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Unitary equivalence of lowest dimensional reproducing formulae of type $\mathcal{E}_2 \subset Sp(2,{\mathbb R})
12 Nov 2018
Abstract
All two-dimensional reproducing formulae, i.e. of $L^2({\mathbb R}^2)$,
resulting out of restrictions of the projective metaplectic representation to
connected Lie subgroups of $Sp(2,{\mathbb R})$ and of type $\mathcal{E}_2$,
were listed and classified up to conjugation within $Sp(2,{\mathbb R})$ in [2],
[3]. A full classification, up to conjugation within ${\mathbb R}^2 \rtimes
Sp(1,{\mathbb R})$, of one-dimensional reproducing formulae, i.e. of
$L^2({\mathbb R})$, resulting out of restrictions of the extended projective
metaplectic representation to connected Lie subgroups of ${\mathbb R}^2 \rtimes
Sp(1,{\mathbb R})$ was obtained in [13], [14]. In dimension one, there are no
reproducing formulae with one-dimensional parametrizations, yet in dimension
two, there are reproducing formulae with two-dimensional parametrizations.
Two-dimensional reproducing subgroups of $Sp(2,{\mathbb R})$ of type
$\mathcal{E}_2$ are a novelty. They exhibit completely new phase space
phenomena. We show, that they are all unitarily equivalent via natural choices
of coordinate systems, and we derive the consequences of this equivalence.
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Details
- Title
- Unitary equivalence of lowest dimensional reproducing formulae of type $\mathcal{E}_2 \subset Sp(2,{\mathbb R})
- Creators
- R BoyerK NowakM Pap
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Computer Science; Mathematics
- Other Identifier
- 991020638373404721