On rapid computation of expansions in ultraspherical polynomials
Recently, \textit{A. Iserles} [Numer. Math. 117, No. 3, 529--553 (2011; Zbl 1211.33001)] has proposed a fast algorithm for the computation of the coefficients in an expansion of an analytic function into Legendre polynomials. In the paper under review, the authors generalize this method. They present an \({\mathcal O}(N\, \log_2 N)\) algorithm for the computation of the first \(N\) coefficients \(f_n\) in the expansion of an analytic function \(f\) into ultraspherical or Jacobi polynomials \(P_n^{(\alpha,\alpha)}\), where \(\alpha > -1\). These polynomials are orthogonal in \([-1,\,1]\) with respect to the weight function \((1-x^2)^{\alpha}\). First the coefficients \(f_n\) are represented as infinite linear combinations of the derivatives \(f^{(k)}(0)\) \((k=0,\,1,\ldots)\) and then as an integral transform with a hypergeometric kernel using the Cauchy integral theorem. The coefficients \(f_n\) are approximately computed with arbitrary accuracy using fast Fourier transform. Numerical experiments are presented, too.
scientific article; zbMATH DE number 6030216