A functional equation from probability theory

Author:

Baker John A.

Abstract

The functional equation ( ( 1 ) ) f ( x ) = j = 1 N [ f ( β j x ) ] γ j \begin{equation}\tag {$(1)$} f(x) = \prod \limits _{j = 1}^N {{{[f({\beta _j}x)]}^{{\gamma _j}}}}\end{equation} has been used by Laha and Lukacs (Aequationes Math. 16 (1977), 259-274) to characterize normal distributions. The aim of the present paper is to study (1) under somewhat different assumptions than those assumed by Laha and Lukacs by using techniques which, in the author’s opinion, are simpler than those employed by the afore-mentioned authors. We will prove, for example, that if 0 > β j > 1 0 > {\beta _j} > 1 and γ j > 0 {\gamma _j} > 0 for 1 j N , j = 1 N β j k γ j = 1 1 \leq j \leq N, \sum {_{j = 1}^N\beta _j^k{\gamma _j} = 1} , where k is a natural number, f : R [ 0 , + ) f:\mathbb {R} \to [0, + \infty ) , (1) holds for x R x \in \mathbb {R} and f ( k ) ( 0 ) {f^{(k)}}(0) exists then either f 0 f \equiv 0 or there exists a real constant c such that f ( x ) = exp ( c x k ) f(x) = \exp (c{x^k}) for all x R x \in \mathbb {R} .

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference7 articles.

1. Functional equations, tempered distributions and Fourier transforms;Baker, John A.;Trans. Amer. Math. Soc.,1989

2. Prace Naukowe Uniwersytetu \'{S}l\polhk askiego w Katowicach [Scientific Publications of the University of Silesia];Jarczyk, Witold,1991

3. Encyclopedia of Mathematics and its Applications;Kuczma, Marek,1990

4. Nonnegative measurable solutions of difference equations;Laczkovich, M.;J. London Math. Soc. (2),1986

5. On a functional equation which occurs in a characterization problem;Laha, R. G.;Aequationes Math.,1977

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