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Simultaneous approximation by greedy algorithms


A 2003 Preprint by D. Leviatan and V. Temlyakov

  • 2003:02
  • We study nonlinear m-term approximation with regard to a redundant dictionary D in a Hilbert space H. It is known that the Pure Greedy Algorithm (or, more generally, the Weak Greedy Algorithm) provides for each $f\in{H}$ and any dictionary D an expansion into a series

    $$f=\sum _ {j=1}^{\infty}c _ j(f)\varphi _ j(f),\ \ \ \ \ \varphi _ j(f)\in{D},\ \ \ \ \ j=1,2,\ldots$$

    with the Parseval property: $\parallel f \parallel = \sum _ j \mid {c} _ j(f) \mid^2$. Following the paper of A. Lutoborski and the second author we study analogs of the above expansions for a given finite number of function $f^1,\ldots,f^N$ with a requirement that the dictionary elements $\varphi _ j$ of these expansions are the same for all $f^i,i=1,\ldots,N$. We study convergence and rate of convergence of such expansions which we call simultaneous expansions.

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