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Quasianalytic Monogenic Solutions of a Cohomological Equation - Marmi, S.
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Quasianalytic Monogenic Solutions of a Cohomological Equation - used book

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We prove that the solutions of a cohomological equation of complex dimension one and in the analytic category have a monogenic dependence on the parameter, and we investigate the question… More...

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Marmi, S. & Sauzin, D.:

Sub-Laplacians with Drift on Lie Groups of Polynomial Volume Growth: Memoirs of the American Mathematical Society, Volume 164, Number 780: July, 2003 - used book

2003, ISBN: 9780821833254

Providence, RI: American Mathematical Society, 2003. 101 clean, unmarked pages; Library of Congress stamp & sticker. 1st. Single Issue Magazine. Very Good. 4 vo., American Mathematica… More...

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Marmi, S. & Sauzin, D.:
Sub-Laplacians with Drift on Lie Groups of Polynomial Volume Growth: Memoirs of the American Mathematical Society, Volume 164, Number 780: July, 2003 - used book

2003

ISBN: 9780821833254

Providence, RI: American Mathematical Society, 2003. 101 clean, unmarked pages; Library of Congress stamp & sticker. 1st. Single Issue Magazine. Very Good. 4 vo., American Mathematica… More...

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S. Marmi; D. Sauzin:
Quasianalytic Monogenic Solutions of a Cohomological Equation (Memoirs of the American Mathematical Society) - Paperback

2003, ISBN: 9780821833254

Amer Mathematical Society, 2003-07-01. Mass Market Paperback. Good., Amer Mathematical Society, 2003-07-01

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Quasianalytic Monogenic Solutions of a Cohomological Equation
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Quasianalytic Monogenic Solutions of a Cohomological Equation - Paperback

2003, ISBN: 9780821833254

Softcover, Buch, [PU: American Mathematical Society]

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Quasianalytic Monogenic Solutions of a Cohomological Equation (Memoirs of the American Mathematical Society, No. 780)

We prove that the solutions of a cohomological equation of complex dimension one and in the analytic category have a monogenic dependence on the parameter, and we investigate the question of their quasi analyticity. This cohomological equation is the standard linearized conjugacy equation for germs of holomorphic maps in a neighborhood of a fixed point. The parameter is the eigenvalue of the linear part, denoted by $q$. Borel's theory of non-analytic monogenic functions has been first investigated by Arnold and Herman in the related context of the problem of linearization of analytic diffeomorphisms of the circle close to a rotation.Herman raised the question whether the solutions of the cohomological equation had a quasi analytic dependence on the parameter $q$. Indeed they are analytic for $q\\in\\mathbb{C}\\setminus\\mathbb{S}^1$, the unit circle $\\S^1$ appears as a natural boundary (because of resonances, i.e. roots of unity), but the solutions are still defined at points of $\\mathbb{S}^1$ which lie 'far enough from resonances'. We adapt to our case Herman's construction of an increasing sequence of compacts which avoid resonances and prove that the solutions of our equation belong to the associated space of monogenic functions; some general properties of these monogenic functions and particular properties of the solutions are then studied.For instance the solutions are defined and admit asymptotic expansions at the points of $\\mathbb{S}^1$ which satisfy some arithmetical condition, and the classical Carleman Theorem allows us to answer negatively to the question of quasi analyticity at these points. But resonances (roots of unity) also lead to asymptotic expansions, for which quasi analyticity is obtained as a particular case of Ecalle's theory of resurgent functions.And at constant-type points, where no quasi analytic Carleman class contains the solutions, one can still recover the solutions from their asymptotic expansions and obtain a special kind of quasi analyticity. Our results are obtained by reducing the problem, by means of Hadamard's product, to the study of a fundamental solution (which turns out to be the so-called $q$-logarithm or 'quantum logarithm'). We deduce as a corollary of our work the proof of a conjecture of Gammel on the monogenic and quasi analytic properties of a certain number-theoretical Borel-Wolff-Denjoy

Details of the book - Quasianalytic Monogenic Solutions of a Cohomological Equation (Memoirs of the American Mathematical Society, No. 780)


EAN (ISBN-13): 9780821833254
ISBN (ISBN-10): 0821833251
Paperback
Publishing year: 2003
Publisher: American Mathematical Society

Book in our database since 2008-02-25T10:33:08-05:00 (New York)
Detail page last modified on 2023-01-29T08:14:26-05:00 (New York)
ISBN/EAN: 0821833251

ISBN - alternate spelling:
0-8218-3325-1, 978-0-8218-3325-4
Alternate spelling and related search-keywords:
Book title: 780, lie groups


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