## Smolyak's Algorithm for Sparse Grid Approximation of -variate Periodic Functions

We consider a continuous periodic -variate function . The main aim is to approximate by sampling operators in the - sense. Such an approximation process uses only discrete information about the function in the following way

where the ,,sampling points , and the functions are fixed. To find proper sampling point distributions and ,,optimal functions we use SMOLYAK's algorithm. Starting with a sequence of sampling operators for the univariate case we come, via tensor product constructions, to a sequence of -variate sampling operators that use a so called sparse grid of sampling points. This construction provides some useful properties. Having more information about , for instance, belonging to some SOBOLEV space with dominating mixed derivative, we are able to prove -error-estimates depending on the size of the used sampling grid.