ε-Entropy and moduli of smoothness in Lp-spaces
Kamont, A.
Studia Mathematica, Tome 103 (1992), p. 277-302 / Harvested from The Polish Digital Mathematics Library
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The asymptotic behaviour of ε-entropy of classes of Lipschitz functions in Lp(d) is obtained. Moreover, the asymptotics of ε-entropy of classes of Lipschitz functions in Lp(ℝd) whose tail function decreases as O(λ-γ) is obtained. In case p = 1 the relation between the ε-entropy of a given class of probability densities on ℝd and the minimax risk for that class is discussed.

Publié le : 1992-01-01
EUDML-ID : urn:eudml:doc:215929 
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     title = {$\epsilon$-Entropy and moduli of smoothness in $L^{p}$-spaces},
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     year = {1992},
     pages = {277-302},
     zbl = {0810.41019},
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Kamont, A. ε-Entropy and moduli of smoothness in $L^{p}$-spaces. Studia Mathematica, Tome 103 (1992) pp. 277-302. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv102i3p277bwm/

[00000] [1] C. de Boor, Splines as linear combinations of B-splines, in: Approximation Theory II, G. G. Lorentz et al. (eds.), Academic Press, New York 1976, 1-47. | Zbl 0343.41011

[00001] [2] Z. Ciesielski, Properties of the orthonormal Franklin system, II, Studia Math. 27 (1966), 289-323. | Zbl 0148.04702

[00002] [3] Z. Ciesielski, Asymptotic nonparametric spline density estimation in several variables, in: Internat. Ser. Numer. Math. 94, Birkhäuser, Basel 1990, 25-53. | Zbl 0715.41015

[00003] [4] Z. Ciesielski and T. Figiel, Spline approximation and Besov spaces on compact manifolds, Studia Math. 75 (1982), 13-36. | Zbl 0601.41017

[00004] [5] Z. Ciesielski and T. Figiel, Spline bases in classical function spaces on compact C∞ manifolds, Part II, ibid. 76 (1983), 95-136.

[00005] [6] L. Devroye and L. Györfi, Nonparametric Density Estimation. The L₁ View, Wiley, New York 1985. | Zbl 0546.62015

[00006] [7] P. Groeneboom, Some current developments in density estimation, in: Mathematics and Computer Science, Proceedings of the CWI symposium, November 1983, J. W. de Bakker, M. Hazewinkel and J. K. Lenstra (eds.), North-Holland, 1986, 163-192.

[00007] [8] A. N. Kolmogorov and V. M. Tikhomirov, ε-Entropy and ε-capacity of sets in function spaces, Uspekhi Mat. Nauk 14 (2) (1959), 3-86 (in Russian); English transl.: Amer. Math. Soc. Transl. (2) 17 (1961), 277-364.

[00008] [9] G. G. Lorentz, Metric entropy and approximation, Bull. Amer. Math. Soc. 72 (1966), 903-937. | Zbl 0158.13603

[00009] [10] I. J. Schoenberg, Cardinal interpolation and spline functions, J. Approx. Theory 2 (1969), 167-206. | Zbl 0202.34803