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:	Valiullin Artur R., Valiullin Albert R.
Formato:	article
Lenguaje:	EN
Publicado:	De Gruyter 2021
Materias:	greedy expansion prescribed coefficients hilbert space greedy approximation convergence 41-xx 41a58 Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/34a04b82f1fd4e4aadd6e1369cce964a
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spelling	oai:doaj.org-article:34a04b82f1fd4e4aadd6e1369cce964a2021-12-05T14:10:52ZSharp conditions for the convergence of greedy expansions with prescribed coefficients2391-545510.1515/math-2021-0006https://doaj.org/article/34a04b82f1fd4e4aadd6e1369cce964a2021-02-01T00:00:00Zhttps://doi.org/10.1515/math-2021-0006https://doaj.org/toc/2391-5455Greedy 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}}.Valiullin Artur R.Valiullin Albert R.De Gruyterarticlegreedy expansionprescribed coefficientshilbert spacegreedy approximationconvergence41-xx41a58MathematicsQA1-939ENOpen Mathematics, Vol 19, Iss 1, Pp 1-10 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	greedy expansion prescribed coefficients hilbert space greedy approximation convergence 41-xx 41a58 Mathematics QA1-939
spellingShingle	greedy expansion prescribed coefficients hilbert space greedy approximation convergence 41-xx 41a58 Mathematics QA1-939 Valiullin Artur R. Valiullin Albert R. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
description	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}}.
format	article
author	Valiullin Artur R. Valiullin Albert R.
author_facet	Valiullin Artur R. Valiullin Albert R.
author_sort	Valiullin Artur R.
title	Sharp conditions for the convergence of greedy expansions with prescribed coefficients
title_short	Sharp conditions for the convergence of greedy expansions with prescribed coefficients
title_full	Sharp conditions for the convergence of greedy expansions with prescribed coefficients
title_fullStr	Sharp conditions for the convergence of greedy expansions with prescribed coefficients
title_full_unstemmed	Sharp conditions for the convergence of greedy expansions with prescribed coefficients
title_sort	sharp conditions for the convergence of greedy expansions with prescribed coefficients
publisher	De Gruyter
publishDate	2021
url	https://doaj.org/article/34a04b82f1fd4e4aadd6e1369cce964a
work_keys_str_mv	AT valiullinarturr sharpconditionsfortheconvergenceofgreedyexpansionswithprescribedcoefficients AT valiullinalbertr sharpconditionsfortheconvergenceofgreedyexpansionswithprescribedcoefficients
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Sharp conditions for the convergence of greedy expansions with prescribed coefficients

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