Approximation results for orthogonal polynomials in Sobolev spaces

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