Geodesic
On an Interpolation Problem with the Smallest $L_2$-Norm of the Laplace Operator
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 28 (2022) no. 4, pp. 143-153 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Mots-clés : interpolation
Keywords: Laplace operator, thin plate splines. 
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S. I. Novikov. On an Interpolation Problem with the Smallest $L_2$-Norm of the Laplace Operator. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 28 (2022) no. 4, pp. 143-153. http://geodesic.mathdoc.fr/item/TIMM_2022_28_4_a13/

[1] Subbotin Yu.N., “Funktsionalnaya interpolyatsiya v srednem s naimenshei $n$-i proizvodnoi”, Tr. MIAN, 88 (1967), 30–60

[2] Subbotin Yu.N., Novikov S.I., Shevaldin V.T., “Ekstremalnaya funktsionalnaya interpolyatsiya i splainy”, Tr. In-ta matematiki i mekhaniki UrO RAN, 24:3 (2018), 200–225 | DOI | MR

[3] Favard J., “Sur l'interpolation”, J. Math. Pures Appl., 19:9 (1940), 281–306 | MR

[4] de Boor S., “How small can one make the derivatives of an interpolating function?”, J. Approx. Theory, 13:2 (1975), 105–116 | DOI | MR

[5] de Boor S., “On “best” interpolation”, J. Approx. Theory, 16:1 (1976), 28–42 | DOI | MR

[6] Pevnyi A.B., “Naturalnye splainy dvukh i trekh peremennykh”, Metody vychislenii, sb. st., v. 14, Kubaturnye formuly i funktsionalnye uravneniya, ed. I.P. Mysovskikh, Izd-vo Leningrad. un-ta, L., 1985, 160–170 | MR

[7] Smolyak S.A., “Optimalnoe vosstanovlenie funktsii i svyazannye s nim geometricheskie kharakteristiki mnozhestv”, Tr. 3-i zimnei shkoly po mat. programmirovaniyu, Izd-vo TsEMI AN SSSR, M., 1971, 509–557

[8] Duchon J., “Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces”, RAIRO Anal. Numer., 10:R3 (1976), 5–12 | MR

[9] Arcangeli R., de Silanes M.C., Torrens Ju., Multidimensinal minimizing splines. Theory and applications, Kluwer Acad. Publ., NY etc., 2004, 261 pp. | MR

[10] Buhmann M.D., “Radial basis functions”, Acta Numer., 9 (2000), 1–38 | DOI | MR

[11] Buhmann M.D., Radial basis functions: Theory and implementations, Cambridge Monogr. Appl. Comput. Math., 12, Cambridge Univ. Press, Cambridge, 2003, 259 pp. | DOI | MR

[12] Mehri B., Jokar S., “Lebesgue function for multivariate interpolation by radial basis functions”, Appl. Math. Comput., 187:1 (2007), 306–314 | DOI | MR

[13] Fikhtengolts G.M., Kurs differentsialnogo i integralnogo ischisleniya, v. 1, Nauka, M., 1970, 608 pp.

[14] Holmes R., Geometric functional analysis and its applications, Springer Verlag, NY etc., 1975, 246 pp. | MR