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On a multiplicative type sum form functional equation and its role in information theory. (English) Zbl 1164.39330

Summary: We obtain all possible general solutions of the sum form functional equations \[ \begin{aligned} \sum _{i=1}^{k}\sum _{j=1}^{\ell }f(p_iq_j)=&\sum _{i=1}^{k}g(p_i) \sum _{j=1}^{\ell }h(q_j)\qquad \text{and} \\ \sum _{i=1}^{k}\sum _{j=1}^{\ell }F(p_iq_j)=&\sum _{i=1}^{k} G(p_i)+\sum _{j=1}^{\ell }H(q_j)+ \lambda \sum _{i=1}^{k}G(p_i)\sum _{j=1}^{\ell }H(q_j) \end{aligned} \] valid for all complete probability distributions \((p_1,\ldots ,p_k)\), \((q_1,\ldots ,q_\ell )\), \(k\geq 3\), \(\ell \geq 3\) fixed integers; \(\lambda \in \mathbb R\), \(\lambda \neq 0\) and \(F\), \(G\), \(H\), \(f\), \(g\), \(h\)  are real valued mappings each having the domain \(I=[0,1]\), the closed unit interval.

MSC:

39B22 Functional equations for real functions
94A15 Information theory (general)
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References:

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