On Rapid Computation of Expansions in Ultraspherical Polynomials

Authors:
María José Cantero;Arieh Iserles
Affiliations:
mjcante@unizar.es;ai@damtp.cam.ac.uk
Venue:
SIAM Journal on Numerical Analysis
Year:
2012

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Abstract

We present an ${\cal O}(N\log_2N)$ algorithm for the computation of the first $N$ coefficients in the expansion of an analytic function in ultraspherical polynomials. We first represent expansion coefficients as an infinite linear combination of derivatives and then as an integral transform with a hypergeometric kernel along the boundary of a Bernstein ellipse. Following a transformation of the kernel, we approximate the coefficients to arbitrary accuracy using the discrete Fourier transform.