On approximation of hypersingular integral operators by bounded ones

TL;DR

This paper addresses the approximation of unbounded hypersingular integral operators by bounded operators, providing sharp inequalities and applications in operator analysis and optimal recovery.

Contribution

It solves the Stechkin approximation problem for hypersingular operators and establishes sharp inequalities and applications in Sobolev spaces.

Findings

Sharp Landau-Kolmogorov inequalities for hypersingular operators

A sharp Ostrowski inequality for multivariate Sobolev classes

Results on the modulus of continuity and optimal recovery of hypersingular integrals

Abstract

We solve the Stechkin problem about approximation of generally speaking unbounded hypersingular integral operators by bounded ones. As a part of the proof, we also solve several related and interesting on their own problems. In particular, we obtain sharp Landau-Kolmogorov type inequalities in both additive and multiplicative forms for hypersingular integral operators and prove a sharp Ostrowski type inequality for multivatiate Sobolev classes. We also give some applications of the obtained results, in particular study the modulus of continuity of the hypersingular integral operators, and solve the problem of optimal recovery of the value of a hypersingular integral operator based on the argument known with an error.