The general solutions of a functional equation related to information theory

Authors

  • Prem Nath Department of Mathematics, University of Delhi Delhi, 110007, INDIA
  • Dhiraj Kumar Singh Department of Mathematics, University of Delhi Delhi, 110007, INDIA

Keywords:

Functional equations, continuous solutions, Lebesgue measurable solutions, the Shannon entropy, the nonadditive entropy of degree ®, multiplicative functions, additive functions.

Abstract

The general solutions of a functional equation, containing two unknown functions, and related to a functional equation characterizing the Shannon entropy and the entropy of degree alpha, are obtained

References

J. Aczzel and Z. Daroczy, Characterisierung der Entropien positiver ordnung und der Shannonschen entropie, Acta Math. Acad. Sci. Hungar., 14 (1963), 95{121.

J. Aczzel and Z. Daroczy, On measures of information and their characteri-zations, Academic Press, New York-San Francisco-London, 1975.

M. Behara and P. Nath, Additive and non-additive entropies of ¯nite measurable partitions, Probability and Information Theory II, Lecture Notes in Math., Vol. 296, Berlin. Heidelberg-New York, 1973, 102

T.W. Chaundy and J.B. Mcleod, On a functional equation, Edinburgh Math. Notes, 43 (1960), 7

Z. Daroczy, On the measurable solutions of a functional equation, Acta Math. Acad. Sci. Hungar., 22 (1971), 11

Z. Daroczy and L. Losonczi, Uber die Erweiterung der auf einer Punktmenge additiven Funktionen, Publ. Math. (Debrecen), 14 (1967), 239{245.

J. Havrda and F. Charvat, Quanti¯cation method of classi¯cation process, Concept of structural ® -entropy, Kybernetika (Prague), 3 (1967), 30{35.

PL. Kannappan, On some functional equations from additive and non-additive measures-I, Proc. of the Edin. Mathematical Society, 23 (1980), 145{150.

PL. Kannappan, On some functional equations from additive and non-additive measures-II, Advances in communication; Second Internat. Conf. on Information Sciences and System (University of Patras, Patras, 1979), 1, 45{50, Reidel, Dordrecht-Boston-Mass., 1980.

PL. Kannappan, On a generalization of some measures in information theory, Glasnik Mat., 9 (29) (1974), 81

L. Losonczi, A characterization of entropies of degree ® , Metrika, 28 (1981), 237-244.

L. Losonczi and Gy. Maksa, On some functional equations of the information theory, Acta Math. Acad. Sci. Hung., 39 (1982), 73-82.

Gy. Maksa, On the bounded solutions of a functional equation, Acta Math. Acad. Sci. Hung., 37 (1981), 445-450.

D.P. Mittal, On continuous solutions of a functional equation, Metrika, 22 (1970), 31-40.

C.E. Shannon, A mathematical theory of communication, Bell Syst. Tech. Jour., 27 (1948), 378-423, 623-656.

Downloads

Published

2005-06-18

Issue

Vol. 15 (2005)

Section

Volume 15-2005

License

This journal provides immediate open access to its content on the principle that making research freely available to the public supports a greater global exchange of knowledge. The journal is committed to real and immediate open access for academic work. All of the journal articles are free to access immediately from the date of publication. There are no charge for readers to download articles and reviews for their own scholarly use. Benefits of open access for authors, include:

  • Free access for all users worldwide
  • Increased visibility and readership
  • Rapid publication
  • No spatial constraints
  • Authors can share their research papers on their personal web-pages (i.e. ResearchGate, etc.)

Ratio Mathematica publishes open access articles under the terms of the Creative Commons Attribution (CC BY) License. The Creative Commons Attribution License (CC-BY) allows users to copy, distribute and transmit an article, adapt the article and make commercial use of the article. The CC BY license permits commercial and non-commercial re-use of an open access article, as long as the author is properly attributed. Copyright on any research article published by Ratio Mathematica is retained by the author(s). Authors grant Ratio Mathematica a license to publish the article and identify itself as the original publisher. Authors also grant any third party the right to use the article freely as long as its original authors, citation details and publisher are identified. 
For further information visit: https://creativecommons.org/licenses/by/3.0/. The following logo will appear in any paper that will be published from the volume number 31/2016 onwards.