On the measurable solution of a functional equation arising in information theory

1979 ◽  
Vol 34 (1-2) ◽  
pp. 105-116 ◽  
Author(s):  
Z. Daróczy ◽  
A. Járai
Keyword(s):  
Information Theory ◽  
Functional Equation ◽  
Measurable Solution

scholarly journals Weighted additive information measures

1990 ◽  
Vol 13 (3) ◽  
pp. 417-423
Author(s):  
Wolfgang Sander
Keyword(s):  
Information Theory ◽  
Functional Equation ◽  
Information Measures ◽  
Measurable Functions

We determine all measurable functionsI,G,L:[0,1]→ℝsatisfying the functional equation∑i=1n∑j=1mI(pi,qj)=∑i=1n∑j=1mG(pi)I(qj)+∑i=1n∑j=1mL(qj)I(pi)forP∈Γn,Q∈Γmand for a fixed pair(n,m),n≥3,m≥3, whereG(0)=L(0)=0andG(1)=L(1)=1. This functional equation has interesting applications in information theory.

scholarly journals Convex solutions of a functional equation arising in information theory

2007 ◽  
Vol 328 (2) ◽  
pp. 1309-1320 ◽  
Author(s):  
J.-B. Hiriart-Urruty ◽  
J.-E. Martínez-Legaz
Keyword(s):  
Information Theory ◽  
Functional Equation

On characterization of a generalized inaccuracy measure in information theory

1973 ◽  
Vol 10 (02) ◽  
pp. 464-468
Author(s):  
Bhu Dev Sharma ◽  
Ram Autar
Keyword(s):  
Information Theory ◽  
Functional Equation ◽  
Boundary Conditions ◽  
Type A ◽  
New Type ◽  
Two Parameters

Sharma and Autar [6] have formed a functional equation in two variables containing two parameters which, as a particular case, includes Daróczy's [1] generalized functional equation used in information theory. The solutions of our functional equation, under suitable boundary conditions, lead to an inaccuracy function of a new type. A characterization of this inaccuracy function and the convexity of Daróczy's entropy of type-ß have been given in this paper.

A Functional Equation for the Cosine

1968 ◽  
Vol 11 (3) ◽  
pp. 495-498 ◽  
Author(s):  
PL Kannappan
Keyword(s):  
Functional Equation ◽  
Measurable Function ◽  
Trivial Solution ◽  
Real Variable ◽  
Real Constant ◽  
Real Numbers ◽  
Measurable Solution ◽  
Image Position ◽  
Complex Valued ◽  
Cosine Functions

It is known [3], [5] that, the complex-valued solutions of(B)(apart from the trivial solution f(x)≡0) are of the form(C)(D)In case f is a measurable solution of (B), then f is continuous [2], [3] and the corresponding ϕ in (C) is also continuous and ϕ is of the form [1],(E)In this paper, the functional equation(P)where f is a complex-valued, measurable function of the real variable and A≠0 is a real constant, is considered. It is shown that f is continuous and that (apart from the trivial solutions f ≡ 0, 1), the only functions which satisfy (P) are the cosine functions cos ax and - cos bx, where a and b belong to a denumerable set of real numbers.

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