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Laguerre Calculus and Its Applications on the Heisenberg Group

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Brand new. We distribute directly for the publisher. For nearly two centuries, the relation between analytic functions of one complex variable, their boundary values, harmonic ... functions, and the theory of Fourier series has been one of the central topics of study in mathematics. The topic stands on its own, yet also provides very useful mathematical applications.This text provides a self-contained introduction to the corresponding questions in several complex variables: namely, analysis on the Heisenberg group and the study of the solutions of the boundary Cauchy-Riemann equations. In studying this material, readers are exposed to analysis in non-commutative compact and Lie groups, specifically the rotation group and the Heisenberg groups--both fundamental in the theory of group representations and physics.Introduced in a concrete setting are the main ideas of the CalderAA³n-Zygmund-Stein school of harmonic analysis. Also considered in the book are some less conventional problems of harmonic and complex analysis, in particular, the Morera and Pompeiu problems for the Heisenberg group, which relates to questions in optics, tomography, and engineering. Read more Show Less

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Overview

For nearly two centuries, the relation between analytic functions of one complex variable, their boundary values, harmonic functions, and the theory of Fourier series has been one of the central topics of study in mathematics. The topic stands on its own, yet also provides very useful mathematical applications. This text provides a self-contained introduction to the corresponding questions in several complex variables: namely, analysis on the Heisenberg group and the study of the solutions of the boundary Cauchy-Riemann equations. In studying this material, readers are exposed to analysis in non-commutative compact and Lie groups, specifically the rotation group and the Heisenberg groups—both fundamental in the theory of group representations and physics. Introduced in a concrete setting are the main ideas of the Calderon-Zygmund-Stein school of harmonic analysis. Also considered in the book are some less conventional problems of harmonic and complex analysis, in particular, the Morera and Pompeiu problems for the Heisenberg group, which relates to questions in optics, tomography, and engineering. The book was borne of graduate courses and seminars held at the University of Maryland (College Park), the University of Toronto (ON), Georgetown University (Washington, DC), and the University of Georgia (Athens). Readers should have an advanced undergraduate understanding of Fourier analysis and complex analysis in one variable.

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Product Details

Table of Contents

Preface
1 The Laguerre Calculus 1
2 Estimates for Powers of the Sub-Laplacian 33
3 Estimates for the Spectrum Projection Operators of the Sub-Laplacian 77
4 Inverse of the Operator [actual symbol not reproducible] 111
5 The explict solution of the [delta]-Neumann problem in a non-isotropic Siegel domain 151
6 Injectivity of the Pompeiu Transform in the Isotropic H[subscript n] 185
7 Morera-type Theorems for Holomorphic H[superscript p] Spaces in H[subscript n] (I) 229
8 Morera-type Theorems for Holomorphic H[superscript P] Spaces in H[subscript n] (II) 257
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