arXiv subjects
Publications and source records attributed to Kh. F. Abu-Helaiel.
This paper is an introduction to the regularity theory of functional equations, motivated by the study of Fréchet's functional equation. Another main goal is to honor the work in functional equations of the Romanian mathematician Tiberiu Popoviciu.
We prove that, if f:R^n\to R satisfies Fréchet's functional equation and f(x_1,...,x_n) is not an ordinary algebraic polynomial in the variables x_1,...,x_n, then f is unbounded on all non-empty open set U of R^n. Furthermore, the closure of its graph contains an unbounded open set.
Recently, the first author of this paper, used the structure of finite dimensional translation invariant subspaces of C(R,C) to give a new proof of classical Montel's theorem, about continuous solutions of Fréchet's functional equation $Δ_h^mf=0$, for real functions (and complex functions) of one real variable. In this paper we use similar ideas to prove a Montel's type theorem for the case of complex valued functions defined over the discrete group Z^d. Furthermore, we also state and demonstrate an improved version of Montel's Theorem for complex functions of several real variables and complex functions of several complex variables.
We study the continuous solutions of several classical functional equations by using the properties of the spaces of continuous functions which are invariant under some elementary linear trans-formations. Concretely, we use that the sets of continuous solutions of certain equations are closed vector subspaces of $C(\mathbb{C}^d,\mathbb{C})$ which are invariant under affine transformations $T_{a,b}(f)(z)=f(az+b)$, or closed vector subspaces of $C(\mathbb{R}^d,\mathbb{R})$ which are translation and dilation invariant. These spaces have been recently classified by Sternfeld and Weit, and Pinkus, respectively, so that we use this information to give a direct characterization of the continuous solutions of the corresponding functional equations.
We prove a version of both Jacobi's and Montel's Theorems for the case of continuous functions defined over the field $\mathbb{Q}_p$ of $p$-adic numbers. In particular, we prove that, if \[ Δ_{h_0}^{m+1}f(x)=0 \ \ \text{for all} x\in\mathbb{Q}_p, \] and $|h_0|_p=p^{-N_0}$ then, for all $x_0\in \mathbb{Q}_p$, the restriction of $f$ over the set $x_0+p^{N_0}\mathbb{Z}_p$ coincides with a polynomial $p_{x_0}(x)=a_0(x_0)+a_1(x_0)x+...+a_m(x_0)x^m$. Motivated by this result, we compute the general solution of the functional equation with restrictions given by {equation} Δ_h^{m+1}f(x)=0 \ \ (x\in X \text{and} h\in B_X(r)=\{x\in X:\|x\|\leq r\}), {equation} whenever $f:X\to Y$, $X$ is an ultrametric normed space over a non-Archimedean valued field $(\mathbb{K},|...|)$ of characteristic zero, and $Y$ is a $\mathbb{Q}$-vector space. By obvious reasons, we call these functions uniformly locally polynomial.
We study discontinuous solutions of the monomial equation $\frac{1}{n!}Δ_h^nf(x)=f(h)$. In particular, we characterize the closure of their graph, $\bar{G(f)}^{\mathbb{R}^2}$, and we use the properties of these functions to present a new proof of the Darboux type theorem for polynomials and of Hamel's theorem for additive functions.
We study some properties of the solutions of the functional equation $$f(x)+f(a_1x)+...+f(a_Nx)=0,$$ which was introduced in the literature by Mora, Cherruault and Ziadi in 1999, for the case $a_k=k+1$, $k=1,2,...,N$ and studied by Mora in 2008 and Mora and Sepulcre in 2009 and 2011.