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From Anaplectic Analysis to Usual Analysis

  • André UnterbergerEmail author
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1935)

Abstract

It is possible to consider anaplectic analysis on the real line as a special case of a oneparameter family of analyses. The parameter \(\nu\) is a complex number mod 2, subject to the restriction that it should not be an integer: anaplectic analysis, as considered until now, corresponds to the case when \(\nu = - \frac{1}{2}\). There is a natural \(\nu\)-anaplectic representation of some cover of SL(2,ℝ) in some space \(\mathfrak{A}_\nu\), compatible in the usual way with the Heisenberg representation; the \(\nu\)-anaplectic representation is pseudounitarizable in the case when \(\nu\) is real. Depending on \(\nu\), the even or odd part of the \(\nu\)-anaplectic representation coincides in this case with a representation taken from the unitary dual of the universal cover of SL(2,ℝ), as completely described by Pukanszky [24]. In \(\nu\)-anaplectic analysis, the spectrum of the harmonic oscillator is the arithmetic sequence \(\nu + \frac{1}{2} + \mathbb{Z}\). Much of the theory subsists in the case when \(\nu \equiv 0 \) mod 2, which leads to a nontrivial enlargement of usual analysis. However, as we shall make clear, while the ascending pseudodifferential calculus extends to the case of \(\nu\)-anaplectic analysis when \(\nu \in \mathbb{C}\backslash\mathbb{Z}\), it is impossible to extend it to the usual analysis environment.

Keywords

Harmonic Oscillator Universal Cover Principal Series Usual Analysis Hermite Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  1. 1.Mathématiques Université de ReimsReims Cedex 2France

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