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

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Ukrainian Mathematical Journal Aims and scope

We study the behavior of functionals of the form

$$ \mathop {\sup }\limits_{l > n} \left( {l - n} \right){\left( {\sum\limits_{k = 1}^l {\frac{1}{{{\psi^r}(k)}}} } \right)^{{{ - 1} \mathord{\left/{\vphantom {{ - 1} r}} \right.} r}}}, $$

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.

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine

    A. L. Shydlich

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  1. A. L. Shydlich
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 10, pp. 1403–1423, October, 2009.

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Shydlich, A.L. Order equalities for some functionals and their application to the estimation of the best n-term approximations and widths. Ukr Math J 61, 1649–1671 (2009). https://doi.org/10.1007/s11253-010-0304-z

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  • DOI: https://doi.org/10.1007/s11253-010-0304-z

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