Using Aichinger's equation to characterize polynomial functions

2022/07/17 by Almira, J. M. · 3 citations
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2207.08248

Abstract

Aichinger's equation is used to give simple proofs of several well-known characterizations of polynomial functions as solutions of certain functional equations. Concretely, we use that Aichinger's equation characterizes polynomial functions to solve, for arbitrary commutative groups, Ghurye-Olkin's functional equation, Wilson's functional equation, the Kakutani-Nagumo-Walsh functional equation, and a general version of Fréchet's unmixed functional equation.

Cited by

• Characterization of Probability Distributions on Locally Compact Abelian Groups by the Property of Identical Distribution of Linear Forms with Random Coefficients
• Characterization of polynomials by their invariance properties
• Heyde characterization theorem for some classes of locally compact Abelian groups

Related

• Aichinger equation on commutative semigroups
• On the equation P(f)=Q(g), where P,Q are polynomials and f,g are entire functions
• Characterization of Polynomials as solutions of certain functional equations
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• Functional composition of polynomials: indecomposability, Diophantine equations and lacunary polynomials