Linear functional equations, differential operators and spectral synthesis

Abstract

Our aim is to describe the solutions of the functional equation $${\sum^{n}_{i=1} a_if(b_ix + c_iy) =0}$$ , where $${a_i,b_i,c_i \in \mathbb{C}}$$ , and the unknown function f is defined on the field $${K = \mathbb{Q} (b_1,\ldots, b_n, c_1.\ldots,c_n )}$$ . Since the set of solutions constitutes a variety on the discrete multiplicative group K* of the field K, our approach is to apply spectral synthesis on K* and on its powers. We prove that spectral synthesis holds in every variety on K* which consists of functions additive on K with respect to addition. As an application we show that the set S 1 of additive solutions of the equation is spanned by $${S_1 \cap \mathcal{D}}$$ , where $${\mathcal{D}}$$ is the set of functions $${\phi \circ D}$$ , where $${\phi}$$ is a field automorphism of $${\mathbb{C}}$$ and D is a differential operator on K. We prove that if V is a variety on the Abelian group (K*) k under multiplication, and every function $${F \in V}$$ is k-additive on K k with respect to addition, then spectral synthesis holds in V. From this we infer that, under some mild conditions on the equation, the set S of all solutions is spanned by $${S\cap \mathcal{A}}$$ , where $${\mathcal{A}}$$ is the algebra generated by $${\mathcal{D}}$$ . This implies that if S is translation invariant with respect to addition, then spectral synthesis holds in S considered as a variety on the additive group of K. We give several applications, and describe the set of solutions of equations having some special properties (e.g. having algebraic coefficients etc.).