Acta Mathematica Hungarica

, Volume 146, Issue 1, pp 128–141

# Algebraic methods for the solution of linear functional equations

• G. Kiss
• A. Varga
• CS. Vincze
Article

## Abstract

The equation
$$\sum^{n}_ {i=0} a_{i}f(b_{i}x + (1 - b_{i})y) = 0$$
belongs to the class of linear functional equations. The solutions form a linear space with respect to the usual pointwise operations. According to the classical results of the theory they must be generalized polynomials. New investigations have been started a few years ago. They clarified that the existence of non-trivial solutions depends on the algebraic properties of some related families of parameters. The problem is to find the necessary and sufficient conditions for the existence of non-trivial solutions in terms of these kinds of properties. One of the earliest results is due to Z. Daróczy [1]. It can be considered as the solution of the problem in case of n = 2. We are going to take more steps forward by solving the problem in case of n = 3.

## Key words and phrases

linear functional equation spectral analysis field homomorphism

## Mathematics Subject Classification

primary 39B22 secondary 39B72

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