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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2020, Volume 26, Number 4, Pages 196–209
(Mi timm1775)
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This article is cited in 1 scientific paper (total in 1 paper)
On a refinement of Marcinkiewicz-Zygmund type inequalities
A. V. Kroó Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences
Abstract:
The main goal of this paper is to verify a refined Marcinkiewicz–Zygmund type inequality with a quadratic error term
$$
\frac{1}{2}\sum_{j=0}^{nm-1}(x_{j+1}-x_{j-1})w(x_j)|t_n(x_{j})|^q=(1+O(m^{-2}))\int\limits_{-\pi}^{\pi}w(x)|t_n(x)|^q\,dx, \quad 2\leq q<\infty,
$$
where $t_n$ is any trigonometric polynomial of degree at most $n, \ -\pi=x_0<x_1<\cdots <x_{mn}=\pi, \max\limits_{0\leq j\leq mn-1}(x_{j+1}-x_{j})=O\Big(\displaystyle\frac{1}{nm}\Big),\ m,n\in\mathbb{N}$, and $w$ is a Jacobi type weight. Moreover, the quadratic error term $O(m^{-2})$ is shown to be sharp, in general. In addition, similar results are given for $q=\infty$ and in the multivariate case.
Keywords:
multivariate polynomials; Marcinkiewicz-Zygmund, Bernstein, and Schur type inequalities; discretization of $L^p$ norm; doubling and Jacobi type weights.
Received: 22.01.2020 Revised: 06.10.2020 Accepted: 12.10.2020
Citation:
A. V. Kroó, “On a refinement of Marcinkiewicz-Zygmund type inequalities”, Trudy Inst. Mat. i Mekh. UrO RAN, 26, no. 4, 2020, 196–209
Linking options:
https://www.mathnet.ru/eng/timm1775 https://www.mathnet.ru/eng/timm/v26/i4/p196
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