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

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Musin I. d., Yakovleva P.

Open Mathematics

2012

tom 10

nr 2

665-692

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.

Ograniczanie wyników

1 Open Mathematics

1 Musin I. d.

1 Yakovleva P.

1 2012

Wyniki wyszukiwania

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