Inequalities for $$$\Lambda$$$-derivatives of functions defined on a metric space and some of their applications

V. Babenko; V. Kolesnyk; O. Kovalenko; N. Parfinovych

doi:10.15421/242511

Inequalities for $$$\Lambda$$$-derivatives of functions defined on a metric space and some of their applications

V. Babenko (Oles Honchar Dnipro National University), https://orcid.org/0000-0001-6677-1914
V. Kolesnyk (Drake University)
O. Kovalenko (Oles Honchar Dnipro National University), https://orcid.org/0000-0002-0446-1125
N. Parfinovych (Oles Honchar Dnipro National University), https://orcid.org/0000-0002-3448-3798

Abstract

We introduce a concept of a $$$\Lambda$$$-derivative operator, which is a certain generalization of hypersingular integral operators, which in turn are used in the definitions of the Marchaud and the Riesz fractional derivatives. We study the problem of sharp Kolmogorov-type inequalities in spaces of continuous bounded functions equipped with a seminorm defined by a certain modulus of continuity for such kind of operators. We also consider an application of the obtained Kolmogorov-type inequality to the Stechkin problem about the best approximation of an operator by bounded ones.

Keywords

Kolmogorov-type inequality; inequality for derivatives; modulus of continuity; fractional derivative; hypersingular integral operator; Stechkin problem

MSC 2020

Pri 26D10, Sec 41A17, 41A44, 41A65

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References

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