Characterizing polynomial functions by a mean value property

Abstract

We generalize Flett's Mean Value Theorem to the case of functions defined in normed spaces. This is a motivation for considering functional equations related to the Flett mean value formula in a quite general setting. We solve them in the case where functions are defined in abelian groups and take values in a rational linear space.

Polynomials and divided differences

“…We note that results related to those in this paper, in the sense of connecting polynomial and similar functions with divided differences, appear inter alia in papers by Aczél and Kuczma [3], Andersen [4], Davies and Rousseau [5], Deeba and Simeonov [6], Haruki [7], Sablik [12,13] and Schwaiger [16]. In [10] and [11], Riedel and Sablik characterize polynomial functions by functional equations derived from Flett's mean value theorem. Further examples and references may be found in the book by Sahoo and Riedel [15].…”

A characterization of polynomials through Flett's MVT

“…They proved that f , g, h satisfy this equation if and only if f (x) = g(x) = ax 2 +bx+c and h(x) = ax+b, for some a, b, c ∈ R. Analogously, T. Riedel and M. Sablik in [5] solved the functional equation ( 1)…”