Computing with Expansions in Gegenbauer Polynomials

In this work, we develop fast algorithms for computations involving finite expansions in Gegenbauer polynomials. We describe a method to convert a linear combination of Gegenbauer polynomials up to degree n into a representation in a different family of Gegenbauer polynomials with generally O(n log(1/ε)) arithmetic operations where ε is a prescribed accuracy. Special cases where source or target polynomials are the Chebyshev polynomials of first kind are particularly important. In combination with (nonequispaced) discrete cosine transforms, we obtain efficient methods for the evaluation of an expansion at prescribed nodes and for the projection onto Gegenbauer polynomials from given function values, respectively. AMS Subject Classification: 42C20, 65T50, 65Y20