Best approximations and moduli of smoothness of functions and their derivatives in $L_p$, $0<p<1$

TL;DR

This paper establishes new inequalities relating moduli of smoothness and approximation errors for functions and their derivatives in $L_p$ spaces with $0<p<1$, advancing understanding of approximation theory in these quasi-normed spaces.

Contribution

It introduces novel inequalities for moduli of smoothness and approximation errors in $L_p$ spaces with $0<p<1$, including for derivatives and various approximation methods.

Findings

Derived inequalities for moduli of smoothness in $L_p$, $0<p<1$

Established bounds for approximation errors of derivatives

Applied results to simultaneous approximation of functions and derivatives

Abstract

Several new inequalities for moduli of smoothness and errors of the best approximation of a function and its derivatives in the spaces $L_p$, $0<p<1$, are obtained. For example, it is shown that for any $0<p<1$ and $k,\,r\in \mathbb{N}$ one has $ \omega_{r+k}(f,\d)_p\leq C({p,k,r})\d^{r+\frac{1}{p}-1}\(\int_0^\d\frac{\omega_{k}(f^{(r)},t)_p^p}{t^{2-p}}{\rm d}t\)^\frac{1}{p},$ where the function $f$ is such that $f^{(r-1)}$ is absolutely continuous. Similar inequalities are obtained for the Ditzian-Totik moduli of smoothness and the error of the best approximation of functions by trigonometric and algebraic polynomials and splines. As an application, positive results about simultaneous approximation of a function and its derivatives by the mentioned approximation methods in the spaces $L_p$, $0<p<1$, are derived.