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On an Exponential Functional Inequality and its Distributional Version

Published online by Cambridge University Press:  20 November 2018

Jaeyoung Chung
Jaeyoung Chung*
Affiliation:
Department of Mathematics, Kunsan National University, Kunsan, 573-701 Korea. e-mail: jychung@kunsan.ac.kr
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Abstract

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Let $G$ be a group and $\mathbb{K}\,=\,\mathbb{C}\,\text{or}\,\mathbb{R}$. In this article, as a generalization of the result of Albert and Baker, we investigate the behavior of bounded and unbounded functions $f\,:\,G\,\to \,\mathbb{K}$ satisfying the inequality

$$\left| f\left( \sum\limits_{k=1}^{n}{{{x}_{k}}} \right)\,-\,\underset{k=1}{\overset{n}{\mathop{\Pi }}}\,f\left( {{x}_{k}} \right) \right|\,\,\le \phi \left( {{x}_{2}},\,.\,.\,.\,,{{x}_{n}} \right),\,\,\,\,\,\,\forall {{x}_{1}},\,.\,.\,.\,,{{x}_{n}}\,\in \,G,$$

Where $\phi :\,{{G}^{n-1}}\,\to \,[0,\,\infty )$. Also as a a distributional version of the above inequality we consider the stability of the functional equation

$$u\,\circ \,S\,-\,\overbrace{u\,\otimes \,.\,.\,.\,\otimes \,u}^{n-\text{times}}\,=\,0,$$

where $u$ is a Schwartz distribution or Gelfand hyperfunction, $\circ$ and $\otimes$ are the pullback and tensor product of distributions, respectively, and $S\left( {{x}_{1}},\,.\,.\,.\,,{{x}_{n}} \right)\,=\,{{x}_{1}}\,+\,.\,.\,.\,+\,{{x}_{n}}$.

Keywords

46F9939B82distributionexponential functional equationGelfand hyperfunctionstability
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 1 , 01 March 2015 , pp. 30 - 43
Copyright
Copyright © Canadian Mathematical Society 2015

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