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JOURNALS // Trudy Instituta Matematiki i Mekhaniki UrO RAN // Archive

Trudy Inst. Mat. i Mekh. UrO RAN, 2022 Volume 28, Number 4, Pages 143–153 (Mi timm1958)

This article is cited in 2 papers

On an Interpolation Problem with the Smallest $L_2$-Norm of the Laplace Operator

S. I. Novikov

N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg

Abstract: The paper is devoted to an interpolation problem for finite sets of real numbers bounded in the Euclidean norm. The interpolation is by a class of smooth functions of two variables with the minimum $L_{2}$-norm of the Laplace operator $\Delta=\partial^{2 }/\partial x^{2}+\partial^{2 }/\partial y^{2}$ applied to the interpolating functions. It is proved that if $N\geq 3$ and the interpolation points $\{(x_{j},y_{j})\}_{j=1}^{N}$ do not lie on the same straight line, then the minimum value of the $L_{2}$-norm of the Laplace operator on interpolants from the class of smooth functions for interpolated data from the unit ball of the space $l_{2}^{N}$ is expressed in terms of the largest eigenvalue of the matrix of a certain quadratic form.

Keywords: interpolation, Laplace operator, thin plate splines.

UDC: 517.5

MSC: 41A05, 41A15

Received: 19.08.2022
Revised: 01.09.2022
Accepted: 05.09.2022

DOI: 10.21538/0134-4889-2022-28-4-143-153


 English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2022, 319, suppl. 1, S193–S203

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© Steklov Math. Inst. of RAS, 2025