Spaces Generated by Orbits of Unitary Representations: A Tribute to Guido Weiss

Tuning Co- and Contra-Variant Transforms: the Heisenberg Group Illustration

“…[5,33,36]. The topics of coherent states and covariant transform (also known under many other names) are extensively covered in the existing literature, e.g., [28,Section 13], [30,Appendix V.2], and [9,18,19,22,34,37,40], and we refer to authoritative surveys [3,15,20] for further references. Our purpose is to present some additional aspects which are commonly shadowed or missing in the existing sources.…”

“…For example, the standard induction (see [16,Chapter 6], [30,Appendix V.2] and [7,20,27]) from a character of the Heisenberg group centre gives the representation (2.5) below, which is different from the commonly used celebrated Fock-Segal-Bargmann (FSB) representation (4.4) in the space of analytic function [15,Section 1.6]. Those two representations are linked by the peeling map which transforms the annihilator of the representation space to the Cauchy-Riemann operator, see Section 4.…”

“…The classical Zak transform [44], also known as Weil-Brezin transform [15, Section 1.10], can be traced back to the works of Gelfand in 1950, see also [40,Section 1.5], [17,Chapter 8], [39,Section 8.1] and [20,21,22] for further applications and historical notes. This transform is an…”

“…Weil [43] defined the abstract Zak transform on arbitrary locally compact abelian (LCA) groups with respect to arbitrary closed subgroups. Subsequently, the Zak transform was reviewed and generalised by many authors, see for example [6,8,20,22,26]. Various connections between the Zak transform and (left) shift-invariant spaces were studied by several authors, cf.…”

“…Various connections between the Zak transform and (left) shift-invariant spaces were studied by several authors, cf. [7,21,24,25], further references may be found in the recent survey paper [20]. Yet, an explicit interpretation of the Zak transform and its inverse as a co-and contravariant transforms respectively, cf.…”