INTERPOLATION WITH MINIMUM VALUE OF \(L_{2}\)-NORM OF DIFFERENTIAL OPERATOR
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 \(N\)...
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Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics
2024-12-01
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| Series: | Ural Mathematical Journal |
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| Online Access: | https://umjuran.ru/index.php/umj/article/view/840 |
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| author | Sergey I. Novikov |
| author_facet | Sergey I. Novikov |
| author_sort | Sergey I. Novikov |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-b4c6aa94f16e4cc5b7bbd0b49add0928 |
| institution | DOAJ |
| issn | 2414-3952 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Ural Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and Mechanics |
| record_format | Article |
| series | Ural Mathematical Journal |
| spelling | doaj-art-b4c6aa94f16e4cc5b7bbd0b49add09282025-08-20T02:54:40ZengUral Branch of the Russian Academy of Sciences and Ural Federal University named after the first President of Russia B.N.Yeltsin, Krasovskii Institute of Mathematics and MechanicsUral Mathematical Journal2414-39522024-12-0110210.15826/umj.2024.2.010219INTERPOLATION WITH MINIMUM VALUE OF \(L_{2}\)-NORM OF DIFFERENTIAL OPERATORSergey I. Novikov0Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108For 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.https://umjuran.ru/index.php/umj/article/view/840interpolation, natural \({\cal l}\)-spline, differential operator, reproducing kernel, quadratic form. |
| spellingShingle | Sergey I. Novikov INTERPOLATION WITH MINIMUM VALUE OF \(L_{2}\)-NORM OF DIFFERENTIAL OPERATOR Ural Mathematical Journal interpolation, natural \({\cal l}\)-spline, differential operator, reproducing kernel, quadratic form. |
| title | INTERPOLATION WITH MINIMUM VALUE OF \(L_{2}\)-NORM OF DIFFERENTIAL OPERATOR |
| title_full | INTERPOLATION WITH MINIMUM VALUE OF \(L_{2}\)-NORM OF DIFFERENTIAL OPERATOR |
| title_fullStr | INTERPOLATION WITH MINIMUM VALUE OF \(L_{2}\)-NORM OF DIFFERENTIAL OPERATOR |
| title_full_unstemmed | INTERPOLATION WITH MINIMUM VALUE OF \(L_{2}\)-NORM OF DIFFERENTIAL OPERATOR |
| title_short | INTERPOLATION WITH MINIMUM VALUE OF \(L_{2}\)-NORM OF DIFFERENTIAL OPERATOR |
| title_sort | interpolation with minimum value of l 2 norm of differential operator |
| topic | interpolation, natural \({\cal l}\)-spline, differential operator, reproducing kernel, quadratic form. |
| url | https://umjuran.ru/index.php/umj/article/view/840 |
| work_keys_str_mv | AT sergeyinovikov interpolationwithminimumvalueofl2normofdifferentialoperator |