Geodesic
Interpolation with minimum value of $L_{2}$-norm of differential operator
Ural mathematical journal, Tome 10 (2024) no. 2, pp. 107-120 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $\Delta_{N}:\ a\leq x_{1}$. 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.
Keywords: Natural ${\mathcal L}$-spline, Differential operator, Reproducing kernel, Quadratic form.
Mots-clés : Interpolation 
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Sergey I. Novikov. Interpolation with minimum value of $L_{2}$-norm of differential operator. Ural mathematical journal, Tome 10 (2024) no. 2, pp. 107-120. http://geodesic.mathdoc.fr/item/UMJ_2024_10_2_a9/

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