Abstract
We study various approximation classes associated
with m-term approximation by elements from a (possibly) redundant dictionary in
a Banach space. The standard approximation class associated with the best m-term
approximation is compared to new classes defined by considering m-term
approximation with algorithmic constraints: thresholding and Chebychev
approximation classes are studied, respectively. We consider embeddings of the
Jackson type (direct estimates) of sparsity spaces into the mentioned
approximation classes. General direct estimates are based on the geometry of the
Banach space, and we prove that assuming a certain structure of the dictionary
is sufficient and (almost) necessary to obtain stronger results. We give
examples of classical dictionaries in L^p spaces and modulation spaces where our
results recover some known Jackson type estimates, and discuss som new estimates
they provide.
Original language | English |
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Journal | Journal of Fourier Analysis and Applications |
Volume | 10 |
Issue number | 1 |
Pages (from-to) | 51-71 |
ISSN | 1069-5869 |
Publication status | Published - 2004 |