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, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation)

Abstract


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 \({\cal L}\)-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


Interpolation, Natural \({\cal L}\)-spline, Differential operator, Reproducing kernel, Quadratic form.

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References


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DOI: http://dx.doi.org/10.15826/umj.2024.2.010

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