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## Open Mathematics

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
Abstrakty
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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EN
Kategorie tematyczne
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Rocznik
Tom
Numer
Strony
665-692
Opis fizyczny
Daty
wydano
2012-04-01
online
2012-01-18
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autor
autor
Bibliografia
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• [3] Carmichael R.D., Kaminski A., Pilipovic S., Notes on Boundary Values in Ultradistribution Spaces, Lecture Notes Series, 49, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, 1999
• [4] Carmichael R.D., Pathak R.S., Pilipovic S., Holomorphic functions in tubes associated with ultradistributions, Complex Variables Theory Appl., 1993, 21(1–2), 49–72
• [5] Carmichael R., Pilipovic S., On the convolution and the Laplace transformation in the space of Beurling-Gevrey tempered ultradistributions, Math. Nachr., 1992, 158(1), 119–132 http://dx.doi.org/10.1002/mana.19921580109
• [6] Edwards R.E., Functional Analysis. Theory and Applications, Holt, Rinehart and Winston, New York-Toronto-London, 1965
• [7] Ehrenpreis L., Fourier Analysis in Several Complex Variables, Pure Appl. Math. (N. Y.), 17, John Wiley & Sons, New York-London-Sydney, 1970
• [8] Hansen S., Localizable analytically uniform spaces and the fundamental principle, Trans. Amer. Math. Soc., 1981, 264(1), 235–250 http://dx.doi.org/10.1090/S0002-9947-1981-0597879-2
• [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
• [10] Komatsu H., Ultradistributions I. Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo Sect. IA Mat., 1973, 20, 25–105
• [11] Komatsu H., Ultradistributions II. The kernel theorem and ultradistributions with support in a submanifold, J. Fac. Sci. Univ. Tokyo Sect. IA Mat., 1977, 24(3), 607–628
• [12] Krivosheev A.S., Napalkov V.V., Complex analysis and convolution operators, Uspekhi Mat. Nauk, 1992, 47(6), 3–58 (in Russian)
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• [14] Michalik S., Laplace ultradistributions supported by a cone, Banach Center Publ., 2010, 88, 229–241 http://dx.doi.org/10.4064/bc88-0-18
• [15] Musin I.Kh., Fedotova P.V., A theorem of Paley-Wiener type for ultradistributions, Mat. Zametki, 2009, 85(6), 894–914 (in Russian)
• [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
• [18] Palamodov V.P., Linear Differential Operators with Constant Coefficients, Grundlehren Math. Wiss., 168, Springer, New York-Berlin, 1970
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• [20] de Roever J.W., Complex Fourier Transformation and Analytic Functionals with Unbounded Carriers, Mathematical Centre Tracts, 89, Mathematisch Centrum, Amsterdam, 1978
• [21] de Roever J.W., Analytic representations and Fourier transforms of analytic functionals in Z′ carried by the real space, SIAM J. Math. Anal., 1978, 9(6), 996–1019 http://dx.doi.org/10.1137/0509081
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• [23] Taylor B.A., Analytically uniform spaces of infinitely differentiable functions, Comm. Pure Appl. Math., 1971, 24(1), 39–51 http://dx.doi.org/10.1002/cpa.3160240105
• [24] Vladimirov V.S., Functions which are holomorphic in tubular cones, Izv. Akad. Nauk SSSR Ser. Mat., 1963, 27, 75–100 (in Russian)
• [25] Vladimirov V.S., Generalized Functions in Mathematical Physics, Mir, Moscow, 1979
• [26] Vladimirov V.S., Drozhzhinov Yu.N., Zavyalov B.I., Multidimensional Tauber Theorems for Generalized Functions, Nauka, Moscow, 1986 (in Russian)
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