A Tool for Approximation in Bivariate Periodic Sobolev Spaces

We give a characterization of Sobolev spaces of bivariate periodic functions with dominating smoothness properties in terms of Sobolev spaces of univariate functions. The mixed Sobolev norm is proved to be a uniform crossnorm. This property can be used as a powerful tool in approximation theory. x1. Introduction Beside the approximation of functions from the usual isotropic periodic Sobo-lev spaces the approximation of bivariate periodic functions with dominating mixed smoothness properties attracted more and more attention. Spaces of functions with dominating mixed smoothness properties are especially well suited for the approximation or numerical integration using sparse grids or a hyperbolic approximation approach (e.g. 1,8]). In this paper, we prove that the Sobolev spaces of bivariate periodic functions with dominating mixed smoothness properties are tensor products of Sobolev spaces of univariate periodic functions. Additionally, the corresponding mixed Sobolev norms turn out to be uniform cross norms. In particular, that means that one can derive error estimates for functions from these spaces from error estimates for the univariate functions in the same simple way as in the Hilbert space case (see e.g. 5]). We recall the deenitions of the periodic Sobolev spaces rst. Let T T d denote the d-dimensional torus represented by the cube T T d := ISBN 1-xxxxx-xxx-x. All rights of reproduction in any form reserved.