Approximation results for orthogonal polynomials in Sobolev spaces

Abstract

We analyze the approximation properties of some interpolation operators and some L ω 2 L_\omega ^2 -orthogonal projection operators related to systems of polynomials which are orthonormal with respect to a weight function ω ( x 1 , … , x d ) \omega ({x_1}, \ldots ,{x_d}) , d ⩾ 1 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 H ω s H_\omega ^s . These results are useful in the numerical analysis of the approximation of partial differential equations by spectral methods.