On Error Bounds for Orthogonal Polynomial Expansions and Gauss-Type Quadrature

Abstract

This paper presents new asymptotics on the coefficients of $f(x)$ expanded in forms of ultraspherical or Jacobi polynomial series, which imply the rate of the decay of the coefficients and derive the truncated error bounds. Moreover, we show that the Chebyshev coefficient $\alpha_n$ decays a factor of $\sqrt{n}$ faster than the Legendre coefficient $u_n$. By using the Chebyshev expansion for $f(x)$ of finite regularity, we give new error bounds for a Gauss-type quadrature and present the aliasing error between the coefficients, which shows the equal accuracy of these quadratures for nonanalytic functions.