Error of approximate integration and interpolation on classes of functions defined by monoids of natural numbers

N. N. Dobrovol'skii   ; N. M. Dobrovol'skii   ; Yu. A. Basalov   ; I. Yu. Rebrova

Čebyševskij sbornik, Tome 25 (2024) no. 4, pp. 74-105

Résumé

The paper provides answers to the following questions: First, the question is in which case the harmonic numbers from the class $M_s^\alpha$ do not fall into the lattice of linear comparison solutions corresponding to the parallelepipedal grid. As a result, a new object of research has appeared – the intersection of the lattice of linear comparison solutions corresponding to a parallelepipedal grid and a multidimensional mononoid defining a class of functions. Secondly, what do the boundary functions of these classes look like for parallelepipedal grids. Here we did not get a simple finite form in the form of an expression from elementary functions, but only an expression in the form of series according to the general theory. The estimation of the error of approximate integration on the class $M_s^\alpha$ is associated with the study of a new number-theoretic object – the hyperbolic zeta function of the intersection of the lattice of solutions of linear comparison and a multidimensional monoid defining a class of functions. Here it was possible to obtain an analogue of the enhanced Bakhvalov–Korobov theorem. Finally, the third question is related to the fact that parallelepipedal grids are interpolation-type grids: what is the error of interpolation polynomials for valid parallelepipedal grids in the case of the monoid $M_{q,1}$. The answer here is as follows: interpolation formulas for parallelepipedal grids are accurate only for some trigonometric polynomials, in which all harmonics fall into the complete system of deductions of the fundamental lattice by the sublattice of solutions of the corresponding linear comparison. In general, the error estimate is similar to the estimates for the Korobov class.

Détail

Keywords: classes of functions, Dirichlet series, zeta function of the monoid of natural numbers.
Mots-clés : quadrature formulas