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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


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
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.

References [Enhancements On Off] (What's this?)

  • [1] J. Boman and L. Hörmander, Classes of infinitely differentiable functions (mimeographed notes), Stockholm, 1962.
  • [2] Joaquim Bruna, On inverse-closed algebras of infinitely differentiable functions, Studia Math. 69 (1980/81), no. 1, 59–68. MR 604354 (82e:46039)
  • [3] B. I. Korenbljum, Conditions of non-triviality of certain classes of functions analytic in a sector and problems of quasi-analyticity, Soviet Math. Dokl. 7 (1966), 232-236.
  • [4] Paul Malliavin, Calcul symbolique et sous-algèbres de 𝐿₁(𝐺). I, II, Bull. Soc. Math. France 87 (1959), 181–186, 187–190. MR 0117505 (22 #8284)
  • [5] S. Mandelbrojt, Séries adhérentes, régularisation des suites, applications, Gauthier-Villars, Paris, 1952 (French). MR 0051893 (14,542f)
  • [6] Walter Rudin, Division in algebras of infinitely differentiable functions, J. Math. Mech. 11 (1962), 797–809. MR 0153796 (27 #3757)
  • [7] J. A. Siddiqi and A. El Koutri, Cartan-Gorny-Kolmogorov type inequalities for generators of analytic semi-groups and cosine operator functions, preprint.

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Additional Information

PII: S 0002-9939(1987)0870798-0
Keywords: Symbolic calculus, inverse-closed algebra, Carleman classes
Article copyright: © Copyright 1987 American Mathematical Society