Sander, Wolfgang On a functional equation arising in information theory. (English) Zbl 0588.39006 Manuscr. Math. 54, 439-452 (1986). 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 Cited in 1 Document MSC: 39B99 Functional equations and inequalities 94A17 Measures of information, entropy Keywords:representation of generalized additive measures of information; sum property; regular generating function; Shannon entropy; sum form equations; measurable solutions PDFBibTeX XMLCite \textit{W. Sander}, Manuscr. Math. 54, 439--452 (1986; Zbl 0588.39006) Full Text: DOI EuDML References: [1] ACZÉL, J.: Lectures on functional equations and their applications, New York-London: Academic Press 1966 · Zbl 0139.09301 [2] ACZÉL, J., DARÓCZY, Z.: On measures of information and their characterizations, New York-San Francisco-London: Academic Press 1975 [3] EBANKS, B. R.: On measures of fuzziness and their representations. J. Math. Anal. Appl.94, 24-37 (1983) · Zbl 0523.94036 · doi:10.1016/0022-247X(83)90003-3 [4] KANNAPPAN, Pl.: On some functional equations from additive and non-additive measures-I. Proc. Edin. Math. Soc.23, 145-150 (1980) · Zbl 0468.39002 · doi:10.1017/S0013091500003023 [5] KANNAPPAN, Pl.: On some functional equations from additive and non-additive measures-V. Utilitas Math.22, 141-147 (1982) · Zbl 0509.39013 [6] KANNAPPAN, Pl.: On a generalization of sum form functional equation-V. Aequationes Math.28, 255-261 (1985) · Zbl 0571.39006 · doi:10.1007/BF02189418 [7] SAHOO, P. K.: On some functional equations connected to sum form information measures on open domains. Utilitas Math.23, 161-175 (1983) · Zbl 0524.94008 [8] SANDER, W.: On a sum form functional equation. Aequationes Math.28, 321-326 (1985) · Zbl 0576.39010 · doi:10.1007/BF02189426 [9] VINCZE, E.: Eine allgemeinere Methode in der Theorie der Funktionalgleichungen II. Publ. Math. Debrecen9, 314-323 (1962) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.