On the best approximations and rate of convergence of decompositions in the root vectors of an operator
We establish upper bounds of the best approximations of elements of a Banach space B by the root vectors of an operator $A$ that acts in B. The corresponding estimates of the best approximations are expressed in terms of a K-functional associated with the operator A. For the operator of differentiat...
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| Datum: | 1997 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Russisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
1997
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/5060 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | We establish upper bounds of the best approximations of elements of a Banach space B by the root vectors of an operator $A$ that acts in B. The corresponding estimates of the best approximations are expressed in terms of a K-functional associated with the operator A. For the operator of differentiation with periodic boundary conditions, these estimates coincide with the classical Jackson inequalities for the best approximations of functions by trigonometric polynomials. In terms of K-functionals, we also prove the abstract Dini-Lipschitz criterion of convergence of partial sums of the decomposition of f from B in the root vectors of the operator A to f |
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