Generalized characteristic equation of branching information measures

Summary

The problem of determining all branching measures of inset information on open domains leads to the functional equation

$$\phi _1 (p,q) + \phi _2 (p + q,r) = \phi _3 (p,r) + \phi _4 (p + r,q)$$

((*))

for (p, q, r) inD ₃ and real-valued mapsφ ₁ (i = 1,⋯, 4) onD ₂. Here, we letJ = ]0, 1[^m for some (fixed) positive integerm, and

$$D_n = \left\{ {(p_1 ,p_2 , \ldots ,p_n )|p_i \in Jforalli,\sum\limits_{i = 1}^n {p_i \in J} } \right\}$$

forn = 2, 3,⋯. The main result of the paper is the general solution of equation (*).

The general solution of (*) in the special case when the four unknown functions are equal and symmetric follows from a 1974 paper of Ng (Representation for measures of information with the branching property, Inform. and Control25, 45–56). The reduction of (*) to this special case is not easy, however, because of the absence of zeros from the domain. The first step is the known reduction of (*) to the two equations

$$\begin{gathered} h(p,q) + k(p + q,r) = h(p,r) + k(p + r,q), \hfill \\ f(p,q) + g(p + q,r) + f(p,r) + g(p + r,q) = 0, \hfill \\ \end{gathered} $$

for (p, q, r) ∈ D ₃ and real-valued mapsf, g, h, k onD ₂. Equation (A) is eventually reduced to the special case with which Ng dealt. Some new methods developed include the general solution of some Sincov-type equations on restricted domains, and the general solution of

$$U(x,y) = \Phi (x + y,q) - \Phi (x,q) - \Phi (y,q) - m(q)$$

for (x, y, q) ∈ D ₃ and real-valued mapsm (onJ),U (onD ₂), and skew-symmetric Ф (onD ₂).