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The stability of Cauchy equations in the space of Schwartz distributions. (English) Zbl 1053.39043

The authors reformulate the classical result of D.H. Hyers [Proc. Natl. Acad. Sci. USA 27, 222-224 (1941; Zbl 0061.26403)] concerning the stability of the Cauchy functional equation $f\left(x+y\right)=f\left(x\right)+f\left(y\right)$. Let $S=S\left({ℝ}^{n}\right)$ denotes the Schwartz space of rapidly decreasing functions on ${ℝ}^{n}$ and ${S}^{\text{'}}$ (its dual) the space of tempered distributions. For $u\in {S}^{\text{'}}$ the pullbacks $u\circ A$, $u\circ {P}_{1}$, $u\circ {P}_{2}$ of $u$ are defined by

$〈u\circ A,\phi \left(x+y\right)〉=〈u,\int \phi \left(x-y,y\right)\phantom{\rule{0.166667em}{0ex}}dy〉;$
$〈u\circ {P}_{1},\phi \left(x+y\right)〉=〈u,\int \phi \left(x,y\right)\phantom{\rule{0.166667em}{0ex}}dy〉;$
$〈u\circ {P}_{2},\phi \left(x+y\right)〉=〈u,\int \phi \left(x,y\right)\phantom{\rule{0.166667em}{0ex}}dx〉$

for all test functions $\phi \in S\left({ℝ}^{2n}\right)$. Finally, for $v\in {S}^{\text{'}}$, $\parallel v\parallel \le \epsilon$ means that $|〈v,\phi 〉|\le {\epsilon \parallel \phi \parallel }_{{L}_{1}}$ for $\phi \in S$. A distribution $u\in {S}^{\text{'}}$ is called $\epsilon$-additive if

$\parallel u\circ A-u\circ {P}_{1}-u\circ {P}_{2}\parallel \le \epsilon ·$

The main result of the paper states that each $\epsilon$-additive distribution $u$ can be written uniquely in the form $u=c·x+h\left(x\right)$ with $c\in {ℂ}^{n}$ and a bounded measurable function $h$ such that ${\parallel h\parallel }_{{L}_{\infty }}\le \epsilon$. Similar investigations are conducted for a counterpart of the Cauchy equation $f\left(x+y\right)=f\left(x\right)f\left(y\right)$ and its superstability.

##### MSC:
 39B82 Stability, separation, extension, and related topics 39B52 Functional equations for functions with more general domains and/or ranges 46F10 Operations with distributions (generalized functions)