Some functional equations characterizing polynomials

Abstract

We present a method of solving functional equations of the type where f, F: P → P are unknown functions acting on an integral domain P and parameteres are given. We prove that under some assumptions on the parameters involved, all solutions to such kind of equations are polynomials. We use this method to solve some concrete equations of this type. For example, the equation (1) for f, F: ℝ → ℝ is solved without any regularity assumptions. It is worth noting that (1) stems from a well-known quadrature rule used in numerical analysis.

Characterization of higher-order monotonicity via integral inequalities

“…(ii) f is continuous and, for all x, y ∈ I with x < y, (iv) f is continuous and, for all x, y ∈ I with x < y, As has recently been recognized [16][17][18], certain functional equations stemming from the Gauss-, Lobatto-and Radau-quadrature rules characterize polynomials under no regularity assumption on the functions involved. Requiring continuity, theorems 3.4 and 3.5 arrive at the same conclusions.…”

On a problem of T. Szostok concerning the Hermite–Hadamard inequalities

“…In all the above mentioned results it turns out that the solution to the considered functional equation has to be continuous. Results in this spirit can be found in the papers [15], [16], [26], [31], [26] and in monograph [25]. All from the above equations are a particular case of the general equation of the form…”