Abstract
We suggest a three-step strategy to find a good basis (dictionary) for non-linear m-term approximation. The first step consists of solving an optimization problem of finding a near best basis for a given function class F, when we optimize over a collection D of bases (dictionaries). The second step is devoted to finding a universal basis (dictionary) D u ∈ D for a given pair (F, D) of collections: F of function classes and D of bases (dictionaries). This means that Du provides near optimal approximation for each class F from a collection F. The third step deals with constructing a theoretical algorithm that realizes near best m-term approximation with regard to D u for function classes from F.
In this paper we work this strategy out in the model case of anisotropic function classes and the set of orthogonal bases. The results are positive. We construct a natural tensor-product-wavelet-type basis and prove that it is universal. Moreover, we prove that a greedy algorithm realizes near best m-term approximation with regard to this basis for all anisotropic function classes.
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Communicated by Ronald A. DeVore.
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Temlyakov, V.N. Universal bases and greedy algorithms for anisotropic function classes. Constr. Approx. 18, 529–550 (2002). https://doi.org/10.1007/s00365-002-0514-1
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DOI: https://doi.org/10.1007/s00365-002-0514-1