On a class of linear functional equations without range condition

Abstract

The main purpose of this work is to provide the general solutions of a class of linear functional equations. Let $n\ge 2$ be an arbitrarily fixed integer, let further $X$ and $Y$ be linear spaces over the field $\mathbb{K}$ and let ${\alpha }_i,{\beta }_i\in \mathbb{K}$, $i=1,\dots ,n$ be arbitrarily fixed constants. We will describe all those functions $f,f_{i,j}:X\times Y\to \mathbb{K}$, $i,j=1,\dots ,n$ that fulfill functional equation \[ f\left(\sum_{i=1}^n α_i x_i, \sum_{i=1}^n β_i y_i\right)= \sum_{i, j=1}^{n}f_{i, j}(x_i, y_j) \qquad \left(x_i \in X, y_i \in Y, i=1, \ldots, n\right). \] Additionally, necessary and sufficient conditions will also be given that guarantee the solutions to be non-trivial.

27 pages, 1 figure