Characterization of Polynomials as solutions of certain functional equations

2017/02/03 by J. M. Almira, Almira, J. M. · 2 citations
Mathematics · #Functional Equations Stability Results #Mathematical and Theoretical Analysis #Advanced Topology and Set Theory

paper · pdf · doi:10.48550/arxiv.1702.00875

Abstract

Recently, the functional equation ∑_i=0^mf_i(b_ix+c_iy)= ∑_i=1ⁿa_i(y)v_i(x) with x,y∈ℝ^d and b_i,c_i\inGL_d(ℂ), was studied by Almira and Shulman, both in the classical context of continuous complex valued functions and in the framework of complex valued Schwartz distributions, where these equations were properly introduced in two different ways. The solution sets of these equations are, typically, exponential polynomials and, in some particular cases, they reduce to ordinary polynomials. In this paper we present several characterizations of ordinary polynomials as the solution sets of certain related functional equations. Some of these equations are important because of their connection with the Characterization Problem of distributions in Probability Theory.

Cited by

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

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