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Nonlinear approximation in finite-dimensional spaces. (English) Zbl 0892.41009

The authors study certain problems of nonlinear approximation which arise in image processing. They take a Banach space \(X\) and a subset \(D\) of \(X\) whose linear span is dense in \(X\), and they consider approximation in finite-dimensional Euclidean spaces equipped with various norms for studying the relationship between the size of \(D\) and its approximation power. Also, they show how to appropriately choose sets \(D\) for which the greedy algorithms achieve estimates similar to those of best \(m\)-term approximation.

MSC:

41A15 Spline approximation
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References:

[1] Davis, G.; Mallat, S.; Avallaneda, M., Adaptive greedy approximations, Constr. Approx., 13, 57-98 (1997) · Zbl 0885.41006
[2] Donahue, M. J.; Gurvits, L.; Darken, C.; Sontag, E., Rates of convex approximation in non-Hilbert spaces, Constr. Approx., 13, 187-220 (1997) · Zbl 0876.41016
[3] Barron, A., Universal approximation bounds for superposition of a sigmoidal function, IEEE Trans. Inform. Theory, 39, 930-945 (1993) · Zbl 0818.68126
[4] DeVore, R. A.; Temlyakov, V. N., Nonlinear approximation by trigonometric sums, J. Fourier Anal. Appl., 2, 29-48 (1995) · Zbl 0886.42019
[5] DeVore, R. A.; Temlyakov, V. N., Some remarks on greedy algorithms, Adv. Comput. Math., 5, 173-187 (1996) · Zbl 0857.65016
[6] Jones, L., A simple lemma on greedy approximation in Hilbert space and convergence rates for projection pursuit regression and neural network training, Ann. Stat., 20, 608-613 (1992) · Zbl 0746.62060
[7] Kashin, B. S.; Temlyakov, V. N., On best \(mL^1\), Math. Notes, 56, 1137-1157 (1994) · Zbl 0836.41008
[8] G. Pisier, Remarques sur un résultat non publié de B. Maurey, Séminaire de’analyse fonctionelle 1980-1981, École Polytechnique, Centre de Mathématiques, Palaiseau; G. Pisier, Remarques sur un résultat non publié de B. Maurey, Séminaire de’analyse fonctionelle 1980-1981, École Polytechnique, Centre de Mathématiques, Palaiseau
[9] Pisier, G., The Volume of Convex Bodies and Banach Space Geometry (1989), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0698.46008
[10] Schütt, C., Entropy numbers of diagonal operators between symmetric Banach spaces, J. Approx. Theory, 40, 121-128 (1984)
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