On an Interpolation Problem with the Smallest $L_2$-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 $L_{2}$-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.