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Czasopismo

1992 | 102 | 3 | 277-302

Tytuł artykułu

ε-Entropy and moduli of smoothness in $L^{p}$-spaces

Autorzy

Treść / Zawartość

Języki publikacji

EN

Abstrakty

EN
The asymptotic behaviour of ε-entropy of classes of Lipschitz functions in $L^p(𝕀^d)$ is obtained. Moreover, the asymptotics of ε-entropy of classes of Lipschitz functions in $L^p(ℝ^d)$ 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 $ℝ^d$ and the minimax risk for that class is discussed.

Twórcy

autor
  • Institute of Mathematics, Polish Academy of Sciences, Abrahama 18, 81-825 Sopot, Poland

Bibliografia

  • [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.
  • [2] Z. Ciesielski, Properties of the orthonormal Franklin system, II, Studia Math. 27 (1966), 289-323.
  • [3] Z. Ciesielski, Asymptotic nonparametric spline density estimation in several variables, in: Internat. Ser. Numer. Math. 94, Birkhäuser, Basel 1990, 25-53.
  • [4] Z. Ciesielski and T. Figiel, Spline approximation and Besov spaces on compact manifolds, Studia Math. 75 (1982), 13-36.
  • [5] Z. Ciesielski and T. Figiel, Spline bases in classical function spaces on compact $C^∞$ manifolds, Part II, ibid. 76 (1983), 95-136.
  • [6] L. Devroye and L. Györfi, Nonparametric Density Estimation. The L₁ View, Wiley, New York 1985.
  • [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.
  • [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.
  • [9] G. G. Lorentz, Metric entropy and approximation, Bull. Amer. Math. Soc. 72 (1966), 903-937.
  • [10] I. J. Schoenberg, Cardinal interpolation and spline functions, J. Approx. Theory 2 (1969), 167-206.

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