×

A Paley-Wiener theorem for convex sets in \({\mathbb{C}}^{n}\). (English) Zbl 1009.32003

From the abstract: The author studies the Laplace transform on Hardy spaces on a class of convex domains in \(\mathbb C^n\) and obtains a Paley-Wiener theorem with a norm that characterizes the entire functions of exponential type which occur as Laplace transforms. This is done by using the Fantappiè transform and the Borel transform to rewrite the Laplace transform and to reduce the problem to known theorems in one complex variable.

MSC:

32A15 Entire functions of several complex variables
44A10 Laplace transform
46E20 Hilbert spaces of continuous, differentiable or analytic functions
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Andersson, M., Topics in Complex Analysis (1996), Springer-Verlag · Zbl 0888.30001
[2] M. Andersson, M. Passare, R. Sigurdsson, Complex convexity and analytic functionals I, Technical Report RH-06-95, Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland, June 1995; M. Andersson, M. Passare, R. Sigurdsson, Complex convexity and analytic functionals I, Technical Report RH-06-95, Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland, June 1995 · Zbl 1057.32001
[3] M. Andersson, M. Passare, R. Sigurdsson, Complex convexity and analytic functionals II, Technical Report RH-20-2000, Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland, September 2000. Also published as Report no 8, 2000, Department of Mathematics, Mid-Sweden University, S-851 70 Sundsvall, Sweden; M. Andersson, M. Passare, R. Sigurdsson, Complex convexity and analytic functionals II, Technical Report RH-20-2000, Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland, September 2000. Also published as Report no 8, 2000, Department of Mathematics, Mid-Sweden University, S-851 70 Sundsvall, Sweden · Zbl 1057.32001
[4] Berndtsson, B., Weighted integral formulas, (Fornaess, J. E., Several Complex Variables: Proceedings of the Mittag-Leffler Institute, 1987-1988 (1993), Princeton University Press), 160-187 · Zbl 0786.32003
[5] Berndtsson, B., An inequality for Fourier-Laplace transforms of entire functions, and the existence of exponential frames in Fock space, J. Funct. Anal., 149, 1, 83-101 (1997) · Zbl 0910.46019
[6] Gindikin, S. G.; Henkin, G. M., Integral geometry for \(∂̄\)-cohomology in \(q\)-linearly concave domains in \(cp^n\), Funktsional. Anal. i Prilozhen., 12, 4, 6-23 (1978) · Zbl 0409.32020
[7] Hansson, T., On Hardy spaces in complex ellipsoids, Ann. Inst. Fourier (Grenoble), 49, 5, 1477-1501 (1999) · Zbl 0944.32004
[8] Hörmander, L., Notions of Convexity (1994), Birkhäuser: Birkhäuser Boston · Zbl 0835.32001
[9] Katsnel’son, V.È., A generalization of the Paley-Wiener theorem on a representation for entire functions of exponential type, Teor. Funktsiı̌. Funktsional. Anal. i Prilozhen., 1, 99-110 (1965), (in Russian)
[10] Kerzman, N.; Stein, E. M., The Szegö kernel in terms of Cauchy-Fantappiè kernels, Duke Math. J., 45, 2, 197-224 (1978) · Zbl 0387.32009
[11] Kiselman, C. O., A differential inequality characterizing weak lineal convexity, Math. Ann., 311, 1-10 (1998) · Zbl 0911.32031
[12] Ja. Levin, B., Distribution of Zeros of Entire Functions. Distribution of Zeros of Entire Functions, Translations of Mathematical Monographs (1964), American Mathematical Society · Zbl 0152.06703
[13] Likht, M. K., A remark about the Paley-Wiener theorem on entire functions of exponential type, Uspekhi Mat. Nauk, 19, 1, 115, 169-171 (1964), (in Russian)
[14] Lindholm, N., Sampling and Fourier-Laplace transforms in several complex variables, PhD thesis (2000), Göteborg University · Zbl 0964.32002
[15] Lutsenko, V. I.; Yulmukhametov, R. S., A generalization of the Paley-Wiener theorem to functionals on Smirnov spaces, Proc. Steklov Inst. Math., 200, 2, 271-280 (1993) · Zbl 0802.46037
[16] Lyubarskiı̌, Yu. I., The Paley-Wiener theorem for convex sets, Soviet J. Contemp. Math. Anal., 23, 64-74 (1988), (English translation. Original article in: Izv. Akad. Nauk. Armyan. SSR, Ser. Mat. 23 (2) (1988) 163-172 · Zbl 0662.30022
[17] Martineau, A., Équations différentielles d’ordre infini, Bull. Soc. Math. France, 95, 109-154 (1967) · Zbl 0167.44202
[18] Range, R. M., Holomorphic Functions and Integral Representations in Several Complex Variables (1986), Springer-Verlag · Zbl 0591.32002
[19] Znamenskiı̆, S. V., A geometric criterion of strong linear convexity, Funktsional. Anal. i Prilozhen., 13, 3, 83-84 (1979), English translation in: Funct. Anal. Appl. 13 (3) (1979) 224-225
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.