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 be the analytic Carleman class of -functions defined in a sector and analytic in its interior such that . In this paper, we give necessary and sufficient conditions in order that be inverse-closed. As a corollary, we obtain a characterization of as an inverse-closed algebra, thus establishing the converse of a theorem of Malliavin [**4**] for the half-line.

**[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****[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****[5]**S. Mandelbrojt,*Séries adhérentes, régularisation des suites, applications*, Gauthier-Villars, Paris, 1952 (French). MR**0051893****[6]**Walter Rudin,*Division in algebras of infinitely differentiable functions*, J. Math. Mech.**11**(1962), 797–809. MR**0153796****[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

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