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JOURNALS // Sibirskii Matematicheskii Zhurnal // Archive

Sibirsk. Mat. Zh., 2022 Volume 63, Number 4, Pages 814–830 (Mi smj7695)

This article is cited in 3 papers

Estimates of solutions to infinite systems of linear equations and the problem of interpolation by cubic splines on the real line

Yu. S. Volkova, S. I. Novikovb

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
b N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg

Abstract: We study the solvability of bi-infinite systems of linear equations whose matrices are diagonally dominant. We prove that the estimates of the norm of the solution in terms of the diagonal dominance value well-known in the case of finite systems of linear equations are also valid for bi-infinite systems of equations. The estimates are used in interpolation by splines on nonuniform meshes on the real line. Using the estimates, we prove the existence and uniqueness of a cubic spline of linear or quadratic growth interpolating data of linear or quadratic growth, without any constraints on node spacing. The familiar estimates of the interpolation error on a segment are carried over to the case of interpolation on the whole real line.

Keywords: bi-infinite system of linear equations, diagonal dominance, norm of solution, splines, interpolation, bi-infinite mesh.

UDC: 517.518.85

MSC: 35R30

Received: 01.12.2021
Revised: 06.03.2022
Accepted: 15.04.2022

DOI: 10.33048/smzh.2022.63.408


 English version:
Siberian Mathematical Journal, 2022, 63:4, 677–690


© Steklov Math. Inst. of RAS, 2025