# On some properties of moduli of smoothness with Jacobi weights

## TL;DR

This paper investigates properties of moduli of smoothness with Jacobi weights, establishing equivalences for polynomial derivatives and applying these to approximation theory, including Jackson and Marchaud inequalities in weighted spaces.

## Contribution

It introduces new properties of Jacobi-weighted moduli of smoothness and applies them to characterize polynomial approximation and inequalities in weighted spaces.

## Key findings

- Equivalence relations for moduli of smoothness and polynomial derivatives.
- Characterization of best polynomial approximation in Jacobi weighted spaces.
- Discussion of sharp Jackson and Marchaud inequalities for 1<p<∞.

## Abstract

We discuss some properties of the moduli of smoothness with Jacobi weights that we have recently introduced and that are defined as \[ \omega_{k,r}^\varphi(f^{(r)},t)_{\alpha,\beta,p} :=\sup_{0\leq h\leq t} \left\| {\mathcal{W}}_{kh}^{r/2+\alpha,r/2+\beta}(\cdot) \Delta_{h\varphi(\cdot)}^k (f^{(r)},\cdot)\right\|_p \] where $\varphi(x) = \sqrt{1-x^2}$, $\Delta_h^k(f,x)$ is the $k$th symmetric difference of $f$ on $[-1,1]$, \[ {\mathcal{W}}_\delta^{\xi,\zeta} (x):= (1-x-\delta\varphi(x)/2)^\xi (1+x-\delta\varphi(x)/2)^\zeta , \] and $\alpha,\beta > -1/p$ if $0<p<\infty$, and $\alpha,\beta \geq 0$ if $p=\infty$. We show, among other things, that for all $m, n\in N$, $0<p\le \infty$, polynomials $P_n$ of degree $<n$ and sufficiently small $t$, \begin{align*} \omega_{m,0}^\varphi(P_n, t)_{\alpha,\beta,p} & \sim t \omega_{m-1,1}^\varphi(P_n', t)_{\alpha,\beta,p} \sim \dots \sim t^{m-1}\omega_{1,m-1}^\varphi(P_n^{(m-1)}, t)_{\alpha,\beta,p} & \sim t^m \left\| w_{\alpha,\beta} \varphi^{m} P_n^{(m)}\right\|_{p} , \end{align*} where $w_{\alpha,\beta}(x) = (1-x)^\alpha (1+x)^\beta$ is the usual Jacobi weight. In the spirit of Yingkang Hu's work, we apply this to characterize the behavior of the polynomials of best approximation of a function in a Jacobi weighted $L_p$ space, $0<p\le\infty$. Finally we discuss sharp Marchaud and Jackson type inequalities in the case $1<p<\infty$.