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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A symbolic calculus for analytic Carleman classes
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by Jamil A. Siddiqi and Mostéfa Ider PDF
Proc. Amer. Math. Soc. 99 (1987), 347-350 Request permission

Abstract:

Let ${\mathcal {C}_M}\left ( {{I_\alpha }} \right )$ be the analytic Carleman class of ${\mathcal {C}^\infty }$-functions $f$ defined in a sector ${I_\alpha } = \left \{ {z \in {\mathbf {C}}:|\arg z| \leqslant \alpha \pi /2} \right \} \cup \left \{ 0 \right \}\left ( {0 \leqslant \alpha \leqslant 1} \right )$ and analytic in its interior such that ${\left \| {{f^{\left ( n \right )}}} \right \|_\infty } \leqslant C{\lambda ^n}{M_n}\left ( {n \geqslant 0} \right ),C = C\left ( f \right ),\lambda = \lambda \left ( f \right )$. 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.
References
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Additional Information
  • © Copyright 1987 American Mathematical Society
  • 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