Interpolation for a system of concentric grids

A. V. Rodionov ; M. N. Dobrovol'skii ; N. N. Dobrovol'skii ; N. M. Dobrovol'skii

Čebyševskij sbornik, Tome 24 (2023) no. 3, pp. 95-121

Résumé

The paper provides an overview of the results of the Tula School of Number Theory on the following issues interpolation of periodic functions of many variables defined in the nodes of a generalized parallelepipedal grid of an integer lattice, and by numerical integration algorithms with a stopping rule. The necessary facts and notations are given in Section 2, which consists of 6 subsections: 2.1. From the geometry of numbers; 2.2. Trigonometric sums of grids and lattices; 2.3. Inequalities for renormalization on the space $E_s^\alpha$; 2.4. Interpolation formulas for the generalized parallelepipedal grid of an integer lattice; 2.5. Properties of the interpolation operator; 2.6. Estimates of the interpolation error. These subsections, along with the known facts and definitions obtained earlier at the Tula School of Number Theory, contain new concepts and facts related to interpolation on shifted parallelepipedal grids. The following section 3. Algorithms of approximate integration and interpolation with the stopping rule contains new definitions related to the transfer of the concept of a concentric algorithm of approximate integration to the case of a multiplicative, concentric algorithm of approximate interpolation. The paper investigates new issues of approximate interpolation with stopping rules. In the 4th section, the most important and interesting case of nested sequences of parallelepipedal grids is considered for practical implementation. An estimate of the norm of the difference between two interpolation operators on a lattice and a sublattice was obtained, which made it possible to take the maximum of the modulus of the difference of these operators at the points of a larger parallelepipedal grid as the stopping rule of the concentric algorithm for approximate interpolation of periodic functions. In conclusion, the task for further research is formulated.

Détail

Keywords: the minimum polynomial of the given algebraic irrationality, residual fractions, continued fractions, TDP-shape, the modules Tue, couple Tue, linear-fractional transformation of the second kind.