Geodesic
Interpolation by functions from a Sobolev space with minimum $L_p$-norm of the Laplace operator
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 4, pp. 212-222 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider an interpolation problem with minimum value of the $L_p$-norm ($1\leq p\infty$) of the Laplace operator of interpolants for a class of interpolated sequences that are bounded in the $l_p$-norm. The data are interpolated at nodes of the grid formed by points from $\mathbb{R}^n$ with integer coordinates. It is proved that, if $1\leq p$$n/2$, then the $L_p$-norm of the Laplace operator of the interpolant can be arbitrarily small for any sequence that is interpolated. Two-sided estimates for the $L_2$-norm of the Laplace operator of the best interpolant are found for the case $n=2$.
Mots-clés : interpolation
Keywords: Laplace operator, Sobolev space, embedding. 
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S. I. Novikov. Interpolation by functions from a Sobolev space with minimum $L_p$-norm of the Laplace operator. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 4, pp. 212-222. http://geodesic.mathdoc.fr/item/TIMM_2015_21_4_a18/

[1] Burenkov V.I., Sobolev spaces on domains, Teubner Texts in Math, 137, B.G.Teubner Verlag GmbH, Stuttgart, 1998, 312 pp. | DOI | MR | Zbl

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

[3] Fisher S., Jerome J., “Minimum norm extremals in function spaces”, Lecture Notes in Math., 479 (1975), 1–209 | DOI | MR

[4] Tikhomirov V.M., Boyanov B.D., “O nekotorykh vypuklykh zadachakh teorii priblizhenii”, Serdica, 5:1 (1979), 83–96 | MR | Zbl

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

[6] Novikov S.I., Shevaldin V.T., “Ob odnoi zadache ekstremalnoi interpolyatsii dlya funktsii mnogikh peremennykh”, Tr. In-ta matematiki i mekhaniki UrO RAN, 7:1 (2001), 144–159

[7] Novikov S.I., “Interpolation in $\mathbb{R}^2$ with minimum value of the uniform norm of the Laplace operator”, J. Math. and System Science, 3:2 (2013), 55–61

[8] Sobolev S.L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, 3-e izd., Nauka, M., 1988, 336 pp. | MR

[9] Fikhtengolts G.M., Kurs differentsialnogo i integralnogo ischisleniya, 3, Nauka, M., 1969, 656 pp.

[10] Stein I.M., Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973, 344 pp. | MR

[11] Tikhomirov V.M., Nekotorye voprosy teorii priblizhenii, Izd-vo MGU, M., 1976, 304 pp. | MR

[12] Iwaniec T., Martin G., “Riesz transforms and related singular integrals”, J. Reine Angew. Math., 473:1 (1996), 25–57 | MR | Zbl

[13] Madych W.R., Potter E.H., “An estimate of multivariate interpolation”, J. Approx. Theory, 43:2 (1985), 132–139 | DOI | MR | Zbl

[14] Maplesoft, Maplesoft [site]: Maple 17 URL: http://www.maplesoft.com/support/help/Maple/view.aspx?path=updates/Maple17