Uniform distribution in the unit cube of weighted nodes of the quadrature formula
https://doi.org/10.22405/2226-8383-2025-26-3-185-219
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 𝐺𝑠 of a sequence of 𝑀𝑛 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 𝐺𝑠 of a sequence of 𝑀𝑛 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 𝑀𝑛 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 𝐺𝑠 of a sequence of nested generalized parallelepiped grids of type II.
The following theorem is proved:
Theorem 2. Let the Fourier series of 𝑓(⃗𝑥) converge absolutely, 𝐶(⃗𝑚) be its Fourier coefficients and 𝑆𝑀,⃗𝜌(⃗𝑚) be the trigonometric sums of the weighted grid, then the following equality holds
and as 𝑁 → ∞ the error 𝑅𝑁[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.
About the Authors
Elena Mikhailovna Rarova
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
Nikolai Nikolaevich Dobrovol’skii
Tula State Lev Tolstoy Pedagogical University; Lomonosov Moscow State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
Irina Yuryevna Rebrova
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
Irina Nikolaevna Balaba
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
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For citations:
Rarova E.M., Dobrovol’skii N.N., Rebrova I.Yu., Balaba I.N. Uniform distribution in the unit cube of weighted nodes of the quadrature formula. Chebyshevskii Sbornik. 2025;26(3):185-219. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-3-185-219
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