## 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 S

MOLYAK'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
S

OBOLEV space with dominating mixed derivative, we are able to prove

-error-estimates depending on the size of the used sampling grid.