Some Error Estimates for Periodic Interpolation on Full and Sparse Grids Curves and Surfaces with Applications in Cagd 355

We give a uniied approach to error estimates for periodic interpolation on full and sparse grids in certain Sobolev spaces. We imposèperiodic' Strang{Fix conditions on the underlying functions in order to obtain error bounds with explicit constants. x1. Introduction The approximation and interpolation of bivariate periodic functions have been studied for some time. While periodic interpolation by translates on full grids is well investigated for several function spaces, especially Sobolev spaces H 2 (T T 2) (see e.g. 2,3,8]), error estimates for periodic interpolation on sparse grids are studied for functions from Korobov spaces E (T T 2) (see e.g. 1,4]). In this paper, we give a uniied approach to error estimates for periodic interpolation on full and sparse grids for functions from the usual Sobolev spaces H 2 (T T 2) and Sobolev spaces S ;; 2;2 H(T T 2) with dominating mixed smoothness. We consider tensor product interpolation and j-th order blending interpolation. Applying the concept of`periodic' Strang{Fix conditions on the underlying univariate fundamental interpolant, we are able to give error estimates with explicit constants. x2. Univariate Periodic Interpolation by Translates Let d 2 IN be xed. Put d j := 2 j d; j 2 IN 0. We consider continuous 2-periodic functions j deened as fundamental interpolants on the equidistant All rights of reproduction in any form reserved.