On rapid computation of expansions in ultraspherical polynomials
type: Item
instance of: scholarly article
zbMATH Open document ID: 1241.42023
DOI: 10.1137/110829568
author: María-José Cantero
author: Arieh Iserles
published in: SIAM Journal on Numerical Analysis
publication date: 2012-05-04T00:00:00Z
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 65T50
zbMATH DE Number: 6030216
zbMATH Keywords: orthogonal expansion
zbMATH Keywords: ultraspherical polynomials
zbMATH Keywords: Jacobi polynomials
zbMATH Keywords: hypergeometric kernel
zbMATH Keywords: rapid computation
zbMATH Keywords: coefficients in orthogonal expansion
zbMATH Keywords: fast Fourier transform
MaRDI profile type: Publication
OpenAlex ID: W2062615235
title: On rapid computation of expansions in ultraspherical polynomials
review text: 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.
reviewed by: Manfred Tasche
description: scientific article; zbMATH DE number 6030216