On the Hyers-Ulam-Rassias stability of the Jensen equation in distributions
Commun. Korean Math. Soc. 2011 Vol. 26, No. 2, 261-271
https://doi.org/10.4134/CKMS.2011.26.2.261
Printed June 1, 2011
Eun Gu Lee and Jaeyoung Chung
Dongyang Mirae University, Kunsan National University
Abstract : We consider the Hyers-Ulam-Rassias stability problem $$ \left\|2u\circ \frac A 2 -u\circ P_1 -u\circ P_2 \right\|\le \varepsilon (|x|^p +|y|^p),\quad x, y \in \Bbb R^n $$ for the Schwartz distributions $u$, which is a distributional version of the Hyers-Ulam-Rassias stability problem of the Jensen functional equation $$ \left|2f\left(\frac{x+y} 2 \right) - f(x) - f(y)\right| \le \varepsilon (|x|^p +|y|^p),\quad x, y \in \Bbb R^n $$ for the function $f :\Bbb R^n \to \Bbb C$.
Keywords : stability, Gauss transforms, heat kernel, distributions, tempered distribution, Jensen functional equation
MSC numbers : 39B82, 46F99
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