Applications of Smolyak quadrature formulas to the numerical integration of Fourier coefficients and in function recovery problems
Let \(f: {\mathbb R}^s \to {\mathbb C}\) be a sufficiently smooth function which is 1-periodic in each variable. Using the Smolyak quadrature formula, which uses only nodes of a sparse grid in \([0,\,1]^s\), see \textit{S. A. Smolyak} [Dokl. Akad. Nauk SSSR 148, 1042--1045 (1963; Zbl 0202.39901)], the authors calculate the Fourier coefficients \[ {\hat f}(n) = \int_{[0,1]^s} f(x)\, e^{-2\pi i \,(n,x)}\, dx \quad (n\in {\mathbb Z}^s). \] If \(f\) belongs to generalized Sobolev or Korobov space, the exact orders of the errors of this numerical integration are determined by the authors. Finally, these results are applied to an approximate recovery of \(f\), where the exact Fourier coefficients \({\hat f}(n)\) of a finite Fourier sum are replaced by their computed values.
function spaces of dominating mixed smoothness
scientific article; zbMATH DE number 5790424