Abstract.
The question of finding an optimal dictionary for nonlinear m -term approximation is studied in this paper. We consider this problem in the periodic multivariate (d variables) case for classes of functions with mixed smoothness. We prove that the well-known dictionary U d which consists of trigonometric polynomials (shifts of the Dirichlet kernels) is nearly optimal among orthonormal dictionaries. Next, it is established that for these classes near-best m -term approximation, with regard to U d , can be achieved by simple greedy-type (thresholding-type) algorithms.
The univariate dictionary U is used to construct a dictionary which is optimal among dictionaries with the tensor product structure.
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June 22, 1998. Date revised: March 26, 1999. Date accepted: March 22, 1999.
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Temlyakov, V. Greedy Algorithms with Regard to Multivariate Systems with Special Structure. Constr. Approx. 16, 399–425 (2000). https://doi.org/10.1007/s003659910017
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DOI: https://doi.org/10.1007/s003659910017