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

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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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     author = {S. I. Novikov},
     title = {Interpolation by functions from a {Sobolev} space with minimum $L_p$-norm of the {Laplace} operator},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {212--222},
     publisher = {mathdoc},
     volume = {21},
     number = {4},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2015_21_4_a18/}
}
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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/