Existence of nontrivial solutions of linear functional equations

In this paper we study the functional equation ∑ i = 1 n a i f ( b i x + c i h ) = 0 ( x , h ∈ C ) where a i , b i , c i are fixed complex numbers and f : C → C is the unknown function. We show, that if there is i such that b i / c i ≠ b j / c j holds for any 1 ≤ j ≤ n , j ≠ i , the functional equat...

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Bibliographic Details
Published inAequationes mathematicae Vol. 88; no. 1-2; pp. 151 - 162
Main Authors Kiss, Gergely, Varga, Adrienn
Format Journal Article
LanguageEnglish
Published Basel Springer Basel 01.09.2014
Springer Nature B.V
Subjects
Analysis
Automorphisms
Combinatorics
Complex numbers
Functions (mathematics)
Linear equations
Mathematical analysis
Mathematics
Mathematics and Statistics
Texts
Primary 39B22
Functional equations
variety
field isomorphisms
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Summary:In this paper we study the functional equation ∑ i = 1 n a i f ( b i x + c i h ) = 0 ( x , h ∈ C ) where a i , b i , c i are fixed complex numbers and f : C → C is the unknown function. We show, that if there is i such that b i / c i ≠ b j / c j holds for any 1 ≤ j ≤ n , j ≠ i , the functional equation has a nonconstant solution if and only if there are field automorphisms ϕ 1 , ... , ϕ k of C such that ϕ 1 ⋯ ϕ k is a solution of the equation.
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ISSN:0001-9054
1420-8903
DOI:10.1007/s00010-013-0212-z