Abstract
The topic of the given commentary is directly related to the papers [1–5]. However, it is natural to consider a somewhat wider cycle in which we shall include [6–10]. The backbone bringing together the entire cycle of articles [1–10] is the notion of entropy.
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
A. N. Kolmogorov, Estimates of the minimal number of elements of e nets in various functional classes and their application to the question of representing functions of several variables as superpositions of functions of a lesser number of variables, Uspekhi Mat. Nauk 101 (1955), 192–194.
A. N. Kolmogorov, On the representation of continuous functions of several variables as superpositions of continuous functions of a lesser number of variables, Dokl. Akad. Nauk SSSR 1082 (1956), 179–182.
A. N. Kolmogorov, On certain asymptotic characteristics of completely bounded metric spaces, Dokl. Akad. Nauk SSSR 1083 (1956), 385–388.
A. N. Kolmogorov, V. M. Tikhomirov, c-entropy and c-capacity of sets in function spaces, Uspekhi Mat. Nauk 142 (1959), 2–86.
A. N. Kolmogorov, Various approaches to the estimate of the complexity of approximations and computations of functions, In: Proceedings of the Itern. Congr. Math. Stockholm (1963).
I. M. Gelfand. A. N. Kolmogorov, A. N. Yaglom, The amount of information and entropy for continuous distributions, Moscow, Academy of Sciences Publishes 3 (1958), 300–320.
A. N. Kolmogorov, The theory of transmission of information.,Plenary Session, Moscow, Academy of Sciences Publishers (1957), 66–99, The Session of the Academy of Sciences of the USSR on Scientific Problems of Industrial Automation.
A. N. Kolmogorov, A new metric invariant of transitive dynamical systems and automorphisms of Lebesgue spaces, Dokl. Akad. Nauk SSSR 119 (1958), 861–864.
A. N. Kolmogorov, On the linear dimension of topological vector spaces,Dokl. Akad. Nauk SSSR 120 (1958), 239–241.
A. N. Kolmogorov, On the entropy in unit time as a metric invariant of automorphisms, Dokl. Akad. Nauk SSSR 124 (1959), 754–755.
V. N. Tikhomirov, The work of A.N. Kolmogorov on c-entropy of functional classes and superpositions of functions, Uspekhi Mat. Nauk 185 (1963), 55–92.
V. N. Tikhomirov, Cross-sectional measures and entropy, Uspekhi Mat. Nauk 384 (1983), 91–99.
Shannon C. E., A mathematical theory of communication, Pt I, II.–Bell Syst. Techn. J. 274 (1948), 623–656.
Renyi A., On the dimension and entropy of probability distributions, Acta math. Acad. Sci. Hung. 101 (1959), 193–215.
Posner E. C., Rodemish E. R., Rumsey H.G., c-entropy of stochastic processes, Ann. Math. Statist. 384 (1967), 1000–1020.
V. M. Tikhomirov, Certain questions of approximation theory, Moscow, Moscow University Press, 1976.
V. G. Boltyansky, I. Ts. Gokhberg, The partition of figures into smaller parts, Moscow, Nauka, 1971.
G. G. Kazaryan, An inequality between derivatives and differential operators and certain of its application to partial differential equations, Doctoral Dissertation (1980), Yerevan.
C. A. Rogers, Packing and covering, Cambridge Univ. Press, 1964.
V. M. Sidelnikov, On the best paving of balls on the surface of a n-dimensional Euclidean sphere and the number of vectors of a binary code with given code distance, Dokl. Akad. Nauk SSSR 2135 (1973), 1029–1032.
V. M. Levenshtein, On the maximal density of fillings of n-dimensional space by different balls, Mat. Zametki 182 (1975), 301–311.
M. Sh. Birman, M. Z. Solomyak, Piece-wise polynomial approximations of functions of classes Wp, M.tem. Sbornik 733 (1977), 331–335.
H. Widom, Rational approximation and n-dimensional diameter, J. Approxim. Theory 51 (1972), 343–361.
L. B. Sofman, Shannon’s problem, Dokl. Akad. Nauk SSSR 2156 (1974), 1313–1316.
A. F. Timan, On the order of growth of c-entropy of spaces of real continuous functionals defined on a connected set, Uspekhi Mat. Nauk 191 (1964), 173–177
A. F. Timan, Funktsional Anal. I Prilozhen. 53 (1971), 104–105
A. F. Timan, Dokl. Akad. Nauk SSSR 2183 (1974), 505–507.
Editor information
Rights and permissions
Copyright information
© 1993 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Tikhomirov, V.M. (1993). ε-Entropy and ε-Capacity. In: Shiryayev, A.N. (eds) Selected Works of A. N. Kolmogorov. Mathematics and Its Applications, vol 27. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2973-4_17
Download citation
DOI: https://doi.org/10.1007/978-94-017-2973-4_17
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-8456-9
Online ISBN: 978-94-017-2973-4
eBook Packages: Springer Book Archive