Linear functional equations, differential operators and spectral synthesis
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| 003 | CHVBK | ||
| 005 | 20210128100639.0 | ||
| 007 | cr unu---uuuuu | ||
| 008 | 210128e20150401xx s 000 0 eng | ||
| 024 | 7 | 0 | |a 10.1007/s00010-014-0269-3 |2 doi |
| 035 | |a (NATIONALLICENCE)springer-10.1007/s00010-014-0269-3 | ||
| 245 | 0 | 0 | |a Linear functional equations, differential operators and spectral synthesis |h [Elektronische Daten] |c [G. Kiss, M. Laczkovich] |
| 520 | 3 | |a 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.). | |
| 540 | |a Springer Basel, 2014 | ||
| 690 | 7 | |a Linear functional equations |2 nationallicence | |
| 690 | 7 | |a Spectral synthesis |2 nationallicence | |
| 690 | 7 | |a Polynomial-exponential functions |2 nationallicence | |
| 700 | 1 | |a Kiss |D G. |u Department of Stochastics, Faculty of Natural Sciences, Budapest University of Technology and Economics, Budapest, Hungary |4 aut | |
| 700 | 1 | |a Laczkovich |D M. |u Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/C, 1117, Budapest, Hungary |4 aut | |
| 773 | 0 | |t Aequationes mathematicae |d Springer Basel |g 89/2(2015-04-01), 301-328 |x 0001-9054 |q 89:2<301 |1 2015 |2 89 |o 10 | |
| 856 | 4 | 0 | |u https://doi.org/10.1007/s00010-014-0269-3 |q text/html |z Onlinezugriff via DOI |
| 898 | |a BK010053 |b XK010053 |c XK010000 | ||
| 900 | 7 | |a Metadata rights reserved |b Springer special CC-BY-NC licence |2 nationallicence | |
| 908 | |D 1 |a research-article |2 jats | ||
| 949 | |B NATIONALLICENCE |F NATIONALLICENCE |b NL-springer | ||
| 950 | |B NATIONALLICENCE |P 856 |E 40 |u https://doi.org/10.1007/s00010-014-0269-3 |q text/html |z Onlinezugriff via DOI | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Kiss |D G. |u Department of Stochastics, Faculty of Natural Sciences, Budapest University of Technology and Economics, Budapest, Hungary |4 aut | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Laczkovich |D M. |u Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/C, 1117, Budapest, Hungary |4 aut | ||
| 950 | |B NATIONALLICENCE |P 773 |E 0- |t Aequationes mathematicae |d Springer Basel |g 89/2(2015-04-01), 301-328 |x 0001-9054 |q 89:2<301 |1 2015 |2 89 |o 10 | ||