Abstract:
In the paper the following problems are considered: 1) behavior of the geometric characteristics of compact operators on interpolation of abstract Banach spaces ($\varepsilon$-entropy, Kolmogorov and Gel'fand diameters); 2) evaluation or estimation of the order of $\varepsilon$-entropy and diameters for the unit ball of a function space of Sobolev–Besov type as a compact set in another function space of that type. The spaces under examination are nonweighted anisotropic spaces as well as nonweighted and weighted isotropic spaces.
Bibliography: 19 titles.
Citation:
H. Triebel, “Interpolation properties of $\varepsilon$-entropy and diameters. Geometric characteristics of imbedding for function spaces of Sobolev–Besov type”, Math. USSR-Sb., 27:1 (1975), 23–37
\Bibitem{Tri75}
\by H.~Triebel
\paper Interpolation properties of $\varepsilon$-entropy and diameters. Geometric charac\-teristics of imbedding for function spaces of Sobolev--Besov type
\jour Math. USSR-Sb.
\yr 1975
\vol 27
\issue 1
\pages 23--37
\mathnet{http://mi.mathnet.ru/eng/sm3668}
\crossref{https://doi.org/10.1070/SM1975v027n01ABEH002496}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=412801}
\zmath{https://zbmath.org/?q=an:0312.46043}
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This publication is cited in the following 14 articles:
A. A. Vasil'eva, “Kolmogorov widths of intersections of weighted Sobolev classes on an interval with conditions on the zeroth and first derivatives”, Izv. Math., 85:1 (2021), 1–23
Vasil'eva A.A., “Kolmogorov Widths of Weighted Sobolev Classes on a Multi-Dimensional Domain With Conditions on the Derivatives of Order R and Zero”, J. Approx. Theory, 269 (2021), 105602
A. A. Vasil'eva, “Linear Widths of Weighted Sobolev Classes with Conditions on the Highest Order and Zero Derivatives”, Russ. J. Math. Phys., 27:4 (2020), 537
R.L.B. Stabile, S.A. Tozoni, “Estimates for entropy numbers of sets of smooth functions on the torus Td”, Journal of Approximation Theory, 235 (2018), 92
M. G. Nasyrova, E. P. Ushakova, “Hardy–Steklov operators and Sobolev-type embedding inequalities”, Proc. Steklov Inst. Math., 293 (2016), 228–254
Romanyuk A.S., “Estimation of the Entropy Numbers and Kolmogorov Widths for the Nikol'skii–Besov Classes of Periodic Functions of Many Variables”, Ukr. Math. J., 67:11 (2016), 1739–1757
Vasil'eva A.A., “Embedding Theorem for Weighted Sobolev Classes with Weights That Are Functions of the Distance to Some H-Set”, Russ. J. Math. Phys., 21:1 (2014), 112–122
Vasil'eva A.A., “Embedding Theorem for Weighted Sobolev Classes on a John Domain with Weights That Are Functions of the Distance to Some H-Set”, Russ. J. Math. Phys., 20:3 (2013), 360–373
A. K. Kushpel, J. Levesley, S. A. Tozoni, “Estimates of n -widths of Besov classes on two-point homogeneous manifolds”, Math Nachr, 282:5 (2009), 748
Yang Y., “Mixing Strategies for Density Estimation”, Ann. Stat., 28:1 (2000), 75–87
Yang Y., Barron A., “Information-Theoretic Determination of Minimax Rates of Convergence”, Ann. Stat., 27:5 (1999), 1564–1599
L. D. Kudryavtsev, S. M. Nikol'skiǐ, Encyclopaedia of Mathematical Sciences, 26, Analysis III, 1991, 1
Carl B., “Entropy Numbers of Diagonal Operators with an Application to Eigenvalue Problems”, J. Approx. Theory, 32:2 (1981), 135–150
Carl B., “Entropy Numbers of Embedding Maps Between Besov-Spaces with an Application to Eigenvalue Problems”, Proc. R. Soc. Edinb. Sect. A-Math., 90:1-2 (1981), 63–70
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