On expansions in orthogonal polynomials

Authors:
María José Cantero;Arieh Iserles
Affiliations:
Departamento de Matemática Aplicada, Centro Politécnico Superior de Ingenieros, Universidad de Zaragoza, Zaragoza, Spain;Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Cambridge, UK
Venue:
Advances in Computational Mathematics
Year:
2013

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Abstract

A recently introduced fast algorithm for the computation of the first N terms in an expansion of an analytic function into ultraspherical polynomials consists of three steps: Firstly, each expansion coefficient is represented as a linear combination of derivatives; secondly, it is represented, using the Cauchy integral formula, as a contour integral of the function multiplied by a kernel; finally, the integrand is transformed to accelerate the convergence of the Taylor expansion of the kernel, allowing for rapid computation using Fast Fourier Transform. In the current paper we demonstrate that the first two steps remain valid in the general setting of orthogonal polynomials on the real line with finite support, orthogonal polynomials on the unit circle and Laurent orthogonal polynomials on the unit circle.