Blending Fourier and Chebyshev interpolation

AbstractOver the rectangle Ω = (−1. 1) × (−π, π) of R2, 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 xj = cos πjM, j = 0,…, M, and the equispaced points yl = (lN − 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 L2-norm of the interpolation error is needed. In this paper, it is shown that this error decays like N−r + Ms provided the interpolation function belongs to the non-isotropic Sobolev space Hr,s(Ω)