×

Continuous algorithms in \(n\)-term approximation and nonlinear widths. (English) Zbl 0951.41011

Let \(X \) be a quasi-normed linear space and \(\Phi = \{\phi_k\}_{k=1}^{\infty} \) a family of elements in \(X\). If the family \(\Phi \) is bounded, i.e., \(\|\phi_k\|\leq C\), \(k=1,2,\dots\), and the span of \(\Phi \) is dense in \(X\), then \(\Phi \) is called a dictionary. Denote by \(M_n(\Phi)\) the set of all linear combinations \(\phi =\sum_{k\in Q}a_k\phi _k\), where \(Q\) is a set of natural numbers with \(|Q|=n\). If \(W\) is a subset \(X\) then the quantity \(\sigma_n(W,\Phi ,X) = \sup_{f\in W}\inf_{\phi \in M_n(\Phi)} \|f-\phi \|\) is called the \(n\)-term approximation of \(W\) by the family \(\Phi \). In this paper the author considers optimal continuous algorithms in \(n\)-term approximation based on various non-linear \(n\)-widths, and the \(n\)-term approximation \(\sigma_n(W,V,X)\) by the dictionary \(V\) formed from the integer translates of the mixed dyadic scales of the tensor product multivariate de la Vallée Poussin kernel, for the unit ball of Sobolev and Besov spaces of functions with a common mixed smoothness. The asymptotic orders of these quantities are given. These orders are achieved by a continuous algorithm of \(n\)-term approximation by \(V\), which is explicitly constructed.

MSC:

41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
41A25 Rate of convergence, degree of approximation
41-02 Research exposition (monographs, survey articles) pertaining to approximations and expansions
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DeVore, R., Nonlinear approximation, Acta Numerica, 7, 51-150 (1998) · Zbl 0931.65007
[2] DeVore, R.; Howard, R.; Micchelli, C., Optimal non-linear approximation, Manuscripta Math., 63, 469-478 (1989)
[3] DeVore, R.; Kyriazis, G.; Leviatan, D.; Tikhomirov, V., Wavelet compression and non-linear \(n\)-widths, Adv. Comp. Math., 1, 194-214 (1993) · Zbl 0824.65145
[4] Dung, Dinh, Approximation by trigonometric polynomials of functions of several variables on the torus, Mat. Sb., 131, 251-267 (1986) · Zbl 0634.42005
[5] Dung, Dinh, On optimal recovery of multivariate periodic functions, (Igari, S., Harmonic Analysis (1991), Springer-Verlag: Springer-Verlag Tokyo/Berlin/Heidelberg), 96-105 · Zbl 0792.42008
[6] Dung, Dinh, Optimal non-linear approximation of functions with a mixed smoothness, East J. Approx., 4, 101-112 (1998)
[7] Dung, Dinh, On nonlinear \(n\)-widths and \(n\)-term approximation, Vietnam J. Math., 26, 165-176 (1998) · Zbl 0921.46027
[8] Dung, Dinh, On best continuous methods in \(n\)-term approximation, Vietnam J. Math., 26, 367-371 (1998) · Zbl 0964.41015
[9] Dung, Dinh; Thanh, Vu Quoc, On non-linear \(n\)-widths, Proc. Amer. Math. Soc., 124, 2757-2765 (1996) · Zbl 0863.41016
[10] Kashin, B.; Temlyakov, V., On best \(m\)-term approximation and the entropy of sets in the space \(L^1\), Mat. Zametki, 56, 57-86 (1994) · Zbl 0836.41008
[11] Mathé, P., \(s\)-Number in information-based complexity, J. Complexity, 6, 41-66 (1990) · Zbl 0723.68047
[12] Nikol’skii, S., Approximation of Functions of Several Variables and Embedding Theorems (1975), Springer-Verlag: Springer-Verlag Berlin
[13] Ratsaby, J.; Maiorov, V., On the value of partial information for learning from examples, J. of Complexity, 13, 509-544 (1997) · Zbl 0894.68063
[14] Stesin, M., Alexandroff widths of finite dimensional sets and classes of smooth functions, Dokl. Akad. Nauk SSSR, 220, 38-41 (1974)
[15] Temlyakov, V., Approximation of functions with bounded mixed derivatives, Trudy Mat. Inst. Steklov, 178, 1-112 (1986)
[16] Temlyakov, V., Nonlinear Kolmogorov’s widths, Mat. Zametki, 63, 891-902 (1998)
[17] V. Temlyakov, Greedy algorithms with regard to the multivariate systems with a special structure, Constr. Approx, to appear.; V. Temlyakov, Greedy algorithms with regard to the multivariate systems with a special structure, Constr. Approx, to appear. · Zbl 0962.41007
[18] Tikhomirov, V., Some Topics in Approximation Theory (1976), Moscow State Univ: Moscow State Univ Moscow
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.