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Entropies of Sets of Functions of Bounded Variation

Published online by Cambridge University Press:  20 November 2018

G. F. Clements
G. F. Clements*
Affiliation:
Syracuse University
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In this paper the entropies of several sets of functions of bounded variation are calculated. The entropy of a metric set, a notion first introduced by Kolmogorov in (2), is a measure of its size in terms of the minimal number of sets of diameter not exceeding 2∊ necessary to cover it. Using this notion, Kolmogorov (4; p. 357) and Vituškin (7) have shown that not all functions of n variables can be represented by functions of fewer variables if only functions satisfying certain smoothness conditions are allowed.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 15 , 1963 , pp. 422 - 432
Copyright
Copyright © Canadian Mathematical Society 1963

References

1. Hausdorff, F., Set theory (New York, 1957).Google Scholar
2. Kolmogorov, A. N., Asymptotic characteristics of some completely bounded metric spaces, Dokl. Akad. Nauk. S.S.S.R., 108 (1956), 585–589 (Russian).Google Scholar
3. Kolmogorov, A. N. and Tihomirov, V. M., e-entropy and e-capacity of sets in function spaces, Uspehi Mat. Nauk, 14 (1959), no. 2 (86), 3–86 (Russian).Google Scholar
4. Kolmogorov, A. N. and Tihomirov, V. M., e-entropy and e-capacity of sets in functional spaces, Amer. Math. Soc. Translations, 17, 2, 277-364. (English translation of 3.)Google Scholar
5. Lorentz, G. G., Metric entropies, widths and superpositions of functions, Amer. Math. Monthly, 69 (1962), 469–485.Google Scholar
6. Riordan, J., An introduction to combinatorial analysis (New York, 1958).Google Scholar
7. Vituskin, A. G., On the 13th problem of Hilbert, Dokl. Akad. Nauk S.S.S.R., 95 (1954), 701–704 (Russian).Google Scholar
8. Vituskin, A. G., Estimation of the complexity of the tabulation problem (Moscow, 1959, Russian).Google Scholar
9. Vituskin, A. G., Theory of the transmission and processing of information (New York, 1961) (English translation of 8.)-Google Scholar