On moduli of smoothness with Jacobi weights

TL;DR

This paper introduces new moduli of smoothness with Jacobi weights for functions in weighted $L_p$ spaces, providing characterizations of smoothness and establishing equivalences with known functionals across different $p$ ranges.

Contribution

It defines and analyzes moduli of smoothness with Jacobi weights, connecting them to weighted $K$-functionals and realization functionals, expanding the tools for smoothness measurement in weighted spaces.

Findings

For $1 \\le p \\le \\infty$, moduli are equivalent to weighted $K$-functionals.

For $0 < p < 1$, moduli are equivalent to realization functionals.

The introduced moduli characterize smoothness of derivatives in weighted $L_p$ spaces.

Abstract

The main purpose of this paper is to introduce moduli of smoothness with Jacobi weights $(1-x)^\alpha(1+x)^\beta$ for functions in the Jacobi weighted $L_p[-1,1]$, $0<p\le \infty$, spaces. These moduli are used to characterize the smoothness of (the derivatives of) functions in the weighted $L_p$ spaces. If $1\le p\le\infty$, then these moduli are equivalent to certain weighted $K$-functionals (and so they are equivalent to certain weighted Ditzian-Totik moduli of smoothness for these $p$), while for $0<p<1$ they are equivalent to certain "Realization functionals".