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On some classes of infinitely decomposable Sobolev-Schwartz test functions. (English) Zbl 0611.46047
Let G(z) \((z=x+iy)\) for a function of exponential type, not identically zero such that
(1) \(| G(z)| \leq e^{\alpha | y|},\)
(2) \(| 1-G(z)| \leq A| z|\) for \(| z| <1,\)
(3) \(| G(t)| \leq \frac{B}{| t|}\) for \(t\leq -1\) and \(t\geq 1,\)
where A, B and \(\alpha\) are positive constants. The author proves that for each choice of a sequence \(\{\lambda_ n\}\subseteq {\mathbb{R}}\) with \(\sum^{\infty}_{n=1}| \lambda_ n| <\infty\) the product \(\prod^{\infty}_{n=1}G(\lambda_ nz)\) represents an entire function, not identically zero which is the Fourier transform of a test-function belonging to the Schwartz space \({\mathcal D}\). Denoting by \({\mathcal D}_ G\subseteq {\mathcal D}\) the class of such test-functions depending on the sequences \(\{\lambda_ n\}\) one observers that any element \(\phi\in {\mathcal D}_ G\) can be represented as the convolution \(\phi =\phi_ 1*\phi_ 2\) of elements \(\phi_ 1,\phi_ 2\in {\mathcal D}_ G\) implying \[ {\mathcal D}_ G={\mathcal D}_ G*{\mathcal D}_ G. \] It is also mentioned that P. Mikusiński constructed another subclass \({\mathcal D}_ 0\subseteq {\mathcal D}\) satisfying the relation \({\mathcal D}_ 0*{\mathcal D}_ 0={\mathcal D}_ 0\) without any help of Fourier transform.
Reviewer: L.Janos
46F05 Topological linear spaces of test functions, distributions and ultradistributions
46F10 Operations with distributions and generalized functions
46F12 Integral transforms in distribution spaces
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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