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

Il’dar Musin; Polina Yakovleva

Open Mathematics (2012)

- Volume: 10, Issue: 2, page 665-692
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topIl’dar Musin, and Polina Yakovleva. "On a space of smooth functions on a convex unbounded set in ℝn admitting holomorphic extension in ℂn." Open Mathematics 10.2 (2012): 665-692. <http://eudml.org/doc/269776>.

@article{Il2012,

abstract = {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.},

author = {Il’dar Musin, Polina Yakovleva},

journal = {Open Mathematics},

keywords = {Ultradistributions; Tempered distributions; Fourier-Laplace transform; Holomorphic functions; Entire functions; ultradistributions; entire functions},

language = {eng},

number = {2},

pages = {665-692},

title = {On a space of smooth functions on a convex unbounded set in ℝn admitting holomorphic extension in ℂn},

url = {http://eudml.org/doc/269776},

volume = {10},

year = {2012},

}

TY - JOUR

AU - Il’dar Musin

AU - Polina Yakovleva

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

JO - Open Mathematics

PY - 2012

VL - 10

IS - 2

SP - 665

EP - 692

AB - 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.

LA - eng

KW - Ultradistributions; Tempered distributions; Fourier-Laplace transform; Holomorphic functions; Entire functions; ultradistributions; entire functions

UR - http://eudml.org/doc/269776

ER -

## References

top- [1] Berenstein C.A., Dostal M.A., Analytically Uniform Spaces and Their Applications to Convolution Equations, Lecture Notes in Math., 256, Springer, Berlin-New York, 1972 Zbl0237.47025
- [2] Berenstein C., Struppa D., Complex analysis and convolution equations, In: Complex Analysis - Several Variables V, Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fundam. Napravl., 54, VINITI, Moscow, 1989, 5–111 (in Russian) Zbl0787.46032
- [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 Zbl0969.46027
- [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 Zbl0809.46036
- [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 Zbl0786.46037
- [6] Edwards R.E., Functional Analysis. Theory and Applications, Holt, Rinehart and Winston, New York-Toronto-London, 1965 Zbl0182.16101
- [7] Ehrenpreis L., Fourier Analysis in Several Complex Variables, Pure Appl. Math. (N. Y.), 17, John Wiley & Sons, New York-London-Sydney, 1970 Zbl0195.10401
- [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 Zbl0482.46023
- [9] Hörmander L., L 2 estimates and existence theorems for the $\overline{\partial}$ operator, Acta Math., 1965, 113(1), 89–152 http://dx.doi.org/10.1007/BF02391775 Zbl0158.11002
- [10] Komatsu H., Ultradistributions I. Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo Sect. IA Mat., 1973, 20, 25–105 Zbl0258.46039
- [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 Zbl0385.46027
- [12] Krivosheev A.S., Napalkov V.V., Complex analysis and convolution operators, Uspekhi Mat. Nauk, 1992, 47(6), 3–58 (in Russian) Zbl0801.46051
- [13] Łusik G., Laplace ultradistributions on a half line and a strong quasi-analyticity principle, Ann. Polon. Math., 1996, 63(1), 13–33
- [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 Zbl1202.46046
- [15] Musin I.Kh., Fedotova P.V., A theorem of Paley-Wiener type for ultradistributions, Mat. Zametki, 2009, 85(6), 894–914 (in Russian) Zbl1186.46037
- [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 Zbl0172.42101
- [18] Palamodov V.P., Linear Differential Operators with Constant Coefficients, Grundlehren Math. Wiss., 168, Springer, New York-Berlin, 1970 Zbl0191.43401
- [19] Robertson A.P., Robertson W., Topological Vector Spaces, Cambridge Tracts in Math., 53, Cambridge University Press, Cambridge-New York, 1980 Zbl0423.46001
- [20] de Roever J.W., Complex Fourier Transformation and Analytic Functionals with Unbounded Carriers, Mathematical Centre Tracts, 89, Mathematisch Centrum, Amsterdam, 1978 Zbl0406.46032
- [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 Zbl0406.46033
- [22] Sebastião e Silva J., Su certe classi di spazi localmente convessi importanti per le applicazioni, Rend. Mat. e Appl., 1955, 14, 388–410
- [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 Zbl0205.41501
- [24] Vladimirov V.S., Functions which are holomorphic in tubular cones, Izv. Akad. Nauk SSSR Ser. Mat., 1963, 27, 75–100 (in Russian) Zbl0149.09602
- [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)
- [27] Yulmukhametov R.S., Entire functions of several variables with given behavior at infinity, Izv. Ross. Akad. Nauk Ser. Mat., 1996, 60(4), 205–224 (in Russian)
- [28] Zharinov V.V., Compact families of locally convex spaces and FS and DFS spaces, Uspekhi Mat. Nauk, 1979, 34(4), 97–131 (in Russian) Zbl0412.46003

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.