MiS Preprint Repository

The Kronecker tensor-product approximation combined with the $\mathcal{H}$-matrix techniques provides an efficient tool to represent integral operators as well as certain functions $F(A)$ of a discrete elliptic operator $A$ in ${\mathbb{R}}^d$ with a high spatial dimension $d$. In particular, we approximate the functions $A^{-1}$ and $sign(A)$ of a finite difference discretisation $A\in {\mathbb{R}}^{N\times N}$ with a rather general location of the spectrum. The asymptotic complexity of our data-sparse representations can be estimated by $\mathcal{O}(n^p{\log }^{q}n)$, $p=1,2$, with $q$ independent of $d$, where $n=N^{1/d}$ is the dimension of the discrete problem in one space direction. In this paper (Part I), we discuss several methods of a separable approximation of multi-variate functions. Such approximations provide the base for a tensor-product representation of operators. We discuss the asymptotically optimal sinc quadratures and sinc interpolation methods as well as the best approximations by exponential sums. These tools will be applied in Part II continuing this paper to the problems mentioned above.