Pseudo-Differential Operators

Pseudodifferential Analysis, Automorphic Distributions in the Plane and Modular Forms

Authors: Unterberger, André

  • Presents pseudodifferential analysis tailored to the needs of number theorists
  • Besides containing novel features of the Weil calculus, it may constitute an approach to non-holomorphic modular form theory, suitable for analysts
  • Gives hints to possible generalizations of the construction of new classes of automorphic functions
  • Explains how and why pseudodifferential analysis should be developed in the adelic setting
  • Series of Kloosterman sums, some of which are of a novel kind, play a major
  • role in several parts of the book
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About this book

Pseudodifferential analysis, introduced in this book in a way adapted to the needs of number theorists, relates automorphic function theory in the hyperbolic half-plane Π to automorphic distribution theory in the plane. Spectral-theoretic questions are discussed in one or the other environment: in the latter one, the problem of decomposing automorphic functions in Π according to the spectral decomposition of the modular Laplacian gives way to the simpler one of decomposing automorphic distributions in R2 into homogeneous components. The Poincaré summation process, which consists in building automorphic distributions as series of g-transforms, for g Î SL(2;Z), of some initial function, say in S(R2), is analyzed in detail. On Π, a large class of new automorphic functions or measures is built in the same way: one of its features lies in an interpretation, as a spectral density, of the restriction of the zeta function to any line within the critical strip.

The book is addressed to a wide audience of advanced graduate students and researchers working in analytic number theory or pseudo-differential analysis.

Reviews

From the reviews:

“The book is devoted to explaining a close relationship between the pseudodifferential analysis and the well-known theory of automorphic functions and modular forms on the upper Poincaré half-plane II, or their generalization as automorphic distributions. … the book is perfectly readable and rich with analytic details for both researchers in pseudodifferential analysis and for number theorists.” (Đȭ Ngọc Diệp, Mathematical Reviews, November, 2013)

“In this book the author explains very beautiful links between pseudodifferential analysis and the theory of nonholomorphic modular forms on the classical modular group … . The book is excellently written and represents an extremely valuable contribution for the two research communities – analysts from PDEs and pseudodifferential operators and number theorists. It exhibits a lot of new and original links between the two research areas. It is self-contained and easily accessible for a broad readership.” (Sören Kraußhar, Zentralblatt MATH, Vol. 1243, 2012)


Table of contents (8 chapters)

  • Introduction

    Unterberger, André

    Pages 1-11

  • The Weyl calculus

    Unterberger, André

    Pages 13-46

  • The Radon transformation and applications

    Unterberger, André

    Pages 47-80

  • Automorphic functions and automorphic distributions

    Unterberger, André

    Pages 81-119

  • A class of Poincaré series

    Unterberger, André

    Pages 121-186

Buy this book

eBook $69.99
price for USA (gross)
  • ISBN 978-3-0348-0166-9
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $99.00
price for USA
  • ISBN 978-3-0348-0165-2
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Pseudodifferential Analysis, Automorphic Distributions in the Plane and Modular Forms
Authors
Series Title
Pseudo-Differential Operators
Series Volume
8
Copyright
2011
Publisher
Birkhäuser Basel
Copyright Holder
Springer Basel AG
eBook ISBN
978-3-0348-0166-9
DOI
10.1007/978-3-0348-0166-9
Softcover ISBN
978-3-0348-0165-2
Series ISSN
2297-0355
Edition Number
1
Number of Pages
VIII, 300
Topics