Convex solutions of a functional equation arising in information theory

J. B. Hiriart-Urruty, J. E. Martínez-Legaz

Research output: Contribution to journalArticleResearchpeer-review

3 Citations (Scopus)

Abstract

Given a convex function f defined for positive real variables, the so-called Csiszár f-divergence is a function If defined for two n-dimensional probability vectors p = (p1, ..., pn) and q = (q1, ..., qn) as If (p, q) : = ∑i = 1n qi f (frac(pi, qi)). For this generalized measure of entropy to have distance-like properties, especially symmetry, it is necessary for f to satisfy the following functional equation: f (x) = x f (frac(1, x)) for all x > 0. In the present paper we determine all the convex solutions of this functional equation by proposing a way of generating all of them. In doing so, existing usual f-divergences are recovered and new ones are proposed. © 2006 Elsevier Inc. All rights reserved.
Original languageEnglish
Pages (from-to)1309-1320
JournalJournal of Mathematical Analysis and Applications
Volume328
DOIs
Publication statusPublished - 15 Apr 2007

Keywords

  • Convex functions
  • Csiszár divergence
  • Functional equations
  • Information theory

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