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Ultradifferentiable functions and Fourier analysis. (English) Zbl 0735.46022
In Beurling’s approach to ultradifferentiable functions, decay properties of the Fourier transform of the functions are imposed rather than growth conditions on the derivatives, as it has been done classically.
In the present article the authors modify Beurling’s approach. They define for nonempty open subsets \(\Omega\) of \(\mathbb{R}^ N\) the spaces \[ \begin{aligned} {\mathcal D}_{(\omega)}(\Omega) &:= \{f\in C_ c(\Omega)\mid\hbox{ for all } k>0 : \int_{\mathbb{R}^ N}| \hat f(t)| e^{k\omega(t)}dt<\infty\} \\ \text{and} {\mathcal D}_{\{\omega\}}(\Omega) &:= \{f\in C_ c(\Omega)\mid\hbox{ there is }\varepsilon>0 : \int_{\mathbb{R}^ N}|\hat f(t)| e^{\varepsilon\omega(t)}dt<\infty\}, \end{aligned} \] where the weight \(\omega\) satisfies \(\omega(2t)=O(\omega(t))\) (instead of the subadditivity assumed by Beurling-Björck), and prove their nontriviality. If \(\omega\) is assumed to be logarithmically convex, these two classes can also be described in terms of bounds on the derivatives and the \(L^ 2\)-techniques of Hörmander can be made use of.
The authors present the theory of ultradifferentiable functions and ultradistributions in this new setting: the analogue of the Paley-Wiener theorem, a representation as sequence spaces, convolutions, the analogue of the Paley-Wiener-Schwartz theorem, and tensor products.
Reviewer: S.Dierolf (Trier)

MSC:
46F05 Topological linear spaces of test functions, distributions and ultradistributions
46E10 Topological linear spaces of continuous, differentiable or analytic functions
46F10 Operations with distributions and generalized functions
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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