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Abstract

Always when a numerical method gives exact results an interesting functional equation arises. And, since no regularity is assumed, some unexpected solutions may appear. Here we deal with equations constructed in this spirit. The vast majority of this paper is devoted to the equation \begin{equation} \label{l} \sum_{i=0}^l(y-x)^i[f_{1,i}(\alpha_{1,i}x+\beta_{1,i}y)+\cdots+f_{{k_i},i}(\alpha_{{k_i},i}x+\beta_{{k_i},i}y)]=0\tag1 \end{equation} and its particular cases.

We use Sablik's lemma to prove that all solutions of (1) are polynomial functions. Since a continuous polynomial function is an ordinary polynomial, the crucial problem throughout the whole paper will be the continuity of solutions of (1).

The first of the particular forms of (1) which we consider is \begin{equation} \label{depex} F(y)-F(x)=(y-x)[a_1f(\alpha_1 x+\beta_1 y)+\cdots +a_nf(\alpha_n x+\beta_n y)]\tag2 \end{equation} and is motivated by the quadrature formulas of numerical integration. Quadrature rules give exact results for polynomials, and therefore the following problem becomes interesting: do equations of the type (2) characterize polynomials? We present new results concerning this equation, in particular, we obtain a general solution of (2) in the case of rational $\alpha_i,\beta_i$, $i=1,\dots,n,$ and we show that if (2) has discontinuous solutions then the equation $$a_1f(\alpha_1 x+\beta_1 y)+\cdots +a_nf(\alpha_n x+\beta_n y)=0$$ has nontrivial solutions. This result allows us to solve functional equations motivated by all classical quadrature rules such as the rule of Simpson (this equation was already solved earlier), Radau, Lobatto and Gauss.

Further we also consider the following equation: \begin{equation} \label{Herm} F(y)-F(x)=(y-x)[a_1f(\alpha_1 x+\beta_1 y)+\cdots +a_nf(\alpha_n x+\beta_n y)]+(y-x)^2[g(y)-g(x)],\tag3 \end{equation} which is connected with Hermite quadrature formulas where on the right-hand side derivatives of $f$ are used; \begin{align}\tag4 \label{Birk} F(y)-F(x) ={}& (y-x) [a_1f(x)+b_1f(\alpha_1x+\beta_1y)+\cdots+b_nf(\alpha_nx+\beta_ny)+a_1f(y)]\\ &{}+ (y-x)^3[c_1g(\alpha_1x+\beta_1y)+\cdots+c_ng(\alpha_nx+\beta_ny)],\notag \end{align} which stems from Birkhoff quadrature rules where $f''$ is involved; and \begin{equation} \label{maindif} g(\alpha x+\beta y)(y-x)^k=a_1f(\alpha_1x+\beta_1y)+\cdots+a_nf(\alpha_nx+\beta_ny),\tag5 \end{equation} which is motivated by formulas used in numerical differentiation. Results concerning (5) are used to obtain new facts about the well known equation $$f[x_1,\dots,x_n]=g(x_1+\cdots+x_n)$$ $(f[x_1,\dots,x_n]$ is the $n$th divided difference of~$f).$

We also present a direct method which may be used to show that solutions of (2) must be polynomial functions and, motivated by this method, we obtain a generalization of the Aczél equation $$F(y)-F(x)=(y-x)g\bigg(\frac{x+y}{2}\bigg).$$ At the end of the paper we present a list of open problems.