Approximation of \(1/\| x-y \|\) by exponentials for wavelet applications
Integrals on high dimensional domains that occur in the solution of partial differential equations can often be transformed into products of lower dimensional integrals if \(1/\| x-y\| \) is approximated by exponentials. This leads to the approximation of the function \(1/\sqrt{x}\) by sums of exponentials. Here polynomial terms are added since this helps to control moments in connection with wavelets. Numerical results for the degree of approximation are provided for small and for large intervals in \(\mathbb R\). The improvement by the polynomial terms is discussed. The latter can be understood as exponential sums with fixed exponents. The tables are useful for a quick view of the error estimates [since the error estimates in the cited papers are less direct and require elliptic functions or the execution of Landen transformations].
scientific article; zbMATH DE number 5009642