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Дни анализа в Сириусе
28 октября 2022 г. 12:40–13:40, Сочи
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Sequences of rational deviations
P. A. Borodin Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
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Эта страница: | 90 |
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Аннотация:
A.A. Pekarskiĭ [1] proved that any strictly monotone sequence realizes as the sequence of the
least rational deviations in the space $C^{\mathbb C}[0,1]$ of complex continuous functions with
the uniform norm. It is not known whether a result of this sort is true for the space
$L_2^{\mathbb C}[0,1]$. However, it turned out that Euclidian norm in general does not allow
rational deviations to be arbitrary. In [2], we have shown that monotone sequences with large
jumps at the beginning cannot be realized as sequences of rational deviations in the Hardy space
$H^2(\Im z>0)$ in the upper half-plane. We consider this problem as a particular case of seeking
an element of a Hilbert space having prescribed $m$-term deviations with respect to a given
dictionary, which in turn is a variation of the well-known Bernstein lethargy problem.
References
[1] Pekarskiĭ A. A., “Existence of a function with given best uniform rational approximations.” Vestsī Akad. Navuk Belarusī Ser. Fīz. Mat. Navuk, 1999, (1): 23–26.
[2] Borodin P., Kopecká E.,
“Sequences of m-term deviations in Hilbert space.” J. Approx. Theory, 2022, 105821.
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