Approximation Results for Orthogonal Polynomials in Sobolev Spaces

Abstract

We analyze the approximation properties of some interpolation operators and some <!-- MATH $L_\omega ^2$ --> -orthogonal projection operators related to systems of polynomials which are orthonormal with respect to a weight function <!-- MATH $\omega ({x_1}, \ldots ,{x_d})$ --> , <!-- MATH $d \geqslant 1$ --> . The error estimates for the Legendre system and the Chebyshev system of the first kind are given in the norms of the Sobolev spaces <!-- MATH $H_\omega ^s$ --> . These results are useful in the numerical analysis of the approximation of partial differential equations by spectral methods.