Sharp conditions for the convergence of greedy expansions with prescribed coefficients
Greedy expansions with prescribed coefficients were introduced by V. N. Temlyakov in a general case of Banach spaces. In contrast to Fourier series expansions, in greedy expansions with prescribed coefficients, a sequence of coefficients {cn}n=1∞{\left\{{c}_{n}\right\}}_{n=1}^{\infty } is fixed in a...
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| Autores principales: | , |
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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
De Gruyter
2021
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/34a04b82f1fd4e4aadd6e1369cce964a |
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| Sumario: | Greedy expansions with prescribed coefficients were introduced by V. N. Temlyakov in a general case of Banach spaces. In contrast to Fourier series expansions, in greedy expansions with prescribed coefficients, a sequence of coefficients {cn}n=1∞{\left\{{c}_{n}\right\}}_{n=1}^{\infty } is fixed in advance and does not depend on an expanded element. During the expansion, only expanding elements are constructed (or, more precisely, selected from a predefined set – a dictionary). For symmetric dictionaries, V. N. Temlyakov obtained conditions on a sequence of coefficients sufficient for a convergence of a greedy expansion with these coefficients to an expanded element. In case of a Hilbert space these conditions take the form ∑n=1∞cn=∞{\sum }_{n=1}^{\infty }{c}_{n}=\infty and ∑n=1∞cn2<∞{\sum }_{n=1}^{\infty }{c}_{n}^{2}\lt \infty . In this paper, we study a possibility of relaxing the latter condition. More specifically, we show that the convergence is guaranteed for cn=o1n{c}_{n}=o\left(\frac{1}{\sqrt{n}}\right), but can be violated if cn≍1n{c}_{n}\hspace{0.33em}\asymp \hspace{0.33em}\frac{1}{\sqrt{n}}. |
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