About three-dimensional nets of Smolyak II
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 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 = + 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 = 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 of weighted grid averages over the Smolyak grid at dimension > 3 normal? 2. what are the true values of the trigonometric sums 𝑆𝑞(𝑚1, . . . ,𝑚𝑠) Smolyak grids with dimension > 3?