Linear functional equations, differential operators and spectral synthesis

We investigate the functional equation ∑n i=1 aif(bix + ciy) = 0, where ai, bi, ci ∈ C, and the unknown function f is defined on the field K = Q(b1, . . . , bn, c1, . . . , cn). (It is easy to see that every solution on K can be extended to C as a solution.) Let S1 denote the set of additive solutions defined on K. We prove that S1 is spanned by S1 ∩ D, where D is the set of the functions φ ◦ D, where φ is a field automorphism of C and D is a differential operator on K. We say that the equation ∑n i=1 aif(bix + ciy) = 0 is normal, if its solutions are generalized polynomials. (The equations ∑n i=1 aif(bix + y) = 0 have this property.) Let S denote the set of solutions of a normal equation ∑n i=1 aif(bix + ciy) = 0 defined on K. We show that S is spanned by S ∩ A, where A is the algebra generated by D. This implies that if S is translation invariant, then spectral synthesis holds in S. The main ingredient of the proof is the observation that if V is a variety on the Abelian group (K∗)k under multiplication, and every function F ∈ V is k-additive on Kk, then spectral synthesis holds in V . We give several applications, and describe the set of solutions of equations having some special properties (e.g. having algebraic coefficients etc.).