2017
Том 69
№ 9

All Issues

Order equalities for some functionals and their application to the estimation of the best $n$-term approximations and widths

Shydlich A. L.

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Abstract

We study the behavior of functionals of the form $\sup_{l>n} (l-n)\left(∑^l_{k=1} \frac1{ψ^r(k)} \right)^{−1/r}$, where $ψ$ is a positive function, as $n → ∞$: The obtained results are used to establish the exact order equalities (as $n → ∞$) for the best $n$-term approximations of $q$-ellipsoids in metrics of the spaces $S^p_{φ}$: We also consider the applications of the obtained results to the determination of the exact orders of the Kolmogorov widths of octahedra in the Hilbert space.

English version (Springer): Ukrainian Mathematical Journal 61 (2009), no. 10, pp 1649-1671.

Citation Example: Shydlich A. L. Order equalities for some functionals and their application to the estimation of the best $n$-term approximations and widths // Ukr. Mat. Zh. - 2009. - 61, № 10. - pp. 1403-1423.

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