Algebraic methods for the solution of linear functional equations

The equation ∑ i = 0 n a i f ( b i x + ( 1 - b i ) y ) = 0 belongs to the class of linear functional equations. The solutions form a linear space with respect to the usual pointwise operations. According to the classical results of the theory they must be generalized polynomials. New investigations...

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Bibliographic Details
Published in:Acta mathematica Hungarica 2015-06, Vol.146 (1), p.128-141
Main Authors: Kiss, G., Varga, A., Vincze, CS
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
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Summary:The equation ∑ i = 0 n a i f ( b i x + ( 1 - b i ) y ) = 0 belongs to the class of linear functional equations. The solutions form a linear space with respect to the usual pointwise operations. According to the classical results of the theory they must be generalized polynomials. New investigations have been started a few years ago. They clarified that the existence of non-trivial solutions depends on the algebraic properties of some related families of parameters. The problem is to find the necessary and sufficient conditions for the existence of non-trivial solutions in terms of these kinds of properties. One of the earliest results is due to Z. Daróczy [ 1 ]. It can be considered as the solution of the problem in case of n = 2. We are going to take more steps forward by solving the problem in case of n = 3.
ISSN:0236-5294
1588-2632
DOI:10.1007/s10474-015-0497-6