About three-dimensional nets of Smolyak II
https://doi.org/10.22405/2226-8383-2021-22-3-100-121
Abstract
This is the second article in a series dedicated to Smolyak grids. The paper relates to analytical number theory and it deals with the application of number theory to problems of approximate analysis.
In this paper, it was shown that for an arbitrary Smolyak grid, the trigonometric sum of the Smolyak grid is 𝑆𝑞(⃗0) = 1. It follows that the norm of the linear functional of approximate integration on the class 𝐸𝛼 𝑠 is equal to the value of the hyperbolic zeta function 𝜁(𝛼|𝑆𝑚(𝑞, 𝑠)) of the resin grid. It is shown that the hyperbolic zeta function 𝜁(𝛼|𝑆𝑚(𝑞, 𝑠)) of the Smolyak grid
is a Dirichlet series. This raises the question of the analytic continuation of the hyperbolic zeta function 𝜁(𝛼|𝑆𝑚(𝑞, 𝑠)) of the Smolyak grid as a function of an arbitrary complex 𝛼 = 𝜎 + 𝑖𝑡.
Since the Smolyak grid belongs to the number of rational grids, it turns out that it has an analytical continuation of the hyperbolic zeta function 𝜁(𝛼|𝑆𝑚(𝑞, 𝑠)) of the Smolyak grid on the entire complex plane except for the point 𝛼 = 1, in which it has a pole of order 𝑠.
It follows from the work that the following questions remain open:
1. is the linear operator 𝐴𝑞 of weighted grid averages over the Smolyak grid at dimension 𝑠 > 3 normal?
2. what are the true values of the trigonometric sums 𝑆𝑞(𝑚1, . . . ,𝑚𝑠) Smolyak grids with dimension 𝑠 > 3?
About the Authors
Nikolai Nikolaevich Dobrovol’skii
Tula State University, Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
Dmitry Viktorovich Gorbachev
Tula State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
Valerii Ivanovich Ivanov
Tula State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
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For citations:
Dobrovol’skii N.N., Gorbachev D.V., Ivanov V.I. About three-dimensional nets of Smolyak II. Chebyshevskii Sbornik. 2021;22(3):100-121. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-3-100-121
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