Alternative pseudodifferential analysis : with an application to modular forms /

Alternative pseudodifferential analysis : with an application to modular forms /

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Bibliographic Details
Author / Creator:	Unterberger, André.
Imprint:	Berlin : Springer, ©2008.
Description:	ix, 118 pages ; 24 cm
Language:	English
Series:	Lecture notes in mathematics, 0075-8434 ; 1935 Lecture notes in mathematics (Springer-Verlag) ; 1935.
Subject:	Pseudodifferential operators. Forms, Modular. Forms, Modular. Pseudodifferential operators.
Format:	Print Book
URL for this record:	http://pi.lib.uchicago.edu/1001/cat/bib/7310444

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ISBN:	9783540779100 3540779108 3540779116 9783540779117
Notes:	Includes bibliographical references and indexes. Also published electronically.
Summary:	"This volume introduces an entirely new pseudodifferential analysis on the line, the opposition of which to the usual (Weyl-type) analysis can be said to reflect that, in representation theory, between the discrete and the (full, non-unitary) principal series of SL(2,R), or that between modular forms of the holomorphic and non-holo-morphic types. In the composition formula, the Rankin-Cohen brackets substitute for the usual Moyal-type brackets. This pseudodifferential analysis relies on the one-dimensional case of the recently introduced anaplectic representation and analysis, a competitor of the metaplectic representation and usual analysis." "Besides researchers and graduate students interested in pseudodifferential analysis, in harmonic analysis and in modular forms, the book may also appeal to analysts in general and physicists: its concepts make it possible to transform the creation-annihilation operators into automorphisms, simultaneously changing the usual scalar product into an indefinite but still non-degenerate one."--Jacket.

Table of Contents:

1. Introduction
2. The Metaplectic and Anaplectic Representations
2.1. The Metaplectic Representation
2.2. Anaplectic Analysis
3. The One-Dimensional Alternative Pseuodifferential Analysis
3.1. Ascending Pseudodifferential Analysis
3.2. Classes of Operators
3.3. The Resolvent of the Lowering Operator
3.4. The Composition Formula
4. From Anaplectic Analysis to Usual Analysis
4.1. The v-Anaplectic Representation
4.2. Ascending Pseudodifferential Calculus in v-Anaplectic Analysis
5. Pseudodifferential Analysis and Modular Forms
5.1. The Eisenstein, Theta, Poincare, and Alternative Poincare Distributions
5.2. Moyal Brackets and Rankin-Cohen Brackets
References
Index
Subject Index