On a class of linear functional equations without range condition

2019/03/19 by Gselmann, Eszter, Kiss, Gergely, Vincze, Csaba
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Primary 39B52 #Secondary 39B22

paper · doi:10.48550/arxiv.1903.07974

Abstract

The main purpose of this work is to provide the general solutions of a class of linear functional equations. Let n≥ 2 be an arbitrarily fixed integer, let further X and Y be linear spaces over the field \mathbbK and let α_i, β_i∈ \mathbbK, i=1, …, n be arbitrarily fixed constants. We will describe all those functions f, f_{i, j}\colon X× Y→ \mathbbK, i, j=1, …, n that fulfill functional equation f(∑_i=1ⁿ α_i x_i, ∑_i=1ⁿ β_i y_i)= ∑_{i, j=1}ⁿf_{i, j}(x_i, y_j) (x_i ∈ X, y_i ∈ Y, i=1, …, n). Additionally, necessary and sufficient conditions will also be given that guarantee the solutions to be non-trivial.

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