On Some Properties of Moduli of Smoothness with Jacobi Weights

Kirill A. Kopotun, Dany Leviatan, Igor A. Shevchuk

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

Abstract

We discuss some properties of the moduli of smoothness with Jacobi weights that we have recently introduced and that are defined as ωk,rφ(f(r),t)α,β,p:=sup0≤h≤t∥Wkhr∕2+α,r∕2+β(⋅)Δhφ(⋅)k(f(r),⋅)∥p,$$\displaystyle{\omega }_{k,r}^\varphi (f^{(r)},t)_{\alpha,\beta,p} :=\sup _{0\leq h\leq t}\left \|{\mathcal {W}}_{kh}^{r/2+\alpha,r/2+\beta }(\cdot )\Delta _{h\varphi (\cdot )}^k (f^{(r)},\cdot )\right \|{ }_{p},$$ where φ(x)=1−x2, Δhk(f,x) is the kth symmetric difference of f on [−1, 1], Wδξ,ζ(x):=(1−x−δφ(x)∕2)ξ(1+x−δφ(x)∕2)ζ,$$\displaystyle{\mathcal {W}}_\delta ^{\xi,\zeta } (x):= (1-x-\delta \varphi (x)/2)^\xi (1+x-\delta \varphi (x)/2)^\zeta,$$ and α, β > −1∕p if 0 < p < ∞, and α, β ≥ 0 if p = ∞. We show, among other things, that for all m, n∈ ℕ, 0 < p ≤∞, polynomials Pn of degree < n and sufficiently small t, ωm,0φ(Pn,t)α,β,p∼tωm−1,1φ(Pn′,t)α,β,p∼⋯∼tm−1ω1,m−1φ(Pn(m−1),t)α,β,p∼tm∥wα,βφmPn(m)∥p,$$\displaystyle\begin {array}{ll} {\omega }_{m,0}^{\varphi }(P_n, t)_{\alpha,\beta,p} & \sim t {\omega }_{m-1,1}^{\varphi }(P_n^{\prime }, t)_{\alpha,\beta,p} \sim \dots \sim t^{m-1}{\omega }_{1,m-1}^{\varphi }(P_n^{(m-1)}, t)_{\alpha,\beta,p} \\ & \sim t^m \left \|w_{\alpha,\beta } \varphi ^{m} P_n^{(m)}\right \|{ }_{p}, \end {array}$$ where wα,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 Lp space, 0 < p ≤∞. Finally we discuss sharp Marchaud and Jackson type inequalities in the case 1 < p < ∞.

Original languageEnglish
Title of host publicationApplied and Numerical Harmonic Analysis
PublisherSpringer International Publishing
Pages19-31
Number of pages13
DOIs
StatePublished - 2019

Publication series

NameApplied and Numerical Harmonic Analysis
ISSN (Print)2296-5009
ISSN (Electronic)2296-5017

Keywords

  • Approximation by polynomials in weighted L-norms
  • Jacobi weights
  • Moduli of smoothness

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