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Sequences of m-term deviations in Hilbert spaceстатья
- Авторы: Borodin Petr A., Kopecká Eva
- Журнал: Journal of Approximation Theory
- Том: 284
- Год издания: 2022
- Издательство: Academic Press
- Местоположение издательства: United States
- Номер статьи: 105821
- DOI: 10.1016/j.jat.2022.105821
- Аннотация: Let $D$ be a dictionary in a Hilbert space $H$, that is, a set of unit elements whose linear combinations are dense in $H$. We consider the least $m$-term deviation $\sigma_m(x)$ of an element $x\in H$: this is the distance from $x$ to the set of all $m$-term linear combinations of elements of $D$. We prove a dichotomy result: for any dictionary $D$, either the sequence $\{\sigma_m(x)\}_{m=0}^{\infty}$ decreases exponentially for every $x\in H$, or the rate of convergence $\sigma_m(x)\to 0$ can be arbitrarily slow. We seek universal dictionaries realizing all strictly decreasing null sequences as sequences of $m$-term deviations. All commonly used dictionaries turn out not to be universal. In particular, the least rational deviations in Hardy space $H^2$ do not form certain strictly monotone null sequences.There are no universal dictionaries in finite dimensional Hilbert spaces. We construct a universal dictionary in every infinite dimensional Hilbert space.
- Добавил в систему: Бородин Петр Анатольевич