Overview
- Examines the different, but fully coherent, analysis one could get when trading the usual L2 scalar product for an indefinite one
- The new analysis developed here is examined in connection with classical elementary notions of representation theory and Lagrangian geometry
- Includes supplementary material: sn.pub/extras
Part of the book series: Progress in Mathematics (PM, volume 250)
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Table of contents (4 chapters)
Keywords
About this book
The fourfold way starts with the consideration of entire functions of one variable satisfying specific estimates at infinity, both on the real line and the pure imaginary line. A major part of classical analysis, mainly that which deals with Fourier analysis and related concepts, can then be given a parameter-dependent analogue. The parameter is some real number modulo 2, the classical case being obtained when it is an integer. The space L2(R) has to give way to a pseudo-Hilbert space, on which a new translation-invariant integral still exists. All this extends to the n-dimensional case, and in the alternative to the metaplectic representation so obtained, it is the space of Lagrangian subspaces of R2n that plays the usual role of the complex Siegel domain. In fourfold analysis, the spectrum of the harmonic oscillator can be an arbitrary class modulo the integers.
Even though the whole development touches upon notions of representation theory, pseudodifferential operator theory, and algebraic geometry, it remains completely elementary in all these aspects. The book should be of interest to researchers working in analysis in general, in harmonic analysis, or in mathematical physics.
Authors and Affiliations
Bibliographic Information
Book Title: The Fourfold Way in Real Analysis
Book Subtitle: An Alternative to the Metaplectic Representation
Authors: André Unterberger
Series Title: Progress in Mathematics
DOI: https://doi.org/10.1007/3-7643-7545-0
Publisher: Birkhäuser Basel
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Birkhäuser Basel 2006
Hardcover ISBN: 978-3-7643-7544-7Published: 17 March 2006
eBook ISBN: 978-3-7643-7545-4Published: 15 June 2006
Series ISSN: 0743-1643
Series E-ISSN: 2296-505X
Edition Number: 1
Number of Pages: X, 222
Topics: Functional Analysis, Topological Groups, Lie Groups, Mathematical Methods in Physics, Functions of a Complex Variable, Abstract Harmonic Analysis