On an Exponential Functional Inequality and its Distributional Version

Canadian mathematical bulletin

Jaeyoung Chung¹ 

[1] Department of Mathematics, Kunsan National University, Kunsan, 573-701 Korea

关键词: distribution;    exponential functional equation;    Gelfand hyperfunction;    stability;   

DOI  :  10.4153/CMB-2014-012-x

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Let $G$ be a group and $mathbb K=mathbb C$ or $mathbbR$.In this article, as a generalization of the result of Albert and Baker, we investigate the behavior of boundedand unbounded functions $fcolon Go mathbb K$ satisfying the inequality$Bigl|f Bigl(sum_{k=1}^n x_k Bigr)-prod_{k=1}^n f(x_k) Bigr|le phi(x_2, dots, x_n),quad forall, x_1, dots,x_nin G,$where $phicolon G^{n-1}o [0, infty)$. Also, as a distributional version of the above inequality we consider the stability ofthe functional equationegin{equation*}ucirc S - overbrace{uotimes cdots otimes u}^{n-ext {times}}=0,end{equation*}where $u$ is a Schwartz distribution or Gelfand hyperfunction, $circ$ and $otimes$ are the pullback and tensor product of distributions, respectively, and $S(x_1, dots, x_n)=x_1+ dots+x_n$.

【 授权许可】

Unknown