A symbolic calculus for analytic Carleman classes
Authors:
Jamil A. Siddiqi and Mostéfa Ider
Journal:
Proc. Amer. Math. Soc. 99 (1987), 347350
MSC:
Primary 46J15; Secondary 30D60, 46E15
DOI:
https://doi.org/10.1090/S00029939198708707980
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 ${I_\alpha } = \left \{ {z \in {\mathbf {C}}:\arg z \leqslant \alpha \pi /2} \right \} \cup \left \{ 0 \right \}\left ( {0 \leqslant \alpha \leqslant 1} \right )$ 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 inverseclosed. As a corollary, we obtain a characterization of ${\mathcal {C}_M}\left ( {{{\mathbf {R}}_ + }} \right )$ as an inverseclosed algebra, thus establishing the converse of a theorem of Malliavin [4] for the halfline.

J. Boman and L. Hörmander, Classes of infinitely differentiable functions (mimeographed notes), Stockholm, 1962.
 Joaquim Bruna, On inverseclosed algebras of infinitely differentiable functions, Studia Math. 69 (1980/81), no. 1, 59–68. MR 604354, DOI https://doi.org/10.4064/sm6915968 B. I. Korenbljum, Conditions of nontriviality of certain classes of functions analytic in a sector and problems of quasianalyticity, Soviet Math. Dokl. 7 (1966), 232236.
 Paul Malliavin, Calcul symbolique et sousalgèbres de $L_{1}(G)$. I, II, Bull. Soc. Math. France 87 (1959), 181–186, 187–190. MR 117505
 S. Mandelbrojt, Séries adhérentes, régularisation des suites, applications, GauthierVillars, Paris, 1952 (French). MR 0051893
 Walter Rudin, Division in algebras of infinitely differentiable functions, J. Math. Mech. 11 (1962), 797–809. MR 0153796 J. A. Siddiqi and A. El Koutri, CartanGornyKolmogorov type inequalities for generators of analytic semigroups and cosine operator functions, preprint.
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Keywords:
Symbolic calculus,
inverseclosed algebra,
Carleman classes
Article copyright:
© Copyright 1987
American Mathematical Society