Duan, Liqin The best \(m\)-term approximations on generalized Besov classes \(M\, B_{q, \theta}^{\Omega}\) with regard to orthogonal dictionaries. (English) Zbl 1209.41007 J. Approx. Theory 162, No. 11, 1964-1981 (2010). The setting of this paper is as follows: Let \({\mathcal D}\) be a dictionary in a Banach space \(X\), (i.e., a system of functions with domain \(X\)), and let the best \(m\)-term approximation with regard to \({\mathcal D}\) be defined by \[ \sigma_n(f, {\mathcal D})_X:=\inf \left\{ \left\| f - \sum_{i=1}^m c_i g_i \right\|\right\}, \] where the infimum is taken over the set of all possible linear combinations of \(m\) functions \(g_i\) in \({\mathcal D}\). For a function class \(F \subset X\) and a collection \(\mathbf D\) of dictionaries, the quantity \(\sigma_n(F, {\mathbf D})_X:=\inf_{{\mathcal D} \in {\mathbf D}} \sup_{f \in F} \sigma_n(f, {\mathcal D})_X\) is the lowest bound of the best \(m\)-term approximation of a given function class \(F\) with regard to any dictionary \({\mathcal D} \in {\mathbf D}\).In this paper, the author investigates \(m\)-term approximation with regard to orthogonal dictionaries. He considers the problem in the periodic multivariate case for generalized Besov classes \(M B^{\Omega}_{q, \theta}\) under the condition \(\Omega(t)=\omega(t_1 \cdot \cdots \cdot t_d)\), where \(\omega(t) \in {\mathbf \Psi}^{\ast}_{\ell}\) is a univariate function, and proves that the well-known dictionary \(U^d\) which consists of trigonometric polynomials (shifts of the Dirichlet kernels) is nearly optimal among orthogonal dictionaries. Moreover, he establishes that for these classes near-best \(m\)-term approximation, with regard to \(U^d\), can be achieved by simple greedy-type algorithms.For related work see e.g. P. Oswald [J. Fourier Anal. Appl. 7, No. 4, 325–341 (2001; Zbl 1003.41020)] Reviewer: Richard A. Zalik (Auburn University) Cited in 5 Documents MSC: 41A50 Best approximation, Chebyshev systems 41A46 Approximation by arbitrary nonlinear expressions; widths and entropy 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems Keywords:best \(m\)-term approximation; orthogonal dictionary; generalized Besov class; greedy algorithm Citations:Zbl 1003.41020 PDFBibTeX XMLCite \textit{L. Duan}, J. Approx. Theory 162, No. 11, 1964--1981 (2010; Zbl 1209.41007) Full Text: DOI References: [1] Amanov, T. I., Representation and imbedding theorems for the function spaces \(S_{p, \theta}^r B(R_n), S_{p, \theta}^{\ast r} B(0 \leq x_j \leq 2 \pi, j = 1, \ldots, n)\), Trudy Mat. Inst. 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