The expansion in ultraspherical polynomials: a simple procedure for the fast computation of the ultraspherical coefficients

Details The expansion in ultraspherical polynomials: a simple procedure for the fast computation of the ultraspherical coefficients The expansion in ultraspherical polynomials: a simple procedure for the fast computation of the ultraspherical coefficients

2011-06-23

arxiv.org/abs/1106.4718v2

Mathematics

Numerical Analysis

7 pages, 1 figure, references added, typos corrected

Scientific paper

We present a simple and fast algorithm for the computation of the coefficients of the expansion of a function f(cos u) in ultraspherical (Gegenbauer) polynomials. We prove that these coefficients coincide with the Fourier coefficients of an Abel-type transform of the function f(cos u). This allows us to fully exploit the computational efficiency of the Fast Fourier Transform, computing the first N ultraspherical coefficients in just O (N log_2 N) operations.

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