ON THE OPTIMAL ESTIMATES AND COMPARISON OF GEGENBAUER EXPANSION COEFFICIENTS.

In this paper, we study optimal estimates and comparison of the coefficients in the Gegenbauer series expansion. We propose aalternative 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 behavior of the ratio of the $n$th Legendre coefficient to the $n$th Chebyshev coefficient is given, which provides us an illuminating insight for the comparison of spectral methods based on Legendre and Chebyshev expansions. [ABSTRACT FROM AUTHOR]