Mixed Moduli of Smoothness in $L_p$, $1<p<\infty$

TL;DR

This survey reviews 25 years of research on mixed fractional moduli of smoothness in $L_p$ spaces, covering key inequalities, properties, and their interrelations with Fourier analysis and approximation theory.

Contribution

It compiles and discusses recent advances, establishing equivalences, sharp inequalities, and properties of mixed moduli of smoothness in $L_p$ spaces.

Findings

Monotonicity properties established

Sharp Jackson, Marchaud, and Ul'yanov inequalities presented

Interrelations between moduli, Fourier coefficients, and approximation analyzed

Abstract

In this paper we survey recent developments over the last 25 years on the mixed fractional moduli of smoothness of periodic functions from $L_p$, $1<p<\infty$. In particular, the paper includes monotonicity properties, equivalence and realization results, sharp Jackson, Marchaud, and Ul'yanov inequalities, interrelations between the moduli of smoothness, the Fourier coefficients, and "angular" approximation. The sharpness of the results presented is discussed.