caa a22 4500 605509018 CHVBK 20210128100639.0 cr unu---uuuuu 210128e20150401xx s 000 0 eng 10.1007/s00010-014-0269-3 doi (NATIONALLICENCE)springer-10.1007/s00010-014-0269-3 Linear functional equations, differential operators and spectral synthesis [Elektronische Daten] [G. Kiss, M. Laczkovich] Our aim is to describe the solutions of the functional equation $${\sum^{n}_{i=1} a_if(b_ix + c_iy) =0}$$ ∑ i = 1 n a i f ( b i x + c i y ) = 0 , where $${a_i,b_i,c_i \in \mathbb{C}}$$ a i , b i , c i ∈ C , and the unknown function f is defined on the field $${K = \mathbb{Q} (b_1,\ldots, b_n, c_1.\ldots,c_n )}$$ K = Q ( b 1 , ... , b n , c 1 . ... , 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 $${S_1 \cap \mathcal{D}}$$ S 1 ∩ D , where $${\mathcal{D}}$$ D is the set of functions $${\phi \circ D}$$ ϕ ∘ D , where $${\phi}$$ ϕ is a field automorphism of $${\mathbb{C}}$$ 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 $${F \in V}$$ F ∈ 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 $${S\cap \mathcal{A}}$$ S ∩ A , where $${\mathcal{A}}$$ A is the algebra generated by $${\mathcal{D}}$$ 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.). Springer Basel, 2014 Linear functional equations nationallicence Spectral synthesis nationallicence Polynomial-exponential functions nationallicence Kiss G. Department of Stochastics, Faculty of Natural Sciences, Budapest University of Technology and Economics, Budapest, Hungary aut Laczkovich M. Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/C, 1117, Budapest, Hungary aut Aequationes mathematicae Springer Basel 89/2(2015-04-01), 301-328 0001-9054 89:2<301 2015 89 10 https://doi.org/10.1007/s00010-014-0269-3 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00010-014-0269-3 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Kiss G. Department of Stochastics, Faculty of Natural Sciences, Budapest University of Technology and Economics, Budapest, Hungary aut NATIONALLICENCE 700 1- Laczkovich M. Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/C, 1117, Budapest, Hungary aut NATIONALLICENCE 773 0- Aequationes mathematicae Springer Basel 89/2(2015-04-01), 301-328 0001-9054 89:2<301 2015 89 10