On some properties of moduli of smoothness with Jacobi weights (1901.03907v1)

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 $1<p<\infty$.