# The Fourfold Way In Real Analysis

## An Alternative to the Metaplectic Representation

Part of the Progress in Mathematics book series (PM, volume 250)

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Part of the Progress in Mathematics book series (PM, volume 250)

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 *L ^{2}(R) *has to give way to a pseudo-Hilbert space, on which a new translation-invariant integral still exists. All this extends to the

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.

Analysis Anaplectic representation Harmonic oscillator Hilbert space Metaplectic representation Operator theory Representation theory Symbolic calculus Symplectic geometry Weyl calculus harmonic analysis mathematical physics

- DOI https://doi.org/10.1007/3-7643-7545-0
- Copyright Information Birkhäuser Verlag 2006
- Publisher Name Birkhäuser Basel
- eBook Packages Mathematics and Statistics
- Print ISBN 978-3-7643-7544-7
- Online ISBN 978-3-7643-7545-4
- Series Print ISSN 0743-1643
- Series Online ISSN 2296-505X
- Buy this book on publisher's site

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