On the non-triviality of certain spaces of analytic functions. Hyperfunctions and ultrahyperfunctions of fast growth

  • Andreas Debrouwere
  • Jasson Vindas
Original Paper


We study function spaces consisting of analytic functions with fast decay on horizontal strips of the complex plane with respect to a given weight function. Their duals, so called spaces of (ultra)hyperfunctions of fast growth, generalize the spaces of Fourier hyperfunctions and Fourier ultrahyperfunctions. An analytic representation theory for their duals is developed and applied to characterize the non-triviality of these function spaces in terms of the growth order of the weight function. In particular, we show that the Gelfand–Shilov spaces of Beurling type \(\mathcal {S}^{(p!)}_{(M_p)}\) and Roumieu type \(\mathcal {S}^{\{p!\}}_{\{M_p\}}\) are non-trivial if and only if
$$\begin{aligned} \sup _{p \ge 2}\frac{(\log p)^p}{h^pM_p} < \infty , \end{aligned}$$
for all \(h > 0\) and some \(h > 0\), respectively. We also study boundary values of holomorphic functions in spaces of ultradistributions of exponential type, which may be of quasianalytic type.


Spaces of analytic functions Hyperfunctions Ultrahyperfunctions Ultradistributions Boundary values Analytic representations Non-triviality Laplace transform Gelfand–Shilov spaces 

Mathematics Subject Classification

Primary 30D60 46E10 46F15 Secondary 46F05 46F12 46F20 


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Copyright information

© Springer-Verlag Italia 2017

Authors and Affiliations

  1. 1.Department of MathematicsGhent UniversityGhentBelgium

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