On the optimal estimates and comparison of Gegenbauer expansion coefficients (Q2810562)

Expansions of given functions in Gegenbauer polynomials are of central importance in approximation theory. Especially, the asymptotic behaviour of the resulting expansion coefficients, e.g. for the expansion in Legendre coefficients is highly relevant, and in this paper, the precise growth rate of the expansion coefficients is identified (that is, optimal estimates are given). The principal tool is a new form of the contour integral representation of the said Gegenbauer coefficients. Error estimates for truncated Gegenbauer expansions are provided too, as is for example a comparison of the aforementioned Legrendre coefficients and Chebyshev coefficients.