ACTA issues

On functional equations characterizing polynomials

Zsolt Páles

Acta Sci. Math. (Szeged) 74:3-4(2008), 581-592

Abstract. In this paper we deal with the linear functional equation $$ \int_0^1 f(x+t(y-x))d\mu(t)=0 ((x,y)\in\Omega ), $$ which can be considered as a generalization of the Fréchet functional equation that characterizes polynomials. Here $f\colon I\to{\msbm R}$ is a continuous function, $\mu $ is a given signed Borel measure on $[0,1]$ and $\Omega\subseteq I\times I$ is a given open set containing the diagonal of $I\times I$. Our main result shows that $f$ is a solution of the above equation if and only if $f$ is a polynomial of degree at most $n-1$, where $n$ is the smallest nonnegative integer such that the $n$th moment of the measure $\mu $ does not vanish. The main result is also used to solve certain composite functional equations.

AMS Subject Classification (1991): 39B22, 39B12

Keyword(s): Linear functional equation, composite functional equation, convolution smoothing

Received July 11, 2008, and in revised form August 27, 2008. (Registered under 6033/2009.)