V.T. Shevaldin. Extremal interpolation in the mean with overlapping averaging intervals and the smallest norm of a linear differential operator ... P. 219-232

The Yanenko—Stechkin—Subbotin problem of extremal functional interpolation in the mean is considered for sequences infinite in both directions on a uniform grid of the numerical axis with the smallest norm in the space $L_p(R)$ $(1 <p<\infty)$ of a linear differential operator ${\cal L}_n$ with constant coefficients. It is assumed that the generalized finite differences of each sequence corresponding to the operator ${\cal L}_n$ are bounded in the space $l_p$, the grid step $h$ and the averaging step $h_1$ are related by the inequality $h<h_1<2h$, and the operator ${\cal L}_n$ is formally self-adjoint. Under these assumptions, in the case of odd $n$, the smallest norm of the operator is found exactly, and the extremal function is a generalized $\cal L$-spline whose knots coincide with the interpolation nodes. This work continues the research of this problem by Yu.N. Subbotin and the author started by Subbotin in 1965.

Keywords: extremal interpolation, splines, uniform grid, formally self-adjoint differential operator, minimum norm, splines

Received January 25, 2023

Revised February 14, 2023

Accepted February 20, 2023

Shevaldin Valerii Trifonovich, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: Valerii.Shevaldin@imm.uran.ru


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Cite this article as: V.T. Shevaldin. Extremal interpolation in the mean with overlapping averaging intervals and the smallest norm of a linear differential operator, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 1, pp. 219–232.