Jackson-Stechkin type inequalities for differentiable functions in weighted Orlicz spaces

TL;DR

This paper establishes Jackson-Stechkin type inequalities for differentiable functions within weighted Orlicz spaces, utilizing advanced harmonic analysis tools to refine approximation and smoothness characterizations.

Contribution

It introduces new Jackson-Stechkin inequalities in weighted Orlicz spaces and develops related extrapolation, multiplier, and Littlewood-Paley theorems for these spaces.

Findings

Refined Jackson type inequalities in weighted Orlicz spaces.

Equivalence between fractional weighted modulus of smoothness and Peetre's K-functional.

Enhanced inverse Marchaud inequalities.

Abstract

In the present work some Jackson Stechkin type direct theorems of trigonometric approximation are proved in Orlicz spaces with weights satisfying some Muckenhoupt's $A_{p}$ condition. To obtain refined version of the Jackson type inequality we prove an extrapolation theorem, Marcinkiewicz multiplier theorem and Littlewood Paley type results. As a consequence refined inverse Marchaud type inequalities are obtained. By means of a realization result we find an equivalence between the fractional order weighted modulus of smoothness and the classical weighted Peetre's K-functional.