Chebyshev expansions on \([-1,1]\), from orthogonality to the convergence theory, interpolation, calculus, quadrature and rootfinding behind Chebfun.
Chebyshev polynomials, their weights and orthogonality relations.
Defines Chebyshev coefficients and proves that Chebyshev expansions are complete in the weighted \(L^2\) space.
Turns finite smoothness into algebraic coefficient decay and uniform truncation error bounds.
Interpolation at Chebyshev points: discrete orthogonality, aliasing, the barycentric formula and logarithmic Lebesgue constants.
Analyticity in a Bernstein ellipse is equivalent to geometric coefficient decay, which gives geometric convergence.
Existence, equioscillation and uniqueness of best uniform approximations, and near-best Chebyshev interpolation.
Clenshaw evaluation, differentiation, integration and multiplication carried out exactly on Chebyshev coefficients.
Clenshaw–Curtis quadrature: polynomial exactness, positive weights and geometric error bounds.
Roots of a Chebyshev expansion as the eigenvalues of its colleague matrix.