Weighted gamma-K-functional and gamma-Modulus of Smoothness on the Semiaxis

TL;DR

This paper explores gamma-relative differentiation and its role in weighted polynomial approximation on the semiaxis, establishing equivalences between Sobolev spaces, K-functionals, and moduli of smoothness, with potential applications and generalizations.

Contribution

It introduces gamma-relative differentiation to improve weighted approximation theory and establishes equivalences among Sobolev spaces, K-functionals, and moduli of smoothness.

Findings

Proves equivalence of first-order K-functionals and moduli of smoothness

Defines generalized Sobolev spaces using gamma-relative differentiation

Provides estimates linking differentiation, approximation, and smoothness

Abstract

In this paper we investigate the gamma-relative differentiation by the motivation of amending the order of the weighted polynomial approximation on the semiaxis for certain functions. With the help of this we give some definitions of generalized Sobolev spaces, K-functionals and moduli of smoothness. We prove theorems for estimating these things with each other, in the case of first order we prove equivalence. We remark some possible applications and other generalizations too.