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

We consider a continuous periodic $ d$-variate function $ f:\mathbb{T}^d \rightarrow \mathbb{C}$. The main aim is to approximate $ f$ by sampling operators in the $ L_p$ - sense. Such an approximation process $ \{A_m\}_m$ uses only discrete information about the function $ f$ in the following way

$\displaystyle A_m\,f(x) = \sum\limits_{k=1}^{N_m}f(x_k)\psi_k(x),\
x\in \mathbb{T}^d,

where the ,,sampling points`` $ x_k \in \mathbb{T}^d$, $ k=1,...,N_m$ and the functions $ \psi_k:\mathbb{T}^d \rightarrow \mathbb{C}$ are fixed. To find proper sampling point distributions and ,,optimal`` functions $ \psi_k$ 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 $ d$-variate sampling operators that use a so called sparse grid of sampling points. This construction provides some useful properties. Having more information about $ f$, for instance, $ f$ belonging to some SOBOLEV space with dominating mixed derivative, we are able to prove $ L_p$-error-estimates depending on the size of the used sampling grid.