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Pseudodifferential Operators with Automorphic Symbols

  • Book
  • © 2015

Overview

Authors:
  1. André Unterberger

    1. Université de Reims Laboratoire de Mathématiques, Reims, France

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  • Features a complete study of pseudodifferential operators with automorphic symbols
  • Covers a complete analysis of link between modular distributions (in the plane) and non-holomorphic modular forms in the half-plane
  • Includes an introduction to the case of holomorphic modular forms, linked to the new anaplectic representation and symbolic calculus
  • Includes supplementary material: sn.pub/extras

Part of the book series: Pseudo-Differential Operators (PDO, volume 11)

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Table of contents (7 chapters)

  1. Front Matter

    Pages i-xvi
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  2. Basic modular distributions

    • André Unterberger
    Pages 7-26

  3. From the plane to the half-plane

    • André Unterberger
    Pages 27-49

  4. A short introduction to the Weyl calculus

    • André Unterberger
    Pages 51-81

  5. Composition of joint eigenfunctions of \(\varepsilon\; \mathrm{and}\ \xi\frac{\partial}{\partial x}\)

    • André Unterberger
    Pages 83-122

  6. The sharp composition of modular distributions

    • André Unterberger
    Pages 123-167

  7. The operator with symbol \(\mathfrak{E}_{\upsilon}\)

    • André Unterberger
    Pages 169-182

  8. From non-holomorphic to holomorphic modular forms

    • André Unterberger
    Pages 183-195

  9. Back Matter

    Pages 197-202
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Keywords

About this book

The main results of this book combine pseudo differential analysis with modular form theory. The methods rely for the most part on explicit spectral theory and the extended use of special functions. The starting point is a notion of modular distribution in the plane, which will be new to most readers and relates under the Radon transformation to the classical one of modular form of the non-holomorphic type. Modular forms of the holomorphic type are addressed too in a more concise way, within a general scheme dealing with quantization theory and elementary, but novel, representation-theoretic concepts.

Reviews

“The book under review is devoted to developing a satisfactory Weyl calculus on the class of tempered automorphic distributions under the group SL(2, Z). … The book is … readable and is an interesting introduction to the subject. The main ideas are explained in detail and illustrated by exact computations.” (Đỗ Ngọc Diệp, Mathematical Reviews, December, 2015)

Authors and Affiliations

  • Université de Reims Laboratoire de Mathématiques, Reims, France

    André Unterberger

Bibliographic Information

  • Book Title: Pseudodifferential Operators with Automorphic Symbols

  • Authors: André Unterberger

  • Series Title: Pseudo-Differential Operators

  • DOI: https://doi.org/10.1007/978-3-319-18657-3

  • Publisher: Birkhäuser Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer International Publishing Switzerland 2015

  • Softcover ISBN: 978-3-319-18656-6Published: 26 June 2015

  • eBook ISBN: 978-3-319-18657-3Published: 22 June 2015

  • Series ISSN: 2297-0355

  • Series E-ISSN: 2297-0363

  • Edition Number: 1

  • Number of Pages: X, 202

  • Topics: Operator Theory, Number Theory

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