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A symbolic calculus for analytic Carleman classes


Authors: Jamil A. Siddiqi and Mostéfa Ider
Journal: Proc. Amer. Math. Soc. 99 (1987), 347-350
MSC: Primary 46J15; Secondary 30D60, 46E15
DOI: https://doi.org/10.1090/S0002-9939-1987-0870798-0
MathSciNet review: 870798
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Abstract: Let $ {\mathcal{C}_M}\left( {{I_\alpha }} \right)$ be the analytic Carleman class of $ {\mathcal{C}^\infty }$-functions $ f$ defined in a sector $ {I_\alpha } = \left\{ {z \in {\mathbf{C}}:\vert\arg z\vert \leqslant \alpha \p... ...\right\} \cup \left\{ 0 \right\}\left( {0 \leqslant \alpha \leqslant 1} \right)$ and analytic in its interior such that $ {\left\Vert {{f^{\left( n \right)}}} \right\Vert _\infty } \leqslant C{\lambda... ...n \geqslant 0} \right),C = C\left( f \right),\lambda = \lambda \left( f \right)$. In this paper, we give necessary and sufficient conditions in order that $ {\mathcal{C}_M}\left( {{I_\alpha }} \right)$ be inverse-closed. As a corollary, we obtain a characterization of $ {\mathcal{C}_M}\left( {{{\mathbf{R}}_ + }} \right)$ as an inverse-closed algebra, thus establishing the converse of a theorem of Malliavin [4] for the half-line.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1987-0870798-0
Keywords: Symbolic calculus, inverse-closed algebra, Carleman classes
Article copyright: © Copyright 1987 American Mathematical Society

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