Geodesic
Greedy expansions in Hilbert spaces
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Orthogonal series, approximation theory, and related problems, Tome 280 (2013), pp. 234-246.

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We study the rate of convergence of expansions of elements in a Hilbert space $H$ into series with regard to a given dictionary $\mathcal D$. The primary goal of this paper is to study representations of an element $f\in H$ by a series $f\sim\sum_{j=1}^\infty c_j(f)g_j(f)$, $g_j(f)\in\mathcal D$. Such a representation involves two sequences: $\{g_j(f)\}_{j=1}^\infty$ and $\{c_j(f)\}_{j=1}^\infty$. In this paper the construction of $\{g_j(f)\}_{j=1}^\infty$ 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 $f\in A_1(\mathcal D)$?" Previously it was believed that the rate of convergence was slower than $m^{-\frac14}$. The qualitative result of this paper is that the best possible rate of convergence of greedy expansions for $f\in A_1(\mathcal D)$ is faster than $m^{-\frac14}$. In fact, we prove it is faster than $m^{-\frac27}$. 
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     title = {Greedy expansions in {Hilbert} spaces},
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}
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J. L. Nelson; V. N. Temlyakov. Greedy expansions in Hilbert spaces. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Orthogonal series, approximation theory, and related problems, Tome 280 (2013), pp. 234-246. http://geodesic.mathdoc.fr/item/TM_2013_280_a15/

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