Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T18:05:35.439Z Has data issue: false hasContentIssue false
  • English
  • Français

Polynomially additive entropies

Published online by Cambridge University Press:  14 July 2016

Bruce R. Ebanks
Bruce R. Ebanks*
Affiliation:
Texas Tech University
*
∗ Postal address: Department of Mathematics, Texas Tech University, Box 4319, Lubbock, TX 79409, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper introduces a property of entropies called polynomial additivity, which includes the additivity properties of the Shannon entropy and of entropies of all degrees. The forms of all such entropies with the sum property and range of infinite cardinality are found. Of these, the only ones with the measurable sum property are affine transformations of the Shannon entropy, the entropies of all degrees, and the length of the distribution.

Keywords

ADDITIVITY PROPERTIESSHANNON ENTROPYENTROPY OF DEGREE ALPHASUM PROPERTYMEASURABLE SUM PROPERTY
Type
Short Communications
Information
Journal of Applied Probability , Volume 21 , Issue 1 , March 1984 , pp. 179 - 185
Copyright
Copyright © Applied Probability Trust 1984 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aczel, J. (1966) Lectures on Functional Equations and their Applications. Academic Press, New York.Google Scholar
[2] Aczel, J. and Daróczy, Z. (1963) Characterisierung der Entropien positiver Ordnung und der Shannonschen Entropie. Acta Math. Acad. Sci. Hungar. 14, 95–121.Google Scholar
[3] Aczel, J. and Daróczy, Z. (1975) On Measures of Information and their Characterizations. Academic Press, New York.Google Scholar
[4] Behara, M. and Nath, P. (1973) Additive and non-additive entropies of finite measurable partitions. In Probability and Information Theory II, Lecture Notes in Mathematics 296, Springer-Verlag, Berlin, 102–138.Google Scholar
[5] Chaundy, T. W. and Mcleod, J. B. (1960) On a functional equation. Edinburgh Math. Notes 43, 7–8.Google Scholar
[6] Daróczy, Z. (1971) On the measurable solutions of a functional equation. Acta Math. Acad. Sci. Hungar. 22, 11–14.Google Scholar
[7] Daróczy, Z. and Járai, A. (1979) On measurable solutions of function equations. Acta Math. Acad. Sci. Hungar. 34, 105–116.CrossRefGoogle Scholar
[8] Kannappan, Pl. (1974) On a generalization of some measures in information theory. Glasnik Mat. 9(29), 81–93.Google Scholar
[9] Losonczi, L. (1981) A characterization of entropies of degree a. Metrika 28, 237–244.CrossRefGoogle Scholar
[10] Losonczi, L. and Maksa, Gy. (1982) On some functional equations of the information theory. Acta Math. Acad. Sci. Hungar. 39, 73–82.Google Scholar
[11] Maksa, Gy. (1981) On the bounded solution of a functional equation. Acta Math. Acad. Sci. Hungar. 37, 445–450.Google Scholar
[12] Mittal, D. P. (1970) On continuous solutions of a functional equation. Metrika 22, 31–40.Google Scholar