6.6 Roadmap and open targets

The results in this chapter are targets for future work. Formalised subcases appear in the relevant subject chapters.

6.6.1 Rational modifications and infinite factorizations

Extend polynomial measure conversion to rational modifications and infinite-dimensional factorizations, with explicit operator domains, invertibility assumptions and finite-section convergence statements.

Reference: [GOS25, Section 2].

6.6.2 Proriol families on triangles and simplices

Develop Proriol families on triangles and their simplex generalizations, including orthogonality, normalizations and structured polynomial operators.

6.6.3 Dunkl–Xu disk polynomials

Develop the Cartesian Koornwinder construction of Dunkl–Xu disk polynomials, including orthogonality, normalization, coordinate-multiplication matrices and weight raising/lowering. Record the basis ordering explicitly and distinguish this construction from the Zernike disk bases when describing operator structure.

Reference: [DX14].

6.6.4 Higher-dimensional measure modification

Build on the multivariate Cholesky conversion foundation to study the resulting block structure and coordinate-multiplication operators. Distinguish sparsity of modified Gram matrices from sparsity of their connection factors. Specific results will receive separate statements and hypotheses as this theory develops.

6.6.5 One-dimensional power-law integral operators

Develop the weighted operator

on \(f(y)=(1-y^2)^{\lambda -1/2}C_n^{(\lambda )}(y)\). Integrability, the left/right decomposition, initial modes, recurrence and the exact compatible coefficient operator are proved above. Continuous amplitudes now have polynomial approximation with locally uniform output convergence, including a common weight for two incompatible powers. Closed-interval stability now has explicit combined-exponent hypotheses, with uniform mass bounds throughout \(\alpha {\gt}-1\), \(\lambda {\gt}-1/2\), \(\alpha +\lambda {\gt}-1/2\). These bounds give a common Gegenbauer \(L^2\) operator and strong finite coefficient-section convergence. The common-space \(L^2\) boundedness threshold is now classified exactly: \(\alpha +\lambda \ge -1/2\), including the critical line by a weighted Schur test. Normalized coefficient-section transport now covers the full boundedness range, using Chebyshev \(T\) at \(\lambda =0\) and Gegenbauer elsewhere. Sufficient two-weight bounds and their coefficient sections are now proved; a sharp two-weight classification remains a later target. In the common-weight space, self-adjointness now holds throughout the boundedness range. In the strict range, compactness and operator-norm section convergence give eventually stable, convergent second-kind solves under an explicit invertibility hypothesis. Compactness is now classified sharply: the critical line is noncompact. Small coupling gives stable, convergent finite solves at every resolution even on that line, and full residuals give error certificates. Critical-line solves under invertibility alone, sharp two-weight results and first-kind inversion remain separate. Keep exact compatible sparsity separate from these approximation statements. This target concerns the integral operator, not equilibrium minimizers.

References: [GCO22, Section 2]; [Gut22, Sections 3.4–3.5] (operator construction only).