nf|\) replacing \(|\operatorname {Re}F-S_Nf|\). They hold for every \(\alpha ,\beta {\gt}-1\) and every \(\lambda {\gt}-1/2\), \(\lambda \ne 0\), respectively, under the open-ellipse hypotheses \(\rho {\gt}1\), analyticity and \(|F|\le M\). The uniform polynomial factors are explicit and nonsharp; no boundary regularity on the ellipse is assumed.

References: [DLMF, 18.18] and [ATAP, Chapter 8].

The bordered equation with right-hand side \((\beta ,\operatorname {repr}_c f)\) is equivalent to \(Bp=\beta \) and \(Lp=f\) for the reconstructed finite expansion. This holds for module bases over a commutative semiring. In particular it applies to polynomial differential operators and their boundary functionals. It asserts neither invertibility nor convergence of a truncated ODE solver.

Reference: [OT13, Sections 2.4 and 3.2].

The full-constant Chebyshev multiplication matrix extends to a bounded operator \(J\) on complex \(\ell ^2\). For a real polynomial \(p\), \(p(J)\) agrees with multiplication by \(p\) on finite polynomial coordinates. The bounded map \(S\) with \((Sa)_0=a_0-a_2/2\) and \((Sa)_j=(a_j-a_{j+2})/2\) for \(j{\gt}0\) agrees with conversion from \(T\) to \(U=C^{(1)}\).

For any measure, an almost-everywhere bound \(|k|\le M\), \(M\ge 0\), gives bounded multiplication on real \(L^2\) with norm at most \(M\). Let \(\mu ,\nu \) be finite measures and let \(k\) be an almost-everywhere strongly measurable real kernel with \(|k|\le M\) on their product. There is a bounded operator \(T:L^2(\nu )\to L^2(\mu )\) such that

Every displayed pairing is absolutely integrable. The kernel may have either sign.

If \(f\) has bounded variation on \([-1,1]\), then for \(n\ge 1\),

In particular, if the variation is at most \(V\ge 0\), then \(|a_n(f)|\le 2V/(\pi n)\). Continuity is not assumed.

Reference: [ATAP, Theorem 7.1], case \(\nu =0\).

Let \(f,g:\mathbb R\to \mathbb R\) have bounded variation on \(\mathbb R\), with \(g\) continuous. Suppose \(h\in L^1(\mathbb R)\) and \(g(v)-g(u)=\int _u^v h(x)\, dx\) for every \(u\le v\). Then \(dg=h\, dx\), and, for \(a\le b\),

In particular, if \(g(a)=g(b)=0\) and \(|g|\le C\) on \([a,b]\), with \(C\ge 0\), then

The density identification also holds for a continuous, locally absolutely continuous function \(g\) of bounded variation and any global integrable derivative representative \(h\in \mathcal D_{dx}(g)\). Jumps of \(f\) are allowed.

Reference: the integration-by-parts step in [ATAP, Chapter 7].

Let \(a\le b\), let \(f\) have bounded variation on \([a,b]\), and let \(g:\mathbb R\to \mathbb R\) be Lipschitz. Suppose \(g'=h\) on \((a,b)\), \(h\) is integrable on \([a,b]\), \(g(a)=g(b)=0\), and \(|g|\le C\) there, with \(C\ge 0\). Then

Reference: the bounded-variation integration step in [ATAP, Chapter 7].

For a continuously differentiable convolution kernel \((g,s)\) as in Definition 6.2.106, \(g\) is differentiable at every \(t\in \mathbb {R}\) and \(g'(t)=s(t)\).

For \(\beta {\gt}1\), the continuously differentiable power kernel has value \(k_\beta (t)=|t|^\beta \) and slope \(s_\beta (t)=\beta |t|^{\beta -1}\operatorname {sign}(t)\) for every \(t\in \mathbb {R}\).

For any ordered basis of a finite polynomial subspace and any measure with finite moments and positive definite polynomial pairing, choose the unique positive upper triangular Gram factor. Its offset support can be checked using any positive upper triangular factor of the same Gram matrix. No separated target orthogonal system is assumed.

Let \(A\ge 0\), \(A+B{\gt}0\), \(A+C\ge 0\), \(C\ne 0\) and \(r\ge 2\). For all sufficiently small positive \(\delta \), the actual canonical Gram factor for the quadratically modified deformed measure has a nonzero first-to-last top-even-sector entry. No \(A+B=1\) normalization remains. The conclusion holds for any complete ordered source basis with the stated sector enumeration; the intermediate measure satisfies the required admissibility conditions.

For the separated polynomial disk modification of Theorem 5.8.5, the coefficient from source \((n,k)\) to target \((m,j)\) can be nonzero only if

If \(\mu \) and \(h\) are also even and \(\deg h\le 2a\), this sharpens to

The unchanged-index coefficient is positive. These are support containments, not assertions that every permitted entry is nonzero.

Assume also that the inner measure \(\nu \) is supported in \([-1,1]\). The Cartesian polynomials are orthonormal and form a basis of every complete finite total-degree space. Their measure is positive definite on the polynomial algebra. No Jacobi formula is assumed for the one-dimensional factors.

Let \(\mu \) and \(\nu \) be measures on \(\mathbb R\) with all polynomial moments finite and positive definite polynomial pairings. Suppose \(\mu \) assigns zero mass to \(\mathbb R\setminus [-1,1]\) and \(\nu \) is invariant under \(x\mapsto -x\). Then there exists a Cartesian disk system for the restriction of \(\mu \) to \([-1,1]\) and the inner measure \(\nu \).

A Cartesian disk system supplies all moments of both one-dimensional factors. If the outer measure is supported almost everywhere on \([-1,1]\), its Cartesian measure has all polynomial moments. More generally, finite moments of the two factors and this outer support condition suffice; no support condition on the inner factor is needed for this conclusion.

Assume the source and target factor measures are supported in \([-1,1]\). On every complete degree space, the connection matrix consists of the mixed integral coefficients of Theorem 5.8.6 and factors the modified Gram matrix as \(R^\top R\). The support bounds of Theorem 5.8.6 therefore apply to these finite matrices uniformly in the truncation degree.

For factor measures \(\mu ,\nu \) on \(\mathbb R\) with \(\nu \) s-finite, evenness of either factor makes \(\mu \times \nu \) invariant under reflection of its corresponding coordinate. For even admissible outer and inner factor measures, the Cartesian polynomial indexed by \((n,k)\) has reflection signs \((-1)^{n-k}\) and \((-1)^k\). A diagonal quadratic density preserves these reflections. Its Gram matrix and canonical positive upper triangular factor have exact zeros between different parity pairs \(((n-k)\bmod 2,k\bmod 2)\). The canonical-factor statement assumes a nonnegative density and the moments and positivity needed to define that factor.

Write \(A(x)=1-x^2\). Let \(h,g\) be nonzero nonnegative polynomial densities for \(\mu ,\nu \), with \(g\) even and \(\deg g\le 2\ell \). The actual bivariate polynomial

changes the Cartesian measure to the pushforward of \(A^\ell h\, d\mu \otimes g\, d\nu \), provided the inner measure \(\nu \) is supported in \([-1,1]\). Under this support assumption the identity holds against all continuous real-valued test functions on \(\mathbb R^2\), in particular for polynomial pairings. The modified positive-leading orthonormal Cartesian system exists even when the density vanishes in the interior.

Suppose \(\sum _{k\ge 0}|a_k|{\lt}\infty \) and \(f(x)=\sum _{k\ge 0}a_kT_k(x)\) on \([-1,1]\). Integration and quadrature commute with this series. If \(N\ge 1\) and \(2m\le N\), then the error has the absolutely convergent expansion

If also \(m\ge 1\), its absolute value is at most \(\frac{32}{15}\sum _{j\ge 0}|a_{2(j+m+1)}|\).

Reference: [ATAP, proof of Theorem 19.3].

For \(N\ge 1\), the folded index \(A_N(k)\) has the same parity as \(k\), and \(Q_NT_k=m_{A_N(k)}\). In particular, \(Q_NT_{2k+1}=0\) for every \(k\ge 0\), even beyond the polynomial exactness degree.

Reference: [ATAP, Chapters 4 and 19].

For \(N\ge 1\) and every \(k\ge 0\), \(|Q_NT_k|\le 2\). For \(k\ge 4\), \(|m_k|\le 2/15\), and hence \(|m_k-Q_NT_k|\le 32/15\).

Reference: [ATAP, proof of Theorem 19.3].

The odd moments vanish, and every positive-degree moment is nonpositive. For every \(m\ge 0\),

In particular, every finite absolute moment sum is strictly below \(3\). For \(N\ge 1\), the endpoint-mode sum satisfies \(\sum _{k=0}^Nb_km_k=2\sum _{k=0}^Nm_k-2-m_N\).

Reference: [ATAP, equation (19.6)].

Let \(\rho {\gt}1\), let \(F\) be analytic on the open Bernstein ellipse \(E_\rho \), and suppose \(|F(z)|\le M\) there. For \(N\ge 2\),

If \(N\) is even, replace \(\rho ^{N-1}\) by \(\rho ^N\). More generally, any \(1\le m\) with \(2m\le N\) gives the denominator \(15\rho ^{2m}(\rho ^2-1)\). The underlying series bound uses \(|a_k|\le Cq^k\), \(0\le q{\lt}1\), to give \(\frac{32}{15}Cq^{2(m+1)}/(1-q^2)\).

Reference: [ATAP, Theorem 19.3, equation (19.7)].

Let \(f\) agree with \(S_a\) on \([-1,1]\), and let \(A_n\) be the primitive of \(T_n\) anchored at \(-1\). If \(\sum _n|a_n|{\lt}\infty \), \(\sum _{n{\lt}N}a_n A_n(x)\) converges to \(\int _{-1}^x f(t)\, dt\) uniformly on \([-1,1]\), with error at most \(2\sum _{n\ge N}|a_n|\) after the first \(N\) modes. If \(|a_n|\le C\rho ^{-n}\), \(\rho {\gt}1\), this is at most \(2C\rho ^{-N}/(1-\rho ^{-1})\).

Reference: [ATAP, Chapter 21].

Write \(A_n\) for the existing primitive of \(T_n\) anchored by \(A_n(-1)=0\). For \(x\in [-1,1]\), \(A_n(x)=\int _{-1}^xT_n(t)\, dt\) and \(|A_n(x)|\le 2\). If \(\sum _n|a_n|{\lt}\infty \) and \(f\) agrees with \(S_a\) on \([-1,1]\), then \(f\) is integrable on every \([-1,x]\), \(x\in [-1,1]\), and

Reference: [ATAP, Chapters 19 and 21].

For a finite expansion \(p=\sum _jv_jT_j\), the four boundary rows are

All sums are over the finite support. More generally, any real linear functional acts by the row of its values on the basis polynomials.

Reference: [OT13, Sections 2.4 and 3.2].

For any commutative ring and any \(x\), use \(a_k=2x\), \(b_k=-1\), \(p_0=1\) and \(p_1=x\). The resulting executable evaluator returns \(\sum _{i=0}^{n-1}c_iT_i(x)\). The coefficient \(c_0\) is the full constant term; unlike [DLMF, 3.11(ii)], it is not halved.

References: [Cle55]; [DLMF, 3.11(ii)].

For every nonzero real polynomial \(p\), define its colleague matrix by applying the comrade construction in the \(T\) basis to \(p/\operatorname {lc}(p)\). Its size is \(\deg p\) and its characteristic polynomial is \(p/\operatorname {lc}(p)\). The entry formula is the final-column correction above, in the full-constant Chebyshev convention. Its real spectrum is the real roots of \(p\); after embedding its entries in \(\mathbb C\), its complex spectrum is all complex roots of \(p\). Repeated roots are allowed.

References: [Goo61]; [ATAP, Theorem 18.1].

For \(w\ne 0\) and \(n\ge 0\), \(T_n(J(w))=(w^n+w^{-n})/2\). Consequently \(|T_n(z)|\le r^n\) for \(r{\gt}1\) and \(z\in E_r\).

Reference: [ATAP, Exercise 8.8].

Over a commutative rational algebra, \(C_n^{(1)}=U_n\) and \(2T_{n+2}=C_{n+2}^{(1)}-C_n^{(1)}\). For \(k,n\ge 0\),

Over a characteristic-zero field, \(D^kT_n=0\) when \(n{\lt}k\). These identities retain the full constant coefficient convention.

References: [DLMF, 18.7.4 and 18.9.9]; [OT13, §2].

For real or complex \(x\) and \(n\in \mathbb {N}\),

where \({}_2F_1\) is the Gauss series of Definition 1.2.6. No hypothesis is needed.

Reference: [DLMF, 18.5.11_2 and 18.5.11_4].

For real or complex \(x\ne 0\) and \(n\in \mathbb {N}\),

and for \(n\ge 1\),

where \({}_2F_1\) is the Gauss series of Definition 1.2.6.

Reference: [DLMF, 18.5.11_1 and 18.5.11_3].

Let \(R\) be a commutative \(\mathbb {Q}\)-algebra and let \(n\in \mathbb {N}\). Then

Reference: [DLMF, 18.7.3–4].

Write \(p=\sum _m a_mT_m\) with a full constant coefficient, and let \(M\) represent multiplication by \(p\) in the \(T\) basis. Then

In particular the positive diagonal is \(a_0+a_{2i}/2\). These are the Toeplitz-plus-Hankel entries with the exceptional row, column and constant coefficient accounted for.

References: [OT13, Section 2.2, equation (2.7)]; [DLMF, 18.18.21].

The polynomials \(T_n\) and \(U_n\) satisfy

These hold over every commutative ring and for every integer \(n\).

Reference: [DLMF, Tables 18.8.1 and 18.6.1].

For \(k\in \mathbb {N}\) and sequences \(a,b\) such that \((a_mb_n)_{|m-n|=k}\) is summable, the product coefficient is

The first sum is finite. The formula includes \(k=0\).

For \(k\in \mathbb {N}\), the set \(\{ (m,n)\in \mathbb {N}^2:m+n=k\} \) is finite, so \(\sum _{m+n=k}a_mb_n\) exists for any sequences \(a,b\). If both sequences are absolutely summable, then \((a_mb_n)_{|m-n|=k}\) is summable for every \(k\), and \((a,b)\) belongs to the product-coefficient domain of Definition 2.7.29.

Over a commutative ring, \(T_0=1\), \(T_1=X\) and \(T_{k+2}=2XT_{k+1}-T_k\) for every integer \(k\).

Reference: [DLMF, Table 18.9.1].

Assume \(\rho {\gt}1\) and \(|a_n|\le C\rho ^{-n}\) for every \(n\). Let \(1{\lt}r{\lt}\rho \) and \(\delta {\gt}0\). If \(z\in E_r\) and the radius-\(\delta \) circle about \(z\) lies in \(E_r\), then

Every compact \(K\subset E_r\) admits one \(\delta {\gt}0\) for which this bound holds for all \(z\in K\) and all \(N\ge 0\), in particular for the real interval.

Reference: [ATAP, Theorem 21.1].

For \(\rho {\gt}1\), \(|a_n|\le C\rho ^{-n}\) and \(k\ge 0\), the function \(S_a|_{E_\rho }\) admits an order-\(k\) derivative chain on \(E_\rho \). The chain starts at \(S_a|_{E_\rho }\) and ends at \(D_{E_\rho }^kS_a\).

If \(|a_n|\le C\rho ^{-n}\), \(\rho {\gt}1\), then on \(E_\rho \),

For \(1{\lt}r{\lt}\rho \), derivatives of partial sums converge uniformly on every compact subset of \(E_r\).

References: [ATAP, Chapter 21]; [DLMF, 18.9.21].

If \(\rho {\gt}1\) and \(|a_n|\le C\rho ^{-n}\), the complex Chebyshev sum \(S_a=\sum _n a_nT_n\) can be differentiated termwise to every order on \(E_\rho \). For \(k\ge 0\) its positive-order derivative is

Reference: [OT13, Section 3].

Assume \(\rho {\gt}1\) and \(|a_n|\le C\rho ^{-n}\) for every \(n\); choose \(1{\lt}r{\lt}\rho \), \(z\in E_r\), and \(\delta {\gt}0\) with its circle about \(z\) contained in \(E_r\). Then

For every compact \(K\subset E_r\) a single positive \(\delta \) works for all \(z\in K\), all derivative orders \(k\), and all cutoffs \(N\).

Reference: [ATAP, Theorem 21.1].

If \(\sum |a_n|\) and \(\sum |b_n|\) are finite, then \(c\) is absolutely summable,

Moreover any \(f\in \mathcal I_k^{\mathbb {C}}\) agreeing with \(S_aS_b\) on \([-1,1]\) has coefficient \(c_k\).

References: [DLMF, 18.18.21]; [OT13, Section 2.2].

If \(\sum _n|a_n|{\lt}\infty \) and \(\sum _n|b_n|{\lt}\infty \), with \(c\) defined by the product-coefficient formula of Definition 2.7.21, uniformly for \(x\in [-1,1]\),

The right side tends to zero.

Reference: [ATAP, Chapter 3].

For \(\rho {\gt}1\), \(|a_n|\le C\rho ^{-n}\) and \(1{\lt}r{\lt}\rho \), the residual and boundary inequalities of Theorem 4.3.17 hold for the Chebyshev truncations with

provided the evaluation points and the specified positive-radius circles lie in \(E_r\).