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On a functional equation arising in information theory. (English) Zbl 0588.39006

The author obtains the measurable solutions of \[ (1)\quad \sum^{n}_{i=1}\sum^{m}_{j=1}I(p_ iq_ j,v_ iw_ j)=\sum^{n}_{i=1}G(p_ i)\sum_{j}I(v_ j,w_ j)+\sum_{j}L(q_ j)\sum_{i}I(p_ i,v_ i) \] holding for some fixed pair (m,n), m,n\(\geq 3\) with \(I(0,0)=0=G(0)=L(0)\) where G,L: [0,1]\(\to {\mathbb{R}}\) are measurable functions and \(I: J\to {\mathbb{R}}\) is measurable in each variable. The equation (1) is a generalization of the well known sum form \[ (2)\sum^{n}_{i=1}\sum^{m}_{j=1}I(p_ iq_ j,v_ iw_ j)=\sum_{i}I(p_ i,v_ i)+\sum_{j}I(q_ j,w_ j) \] investigated by several authors including the reviewer. The measurable solutions are too complicated to be reproduced and they are nine in number.
Reviewer: Pl.Kannappan

MSC:

39B99 Functional equations and inequalities
94A17 Measures of information, entropy
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References:

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