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

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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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     author = {S. I. Novikov},
     title = {On an {Interpolation} {Problem} with the {Smallest} $L_2${-Norm} of the {Laplace} {Operator}},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {143--153},
     publisher = {mathdoc},
     volume = {28},
     number = {4},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2022_28_4_a13/}
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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/