The stability of Cauchy equations in the space of Schwartz distributions
The authors reformulate the classical result of \textit{D.H. Hyers} [Proc. Natl. Acad. Sci. USA 27, 222-224 (1941; Zbl 0061.26403)] concerning the stability of the Cauchy functional equation \(f(x+y)=f(x)+f(y)\). Let \(S=S(\mathbb R^n)\) denotes the Schwartz space of rapidly decreasing functions on \(\mathbb R^n\) and \(S'\) (its dual) the space of tempered distributions. For \(u\in S'\) the pullbacks \(u\circ A\), \(u\circ P_1\), \(u\circ P_2\) of \(u\) are defined by \[ \langle u\circ A,\varphi(x+y)\rangle=\left\langle u, \int\varphi(x-y,y)\,dy\right\rangle; \] \[ \langle u\circ P_1,\varphi(x+y)\rangle=\left\langle u, \int\varphi(x,y)\,dy\right\rangle; \] \[ \langle u\circ P_2,\varphi(x+y)\rangle=\left\langle u, \int\varphi(x,y)\,dx\right\rangle \] for all test functions \(\varphi\in S(\mathbb R^{2n})\). Finally, for \(v\in S'\), \(\| v\| \leq\varepsilon\) means that \(| \langle v,\varphi\rangle| \leq\varepsilon\| \varphi\| _{L_1}\) for \(\varphi\in S\). A distribution \(u\in S'\) is called \(\varepsilon\)-additive if \[ \| u\circ A-u\circ P_1-u\circ P_2\| \leq\varepsilon. \] The main result of the paper states that each \(\varepsilon\)-additive distribution \(u\) can be written uniquely in the form \( u=c\cdot x +h(x) \) with \(c\in\mathbb{C}^n\) and a bounded measurable function \(h\) such that \(\| h\| _{L_{\infty}}\leq\varepsilon\). Similar investigations are conducted for a counterpart of the Cauchy equation \(f(x+y)=f(x)f(y)\) and its superstability.
scientific article; zbMATH DE number 2093796
scientific article