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Results in Mathematics

, 73:74

# On Functional Equations Characterizing Derivations: Methods and Examples

Article

## Abstract

Functional equations satisfied by additive functions have a special interest not only in the theory of functional equations, but also in the theory of (commutative) algebra because the fundamental notions such as derivations and automorphisms are additive functions satisfying some further functional equations as well. It is an important question that how these morphisms can be characterized among additive mappings in general. The paper contains some multivariate characterizations of higher order derivations. The univariate characterizations are given as consequences by the diagonalization of the multivariate formulas. This method allows us to refine the process of computing the solutions of univariate functional equations of the form
\begin{aligned} \sum _{k=1}^{n}x^{p_{k}}f_{k}(x^{q_{k}})=0, \end{aligned}
where $$p_k$$ and $$q_k$$ ($$k=1, \ldots , n$$) are given nonnegative integers and the unknown functions $$f_{1}, \ldots , f_{n}:R\rightarrow R$$ are supposed to be additive on the ring R. It is illustrated by some explicit examples too. As another application of the multivariate setting we use spectral analysis and spectral synthesis in the space of the additive solutions to prove that it is spanned by differential operators. The results are uniformly based on the investigation of the multivariate version of the functional equations.

## Keywords

Linear function derivation multiadditive function spectral analysis and synthesis

## Mathematics Subject Classification

39B50 13N15 43A45

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## Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

## Authors and Affiliations

1. 1.Institute of MathematicsUniversity of DebrecenDebrecenHungary
2. 2.Faculté des Sciences, de la Technologie et de la Communication Maison du NombreUniversité du LuxembourgEsch-sur-AlzetteLuxembourg