Generalized sparse grid interpolation based on the fast discrete Fourier transform

Abstract

In [9], an algorithm for trigonometric interpolation involving only so-called <em>standard information</em> of multivariate functions on generalized sparse grids has been suggested and a study on its application for the interpolation of functions in periodic Sobolev spaces of dominating mixed smoothness has been presented. In this complementary paper, we now give a slight modification of the proofs, which yields an extension from the pairing (<em>&hamilt;^s</em>, <em>&hamilt;^t_mix</em>) to the more general pairing (<em>&hamilt;^s</em>, <em>&hamilt;^t,r_mix</em>) and which in addition results in an improved estimate for the interpolation error. The improved (constructive) upper bound is in particular consistent with the lower bound for sampling on regular sparse grids with <em>r</em> = 0 and <em>s</em> = 0 given in [4, 5].