From Anaplectic Analysis to Usual Analysis

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.