Blending Fourier and Chebyshev interpolation

Abstract

Over the rectangle Ω = (−1. 1) × (−π, π) of R 2, interpolation involving algebraic polynomials of degree M in the x direction, and trigonometric polynomials of degree N in the y direction is analyzed. The interpolation nodes are Cartesian products of the Chebyshev points x j = cos πj M , j = 0,…, M , and the equispaced points y l = ( l N − 1)π, l = 0,…,2N − 1 . This interpolation process is the basis of those spectral collocation methods using Fourier and Chebyshev expansions at the same time. For the convergence analysis of these methods, an estimate of the L 2 -norm of the interpolation error is needed. In this paper, it is shown that this error decays like N − r + M s provided the interpolation function belongs to the non-isotropic Sobolev space H r,s(Ω) .