For second-order linear differential operators $\mathcal L_2(D)$ of the form $D^2$, $D^2+\alpha^2$, $D^2-\beta^2$ $(\alpha,\beta>0)$, the Yanenko-Stechkin-Subbotin problem of extremal interpolation of numerical sequences by twice differentiable functions $f$ with the smallest value of the norm of the function $\mathcal L_2(D)f$ in the space $L_p$ $(1\le p\le \infty)$ is considered on a grid of nodes of the numerical axis that is infinite in both directions. Subbotin's parabolic splines and their analogs for the operators $D^2+\alpha^2$ and $D^2-\beta^2$ (with knots lying in the middle between consecutive interpolation nodes) are used to derive upper bounds for the values of the smallest norm in terms of grid steps for any value of $p$, $1\le p\le \infty$.
Keywords: Subbotin's splines, interpolation, infinite grid, second-order differential operator
Received August 23, 2021
Revised September 22, 2021
Accepted September 27, 2021
Funding Agency: This study is a part of the research carried out at the Ural Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-02-2021-1383).
Valerii Trifonovich Shevaldin, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics 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. Subbotin’s splines in the problem of extremal interpolation in the space $L_p$ for second-order linear differential operators, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 4, pp. 255–262.
