Chung, J.; Chung, S.-Y.; Kim, D. Real version of Paley-Wiener theorem for hyperfunctions and ultradistributions. (English) Zbl 0984.46026 Integral Transforms Spec. Funct. 10, No. 3-4, 201-210 (2000). 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. Cited in 1 Document MSC: 46F15 Hyperfunctions, analytic functionals 46F12 Integral transforms in distribution spaces 46F05 Topological linear spaces of test functions, distributions and ultradistributions Keywords:Fourier-Laplace transform; ultradistribution; compact support; Beurling type; Roumieu type; quasianalytic and non-quasianalytic cases; Paley-Wiener-Ehrenpreis theorem PDFBibTeX XMLCite \textit{J. Chung} et al., Integral Transforms Spec. Funct. 10, No. 3--4, 201--210 (2000; Zbl 0984.46026) Full Text: DOI References: [1] Hö,r,amder L., The analysis of linear partial differential operator 1 (1983) [2] Komatsu H., J. Fac. Sci. Univ. Todyo,Sect.AI 20 pp 25– (1973) [3] Mandache N., Rev. Roumaine Math. Pure Appl. 35 pp 321– (1990) [4] DOI: 10.1007/BF02590896 · Zbl 0172.42101 · doi:10.1007/BF02590896 [5] Treves F., Lectures on linear partial differential equation with compact support 7 (1961) · Zbl 0129.06905 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.