Uniform distribution in the unit cube of weighted nodes of the quadrature formula

E. M. Rarova^a, N. N. Dobrovol'skii^ba, I. Yu. Rebrova^a, I. N. Balaba<sup>a

a</sup> Tula State Lev Tolstoy Pedagogical University (Tula)
^b Lomonosov Moscow State University (Moscow)

Abstract: The paper defines a uniform distribution in a unit $s$-dimensional cube of a sequence of nested generalized parallelepiped grids of type II with a weight function. In addition, a definition of a uniform distribution in a unit $s$-dimensional cube $G_s$ of a sequence of $ M_n$ grids with a weight function is given.
A proof is given of an analogue of the generalized G. Weyl criterion on necessary and sufficient conditions for a uniform distribution in a unit $s$-dimensional cube $G_s$ of a sequence of $ M_n$ grids with weights.
Since the definition of the uniform distribution of a sequence of nested generalized parallelepiped grids of type II with a weight function differs from the definition of the uniform distribution of a sequence of grids $ M_n$ with a weight function, the paper proves the second analogue of the Weyl criterion on the necessary and sufficient conditions for the uniform distribution in a unit $s$-dimensional cube $G_s$ of a sequence of nested generalized parallelepiped grids of type II.
The following theorem is proved:
Theorem 1. Let the Fourier series of $f(\vec x)$ converge absolutely, $C(\vec m)$ be its Fourier coefficients and $S_{M,\vec\rho}(\vec m)$ be the trigonometric sums of the weighted grid, then the following equality holds
\begin{gather*} R_N[f]=C(\vec 0)\left(\dfrac{1}{N}S_{M,\vec\rho}(\vec 0)-1\right)+\dfrac{1}{N}\mathop{{\sum}'}\limits_{m_1, \ldots, m_s=-\infty}^{\infty}C(\vec m)S_{M,\vec\rho}(\vec m)=\cr=C(\vec 0)\left(S_{M,\vec\rho}^*(\vec 0)-1\right)+\mathop{{\sum}'}\limits_{m_1, \ldots, m_s=-\infty}^{\infty}C(\vec m)S_{M,\vec\rho}^*(\vec m) \end{gather*}
and as $N\to\infty$ the error $R_N[f]$ will tend to zero if and only if the weighted nodes of the quadrature formula are uniformly distributed in the unit $s$–dimensional cube.

Keywords: algebraic lattices, algebraic net, trigonometric sums of algebraic net with weights, weight functions.

UDC: 511.3

Received: 18.03.2025
Revised: 27.08.2025

DOI: 10.22405/2226-8383-2025-26-3-185-219