Some functional equations characterizing polynomials

Rok, strany: 2009, 27 - 40

We present a method of solving functional equations of the type

$$ F(x)-F(y)=(x-y)\bl[b₁f(α₁x+β₁y) +…+b_nf(α_nx+β_ny)\br], $$

where $f,F\colon P\to P$ are unknown functions acting on an integral domain $P$ and parameters $b₁,…,b_n;α₁,…,α_n;β₁,…,β_n\in P$ 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 \begin{equation} 8\bl[F(x)-F(y)\br]=(x-y)\Bgl[f(x)+3f(((x+2y) / (3)))+ 3f(((2x+y) / (3)))+f(y)\Bgr] \label{simp} \end{equation} for $f,F\colon \Rz\to\Rz$ is solved without any regularity assumptions. It is worth noting that (\ref{simp}) stems from a well-known quadrature rule used in numerical analysis.

ISO 690:

Koclęga-Kulpa, B., Szostok, T., Wąsowicz, S. 2009. Some functional equations characterizing polynomials. In Tatra Mountains Mathematical Publications, vol. 44, no.3, pp. 27-40. 1210-3195.

APA:

Koclęga-Kulpa, B., Szostok, T., Wąsowicz, S. (2009). Some functional equations characterizing polynomials. Tatra Mountains Mathematical Publications, 44(3), 27-40. 1210-3195.