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Fundamental equation of information revisited. (English) Zbl 0842.39008
Let \(J\) be the \(k\)-th Cartesian power of the open interval \(]0,1[\). The fundamental equation of information measures of multiplicative type, depending upon \(k\) probability distributions with nonzero probabilities, is \(f(x)+M(1-x)f(y/(1-x))=f(y)+M(1-y)f(x/(1-y))\) (whenever \(x\), \(y\) and also \(x+y\) are in \(J\)), where \(M: J\to \mathbb{R}\) is multiplicative. This equation has been completely solved by C. T. Ng and the reviewer [Linear Algebra Appl. 52-53, 1-30 (1983; Zbl 0517.39006)].
The author gives here an alternative proof for the case where \(M\) is also additive (as is the case for the most applied information measures) which uses a result of B. Jessen, J. Karpf and A. Thorup [Math. Scand. 22, 257-265 (1968; Zbl 0183.04004)] just once (instead of twice) and is more similar to the proof for the nonadditive case in the 1983 paper. Finally, generalizations of (part of) the result to more general domains (ordered commutative rings with unit, in particular real-closed fields and positive cones in ordered fields) are offered.
39B22 Functional equations for real functions
94A17 Measures of information, entropy
39B52 Functional equations for functions with more general domains and/or ranges
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[1] Aczél, J. andDaróczy, Z.,Measures of information and their characterizations. Academic Press, New York, 1975. · Zbl 0345.94022
[2] Aczél, J. andNg, C. T.,Determination of all semisymmetric recursive information measures of multiplicative type on n positive discrete probability distributions. Linear Alg. Appl.52 (1983), 1–30. · Zbl 0517.39006
[3] Jacobson, N.,Lectures on abstract algebra, vol. III. D. van Nostrand, Princeton, 1964. · Zbl 0124.27002
[4] Jessen, B., Karpf, J. andThorup, A.,Some functional equations on groups and rings. Math. Scand.22 (1968), 257–265. · Zbl 0183.04004
[5] Kuczma, M.,Note on additive functions of several variables. Prace Nauk. Uniw. Slask. w Katowicach12 (1972), 49–51. · Zbl 0247.39005
[6] Lang, S.,Algebra. Addison-Wesley, Reading, MA, 1965.
[7] Maksa, Gy.,Solution on the open triangle of the generalized fundamental equation of information with four unknown functions. Utilitas Math.21 (1982), 267–282. · Zbl 0497.94003
[8] Ng, C. T.,Representation for measures of information with the branching property. Information and Control25 (1974), 45–56. · Zbl 0279.94018
[9] Rédei, L.,Algebra, vol. I. Pergamon Press, Oxford, 1967.
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