We deal with the functional equationmotivated by quadrature rules of approximate integration. In previous results the solutions of this equation were found only in some particular cases. For example the coefficients λi were supposed to be rational or the equation in question was solved only for n = 2. In the current paper we do not assume any particular form of coefficients occurring in this equation and we allow n to be any positive integer. Moreover, we obtain a solution of our equation without any regularity assumptions concerning the functions f and F .

No abstract

The functional equations of the form are considered. They are connected with quadrature rules of the approximate integration. We show that such equations characterize polynomials in the class of continuous functions. It is also shown that if the number of components is sufficiently small, then the continuity is forced by the equation itself. Unique solvability of the considered problem are established.

We ask for all functions having properties of this kind. Moreover, we consider some generalizations of the problem for functions acting on more general structures than R (integral domains).

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.