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The Fourfold Way In Real Analysis [electronic resource] : An Alternative to the Metaplectic Representation / by André Unterberger.

By: Unterberger, André [author.].
Contributor(s): SpringerLink (Online service).
Material type: TextTextSeries: Progress in Mathematics: 250Publisher: Basel : Birkhäuser Basel, 2006Description: X, 222 p. online resource.Content type: text Media type: computer Carrier type: online resourceISBN: 9783764375454.Subject(s): Functional analysis | Topological Groups | Mathematical physics | Functions of complex variables | Harmonic analysis | Functional Analysis | Topological Groups, Lie Groups | Mathematical Methods in Physics | Functions of a Complex Variable | Abstract Harmonic AnalysisAdditional physical formats: Printed edition:: No title; Printed edition:: No titleDDC classification: 515.7 Online resources: Click here to access online
Contents:
The One-dimensional Anaplectic Representation -- The n-dimensional Anaplectic Analysis -- Towards the Anaplectic Symbolic Calculi -- The One-dimensional Case Revisited.
In: Springer eBooksSummary: 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.
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The One-dimensional Anaplectic Representation -- The n-dimensional Anaplectic Analysis -- Towards the Anaplectic Symbolic Calculi -- The One-dimensional Case Revisited.

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.

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Fourfold way in real analysis by Unterberger Andre
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