On the optimal estimates and comparison of Gegenbauer expansion coefficients

TL;DR

This paper derives optimal estimates for Gegenbauer series coefficients, including Legendre coefficients, and compares the decay rates of Chebyshev and Legendre coefficients to inform spectral method choices.

Contribution

It introduces an alternative contour integral representation for Gegenbauer coefficients and uses it to establish optimal bounds and compare decay rates of spectral coefficients.

Findings

Optimal estimates for Gegenbauer and Legendre coefficients

Bounds for truncated Gegenbauer series

Asymptotic ratio of Legendre to Chebyshev coefficients

Abstract

In this paper, we study optimal estimates and comparison of the coefficients in the Gegenbauer series expansion. We propose an alternative derivation of the contour integral representation of the Gegenbauer expansion coefficients which was recently derived by Cantero and Iserles [SIAM J. Numer. Anal., 50 (2012), pp.307-327]. With this representation, we show that optimal estimates for the Gegenbauer expansion coefficients can be derived, which in particular includes Legendre coefficients as a special case. Further, we apply these estimates to establish some rigorous and computable bounds for the truncated Gegenbauer series. In addition, we compare the decay rates of the Chebyshev and Legendre coefficients. For functions whose singularity is outside or at the endpoints of the expansion interval, asymptotic behaviour of the ratio of the nth Legendre coefficient to the nth Chebyshev coefficient is given, which provides us an illuminating insight for the comparison of the spectral methods based on Legendre and Chebyshev expansions.