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Hyers-Ulam-Rassias stability of Cauchy equation in the space of Schwartz distributions. (English) Zbl 1066.39028

The Hyers-Ulam-Rassias stability problem is considered in the spaces of Schwartz tempered distributions and Fourier hyperfunctions. For all \(x,y\in{\mathbb R}^n\) we define \(A(x+y)=x+y\), \(P_1(x,y)=x\), \(P_2(x,y)=y\). Then the pullbacks \(u\circ A\), \(u\circ P_1\), \(u\circ P_2\) of a tempered distribution (or of a Fourier hyperfunction) \(u\) can be written in the way \[ \begin{aligned} \bigl\langle u\circ A,\varphi(x,y)\bigr\rangle&= \Bigl\langle u,\int\varphi(x-y,y)dy\Bigr\rangle,\\ \bigl\langle u\circ P_1,\varphi(x,y)\bigr\rangle&= \Bigl\langle u,\int\varphi(x,y)dy\Bigr\rangle,\\ \bigl\langle u\circ P_2,\varphi(x,y)\bigr\rangle&= \Bigl\langle u,\int\varphi(x,y)dx\Bigr\rangle \end{aligned} \] for all rapidly decreasing functions \(\varphi\) in \({\mathbb R}^n\).
The main result states that if a tempered distribution (or a Fourier hyperfunction) \(u\) satisfies an inequality \[ \| u\circ A-u\circ P_1-u\circ P_2\| \leq \varepsilon\bigl(\| x\| ^{2p}+\| y\| ^{2p}\bigr) \] for some \(\varepsilon>0\) and for some integer \(p>1\), then there exists a unique \(a\in{\mathbb C}^n\) such that \[ \| u-a\cdot x\| \leq\frac{2\varepsilon}{4^p-2}\| x\| ^{2p}. \] In the proof the heat kernels and Gauss transforms are used.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
35K05 Heat equation
39B52 Functional equations for functions with more general domains and/or ranges
46F10 Operations with distributions and generalized functions
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References:

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