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

Abstract

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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 ∂ ¯ operator. Also, some new facts on the Fourier-Laplace transform of tempered distributions complementing some well-known results of V.S. Vladimirov are employed.

How to cite

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Il’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

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