On a functional equation arising in information theory
type: Item
instance of: scholarly article
title: On a functional equation arising in information theory
zbMATH Open document ID: 0588.39006
DOI: 10.1007/BF01168486
author: Wolfgang Sander
published in: Manuscripta Mathematica
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155122
review text: 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.
Mathematics Subject Classification ID: 39B99
Mathematics Subject Classification ID: 94A17
zbMATH DE Number: 3944445
zbMATH Keywords: representation of generalized additive measures of information
zbMATH Keywords: sum property
zbMATH Keywords: regular generating function
zbMATH Keywords: Shannon entropy
zbMATH Keywords: sum form equations
zbMATH Keywords: measurable solutions
reviewed by: Pl. Kannappan
MaRDI profile type: Publication
cites work: Q5514010
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OpenAlex ID: W2089045310
description: scientific article; zbMATH DE number 3944445