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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 

 

Metric entropy and approximation


Author: G. G. Lorentz
Journal: Bull. Amer. Math. Soc. 72 (1966), 903-937
MathSciNet review: 0203320
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  • 1. N. I. Achieser, Theory of approximation, Translated by Charles J. Hyman, Frederick Ungar Publishing Co., New York, 1956. MR 0095369
  • 2. S. Ja. Al′per, On 𝜖-entropy of certain classes of functions, Soviet Math. Dokl. 1 (1960), 667–669. MR 0126697
  • 3. T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, Springer-Verlag, Berlin-New York, 1974 (German). Berichtigter Reprint. MR 0344997
  • 4. Ju. A. Brudnyĭ and B. D. Kotljar, The order of growth of 𝜖-entropy on certain classes of functions, Dokl. Akad. Nauk SSSR 148 (1963), 1001–1004 (Russian). MR 0146580
  • 5. Ju. A. Brudnyĭ and A. F. Timan, Constructive charateristics of compact sets in Banach spaces and 𝜖-entropty, Dokl. Akad. Nauk SSSR 126 (1959), 927–930 (Russian). MR 0112033
  • 6. G. F. Clements, Entropies of sets of functions of bounded variation, Canad. J. Math. 15 (1963), 422–432. MR 0158969
  • 7. G. F. Clements, Entropies of several sets of real valued functions, Pacific J. Math. 13 (1963), 1085–1095. MR 0158970
  • 8. Philip J. Davis, Interpolation and approximation, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1963. MR 0157156
  • 9. V. K. Dzyadyk, On a problem of S. M. Nikol′skiĭ in a complex region, Izv. Akad. Nauk SSSR. Ser. Mat. 23 (1959), 697–736 (Russian). MR 0109982
  • 10. V. K. Dzjadik, On the theory of approximation of continuous functions in closed regions and on a problem of S. M. Nikol′skiĭ, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), 1135–1164 (Russian). MR 0154995
  • 11. V. D. Erohin, On conformal transformations of rings and the fundamental basis of the space of functions analytic in an elementary neighbourhood of an arbitrary continuum, Dokl. Akad. Nauk SSSR 120 (1958), 689–692 (Russian). MR 0102743
  • 12. V. D. Erohin, Asymptotic theory of the 𝜖-entropy of analytic functions, Dokl. Akad. Nauk SSSR 120 (1958), 949–952 (Russian). MR 0102744
  • 13. K. K. Golovkin, The 𝜖-entropy of certain compact sets of differentiable functions in spaces with monotone norm, Dokl. Akad. Nauk SSSR 158 (1964), 261–263 (Russian). MR 0169036
  • 14. A. Ja. Helemskiĭ and G. M. Henkin, Embeddings of compacta into ellipsoids, Vestnik Moskov. Univ. Ser. I Mat. Meh. 1963 (1963), no. 2, 3–12 (Russian, with English summary). MR 0157212
  • 15. G. M. Henkin, Linear superpositions of continuously differentiable functions, Dokl. Akad. Nauk SSSR 157 (1964), 288–290 (Russian). MR 0166319
  • 16. A. N. Kolmogorov, On certain asymptotic characteristics of completely bounded metric spaces, Dokl. Akad. Nauk SSSR (N.S.) 108 (1956), 385–388 (Russian). MR 0080904
  • 17. A. N. Kolmogorov, On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition, Dokl. Akad. Nauk SSSR 114 (1957), 953–956 (Russian). MR 0111809
  • 18. A. N. Kolmogorov, On linear dimensionality of topological vector spaces, Dokl. Akad. Nauk SSSR 120 (1958), 239–241 (Russian). MR 0097704
  • 19. A. N. Kolmogorov and V. M. Tihomirov, 𝜖-entropy and 𝜖-capacity of sets in function spaces, Uspehi Mat. Nauk 14 (1959), no. 2 (86), 3–86 (Russian). MR 0112032
  • 20. B. D. Kotljar, The order of growth of 𝜖-entropy on the class of quasi-smooth functions, Uspehi Mat. Nauk 18 (1963), no. 2 (110), 135–138 (Russian). MR 0151563
  • 21. G. G. Lorentz, Lower bounds for the degree of approximation, Trans. Amer. Math. Soc. 97 (1960), 25–34. MR 0117487, 10.1090/S0002-9947-1960-0117487-3
  • 22. G. G. Lorentz, Metric entropy, widths, and superpositions of functions, Amer. Math. Monthly 69 (1962), 469–485. MR 0141926
  • 23. G. G. Lorentz, Entropy and its applications, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 1 (1964), 97–103. MR 0179533
  • 24. G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York-Chicago, Ill.-Toronto, Ont., 1966. MR 0213785
  • 25. B. S. Mitjagin, Approximate dimension and bases in nuclear spaces, Uspehi Mat. Nauk 16 (1961), no. 4 (100), 63–132 (Russian). MR 0152865
  • 26. O. A. Oleĭnik, Estimates of the Betti numbers of real algebraic hypersurfaces, Mat. Sbornik N.S. 28(70) (1951), 635–640 (Russian). MR 0044864
  • 27. A. M. Olevskiĭ, On a problem of P. L. Ul′janov, Uspehi Mat. Nauk 20 (1965), no. 2 (122), 197–202 (Russian). MR 0184017
  • 28. A. Pełczyński, On the approximation of 𝑆-spaces by finite dimensional spaces, Bull. Acad. Polon. Sci. Cl. III. 5 (1957), 879–881, LXXV (English, with Russian summary). MR 0092114
  • 29. B. Penkov and Bl. Sendov, Entropy of the set of continuous functions of several variables, C. R. Acad. Bulgare Sci. 17 (1964), 335–337 (Russian). MR 0167766
  • 30. S. Rolewicz, On spaces of holomorphic functions, Studia Math. 21 (1961/1962), 135–160. MR 0154146
  • 31. Harold S. Shapiro, Some negative theorems of approximation theory, Michigan Math. J. 11 (1964), 211–217. MR 0167769
  • 32. S. A. Smoljak, 𝜖-entropy of the classes 𝐸_{𝑠^{𝛼,𝑘}}(𝐵) and 𝑊_{𝑠^{𝛼}}(𝐵) in the metric of 𝐿₂, Soviet Math. Dokl. 1 (1960), 192–196. MR 0123130
  • 33. V. M. Tihomirov, Diameters of sets in functional spaces and the theory of best approximations, Russian Math. Surveys 15 (1960), no. 3, 75–111. MR 0117489
  • 34. V. M. Tihomirov, The 𝜖-entropy of certain classes of periodic functions., Uspehi Mat. Nauk 17 (1962), no. 6 (108), 163–169 (Russian). MR 0144150
  • 35. A. F. Timan, Theory of approximation of functions of a real variable, Translated from the Russian by J. Berry. English translation edited and editorial preface by J. Cossar. International Series of Monographs in Pure and Applied Mathematics, Vol. 34, A Pergamon Press Book. The Macmillan Co., New York, 1963. MR 0192238
  • 36. A. F. Timan, The order of growth of 𝜖-entropy of spaces of real continuous functionals defined on a connected compactum, Uspehi Mat. Nauk 19 (1964), no. 1 (115), 173–177 (Russian). MR 0162909
  • 37. A. G. Vituškin, Theory of the transmission and processing of information, Translated from the Russian by Ruth Feinstein; translation editor A. D. Booth, Pergamon Press, New York-Oxford-London-Paris, 1961. MR 0132342
  • 38. A. G. Vituškin, Some properties of linear superpositions of smooth functions, Dokl. Akad. Nauk SSSR 156 (1964), 1003–1006 (Russian). MR 0162907
  • 39. A. G. Vituškin, A proof of the existence of analytic functions of several variables not representable by linear superpositions of continuously differentiable functions of fewer variables, Dokl. Akad. Nauk SSSR 156 (1964), 1258–1261 (Russian). MR 0165138
  • 40. A. C. Vosburg, Metric entropy of certain classes of Lipschitz functions, Proc. Amer. Math. Soc. 17 (1966), 665–669. MR 0193624, 10.1090/S0002-9939-1966-0193624-0


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9904-1966-11586-0