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Interpolation with minimum value of $L_{2}$-norm of differential operator
Sergey I. Novikov Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences
Аннотация:
For the class of bounded in $l_{2}$-norm interpolated data, we consider a problem of interpolation on a finite interval $[a,b]\subset\mathbb{R}$ with minimal value of the $L_{2}$-norm of a differential operator applied to interpolants. Interpolation is performed at knots of an arbitrary $N$-point mesh $\Delta_{N}:\ a\leq x_{1}<x_{2}<\cdots <x_{N}\leq b$. The extremal function is the interpolating natural ${\mathcal L}$-spline for an arbitrary fixed set of interpolated data. For some differential operators with constant real coefficients, it is proved that on the class of bounded in $l_{2}$-norm interpolated data, the minimal value of the $L_{2}$-norm of the differential operator on the interpolants is represented through the largest eigenvalue of the matrix of a certain quadratic form.
Ключевые слова:
Interpolation, Natural ${\mathcal L}$-spline, Differential operator, Reproducing kernel, Quadratic form.
Образец цитирования:
Sergey I. Novikov, “Interpolation with minimum value of $L_{2}$-norm of differential operator”, Ural Math. J., 10:2 (2024), 107–120
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/umj238 https://www.mathnet.ru/rus/umj/v10/i2/p107
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