On a refinement of Marcinkiewicz-Zygmund type inequalities

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.