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Real version of Paley-Wiener theorem for hyperfunctions and ultradistributions. (English) Zbl 0984.46026

Summary: We prove that the following conditions (i) and (ii) are equivalent for an finitely differentiable function \(\widehat u(\xi)\) on \(\mathbb{R}^n\):
(i) \(\widehat u(\xi)\) is the Fourier-Laplace transform of an ultradistribution \(u\) with compact support of Beurling type \((M_p)\) (resp. of Roumieu type \(\{M_p\}\).
(ii) For every \(\varepsilon> 0\) there are positive constants \(L\) and \(C\) (resp. for every \(\varepsilon> 0\) there is a constant \(C\)) such that \[ |\partial^\alpha\widehat u(\xi)|\leq CB_{K,\varepsilon,\alpha} \exp M(L|\xi|) (\text{resp. }|\partial^\alpha\widehat u(\xi)|\leq CB_{K,\varepsilon,\alpha} \exp M(\varepsilon|\xi|)), \] where \(B_{K,\varepsilon,\alpha}= \inf_{\rho\in \mathbb{R}^n_+}\alpha!\exp(H_K(\rho)+ \varepsilon|\rho|)/\rho^\alpha\), \(M(\rho)\) is the associated function of \(M_p\) and \(H_K\) is the supporting function of \(K\).
Since our result includes both quasianalytic and non-quasianalytic cases as a corollary we also obtain a real version of the Paley-Wiener-Ehrenpreis theorem for hyperfunctions with support in a compact set.

MSC:

46F15 Hyperfunctions, analytic functionals
46F12 Integral transforms in distribution spaces
46F05 Topological linear spaces of test functions, distributions and ultradistributions
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References:

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