Convolution operators on spaces of real analytic functions

Let \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$I\subset \mathbb {R}$\end{document} be an open interval and let \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mu \in A(\mathbb {R})^{\prime }$\end{document} and \documentclass{article}\us...

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Bibliographic Details
Published in:Mathematische Nachrichten 2013-06, Vol.286 (8-9), p.908-920
Main Author: Langenbruch, Michael
Format: Article
Language:English
Subjects:
46F15
46M18
convolution operator
Fourier transforms
Real analytic functions
right inverse MSC Primary: 35R50
Secondary: 26E05
surjectivity
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Summary:Let \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$I\subset \mathbb {R}$\end{document} be an open interval and let \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mu \in A(\mathbb {R})^{\prime }$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$G:=\mbox{conv}(\mbox{supp}(\mu ))$\end{document}. We characterize the surjectivity of the convolution operator Tμ: A(I − G) → A(I) by means of a new estimate from below for the Fourier transform \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\widehat{\mu }$\end{document} valid on conical subsets of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {C}\setminus \mathbb {R}$\end{document}. We also characterize when Tμ admits a continuous linear right inverse.
ISSN:0025-584X
1522-2616
DOI:10.1002/mana.201100155