On an Interpolation Problem with the Smallest subscript𝐿2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{2}$$\end{document}-Norm of the Laplace Operator

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 subscript𝐿2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{2}$$\end{document}-norm of the Laplace operator Δsuperscript2superscript𝑥2superscript2superscript𝑦2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta=\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}$$\end{document} applied to the interpolating functions. It is proved that if 𝑁3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\geq 3$$\end{document} and the interpolation points superscriptsubscriptsubscript𝑥𝑗subscript𝑦𝑗𝑗1𝑁\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{(x_{j},y_{j})\}_{j=1}^{N}$$\end{document} do not lie on the same straight line, then the minimum value of the subscript𝐿2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{2}$$\end{document}-norm of the Laplace operator on interpolants from the class of smooth functions for interpolated data from the unit ball of the space superscriptsubscript𝑙2𝑁\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_{2}^{N}$$\end{document} is expressed in terms of the largest eigenvalue of the matrix of a certain quadratic form. .