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2012 | 10 | 2 | 665-692
Tytuł artykułu

On a space of smooth functions on a convex unbounded set in ℝn admitting holomorphic extension in ℂn

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EN
For some given logarithmically convex sequence M of positive numbers we construct a subspace of the space of rapidly decreasing infinitely differentiable functions on an unbounded closed convex set in ℝn. Due to the conditions on M each function of this space admits a holomorphic extension in ℂn. In the current article, the space of holomorphic extensions is considered and Paley-Wiener type theorems are established. To prove these theorems, some auxiliary results on extensions of holomorphic functions satisfying some weighted L 2-bounds in a domain of holomorphy in ℂn are obtained with the aid of L. Hörmander’s method of L 2-bounds for the $$\bar \partial$$ operator. Also, some new facts on the Fourier-Laplace transform of tempered distributions complementing some well-known results of V.S. Vladimirov are employed.
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Bibliografia
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  • [9] Hörmander L., L 2 estimates and existence theorems for the \(\bar \partial\) operator, Acta Math., 1965, 113(1), 89–152 http://dx.doi.org/10.1007/BF02391775
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  • [16] Musin I.Kh., Yakovleva P.V., On a space of rapidly decreasing infinitely differentiable functions on an unbounded convex set in ℝn and its dual, preprint available at http://arxiv.org/abs/1003.3302
  • [17] Neymark M., On the Laplace transform of functionals on classes of infinitely differentiable functions, Ark. Mat., 1969, 7(6), 577–594 http://dx.doi.org/10.1007/BF02590896
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  • [24] Vladimirov V.S., Functions which are holomorphic in tubular cones, Izv. Akad. Nauk SSSR Ser. Mat., 1963, 27, 75–100 (in Russian)
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Typ dokumentu
Bibliografia
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bwmeta1.element.doi-10_2478_s11533-011-0142-8
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