naa a22 4500 51081171X CHVBK 20180411083445.0 cr unu---uuuuu 180411e20130401xx s 000 0 eng 10.1134/S0081543813010161 doi (NATIONALLICENCE)springer-10.1134/S0081543813010161 Greedy expansions in Hilbert spaces [Elektronische Daten] [J. Nelson, V. Temlyakov] We study the rate of convergence of expansions of elements in a Hilbert space H into series with regard to a given dictionary D. The primary goal of this paper is to study representations of an element f ∈ H by a series f ∼ ∑ j=1 ∞ c j (f)g j (f), $$g_j \left( f \right) \in \mathcal{D}$$ . Such a representation involves two sequences: {g j (f)} j=1 ∞ and {c j (f) j=1 ∞ . In this paper the construction of {g j (f)} j=1 ∞ is based on ideas used in greedy-type nonlinear approximation, hence the use of the term greedy expansion. An interesting open problem questions, "What is the best possible rate of convergence of greedy expansions for f ∈ A 1(D)?” Previously it was believed that the rate of convergence was slower than $$m^{ - \tfrac{1} {4}}$$ . The qualitative result of this paper is that the best possible rate of convergence of greedy expansions for $$f \in A_1 \left( \mathcal{D} \right)$$ is faster than $$m^{ - \tfrac{1} {4}}$$ . In fact, we prove it is faster than $$m^{ - \tfrac{2} {7}}$$ . Pleiades Publishing, Ltd., 2013 Nelson J. Mathematics Department, University of South Carolina, 1523 Greene Street, 29208, Columbia, SC, USA aut Temlyakov V. Mathematics Department, University of South Carolina, 1523 Greene Street, 29208, Columbia, SC, USA aut Proceedings of the Steklov Institute of Mathematics SP MAIK Nauka/Interperiodica 280/1(2013-04-01), 227-239 0081-5438 280:1<227 2013 280 11501 https://doi.org/10.1134/S0081543813010161 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1134/S0081543813010161 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Nelson J. Mathematics Department, University of South Carolina, 1523 Greene Street, 29208, Columbia, SC, USA aut NATIONALLICENCE 700 1- Temlyakov V. Mathematics Department, University of South Carolina, 1523 Greene Street, 29208, Columbia, SC, USA aut NATIONALLICENCE 773 0- Proceedings of the Steklov Institute of Mathematics SP MAIK Nauka/Interperiodica 280/1(2013-04-01), 227-239 0081-5438 280:1<227 2013 280 11501 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer