On Some Properties of Moduli of Smoothness with Jacobi Weights

Abstract

We discuss some properties of the moduli of smoothness with Jacobi weights that we have recently introduced and that are defined as

$$\displaystyle{\omega }_{k,r}^\varphi (f^{(r)},t)_{\alpha ,\beta ,p} :=\sup _{0\leq h\leq t}\left \|{\mathcal {W}}_{kh}^{r/2+\alpha ,r/2+\beta }(\cdot )\Delta _{h\varphi (\cdot )}^k (f^{(r)},\cdot )\right \|{ }_{p},$$

where $\varphi (x) = \sqrt {1-x^2}$, $\Delta _h^k(f,x)$ is the kth symmetric difference of f on [−1, 1],

$$\displaystyle{\mathcal {W}}_\delta ^{\xi ,\zeta } (x):= (1-x-\delta \varphi (x)/2)^\xi (1+x-\delta \varphi (x)/2)^\zeta ,$$

and α, β > −1∕p if 0 < p < ∞, and α, β ≥ 0 if p = ∞.

We show, among other things, that for all $m, n\in \mathbb N$, 0 < p ≤∞, polynomials P _n of degree < n and sufficiently small t,

$$\displaystyle\begin {array}{ll} {\omega }_{m,0}^{\varphi }(P_n, t)_{\alpha ,\beta ,p} & \sim t {\omega }_{m-1,1}^{\varphi }(P_n^{\prime }, t)_{\alpha ,\beta ,p} \sim \dots \sim t^{m-1}{\omega }_{1,m-1}^{\varphi }(P_n^{(m-1)}, t)_{\alpha ,\beta ,p} \\ & \sim t^m \left \|w_{\alpha ,\beta } \varphi ^{m} P_n^{(m)}\right \|{ }_{p} , \end {array}$$

where w _α,β(x) = (1 − x) ^α(1 + x)^β 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 ≤∞. Finally we discuss sharp Marchaud and Jackson type inequalities in the case 1 < p < ∞.

Dedicated to the memory of our friend, colleague, and collaborator Yingkang Hu (July 6, 1949–March 11, 2016)