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arXiv 1702.00875math.CA
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多项式作为某些函数方程解的特征

Characterization of Polynomials as solutions of certain functional equations

J. M. Almira

备注：
13 pages, submitted to a Journal. arXiv admin note: formerly part of arXiv:1612.03756
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摘要：

最近，函数方程\[ \sum_{i=0}^mf_i(b_ix+c_iy)= \sum_{i=1}^na_i(y)v_i(x) \]与$x,y\in\mathbb{R}^d$和$b_i,c_i\in\mathbf{GL}_d(\mathbb{C})$，由Almira和Shulman研究，无论是在经典的连续复值函数的背景下，并在复值施瓦茨分布的框架，这些方程适当地介绍了两种不同的方式。这些方程的解集通常是指数多项式，在某些特定情况下，它们可以简化为普通多项式。本文给出了普通多项式作为某些相关函数方程解集的几个刻画。其中一些方程很重要，因为它们与概率论中分布的特征化问题有关。

英文摘要：

Recently, the functional equation \[ \sum_{i=0}^mf_i(b_ix+c_iy)= \sum_{i=1}^na_i(y)v_i(x) \] with $x,y\in\mathbb{R}^d$ and $b_i,c_i\in\mathbf{GL}_d(\mathbb{C})$, 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.

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