Best Recovery of the Laplace Operator of a Function and Sharp Inequalities

Abstract

The paper is concerned with the problem of optimal recovery of fractional powers of the Laplace operator in the uniform norm on the multivariate generalized Sobolev class of functions from incomplete data on the Fourier transform of these functions on a ball of radius r centered at the origin. An optimal recovery method is constructed, and a number \( \widehat{r} \)> 0 is specified such that for r ≤\( \widehat{r} \) the method makes use of all the information on the Fourier transform, smoothing thereof; and if r >\( \widehat{r} \), then the information on the Fourier transform proves superfluous and hence is not used by the optimal method. For fractional powers of the Laplace operator, a sharp inequality is proved. This inequality turns out to be closely related to the recovery problem and is an analogue of Kolmogorov-type inequalities for derivatives.