On a refinement of Marcinkiewicz-Zygmund type inequalities

A. V. Kroó

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 4, pp. 196-209

Résumé

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 $n, \ -\pi=x_0$, 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.

Détail

Keywords: multivariate polynomials; Marcinkiewicz-Zygmund, Bernstein, and Schur type inequalities; discretization of $L^p$ norm; doubling and Jacobi type weights.