# On a class of linear functional equations without range condition

## TL;DR

This paper characterizes all solutions to a broad class of linear functional equations involving sums over fixed coefficients, providing explicit descriptions and conditions for non-trivial solutions in linear spaces.

## Contribution

It offers a comprehensive solution framework for a class of linear functional equations without range restrictions, including necessary and sufficient conditions for non-trivial solutions.

## Key findings

- Explicit general solutions for the functional equations
- Necessary and sufficient conditions for non-trivial solutions
- Framework applicable to linear spaces over any field

## Abstract

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