Zak transforms on binary fields
type: Item
instance of: scholarly article
title: Zak transforms on binary fields
zbMATH Open document ID: 0945.42024
published in: Journal of Approximation Theory
publication date: 2000-04-02T00:00:00Z
review text: Since the classic Acta paper by \textit{A. Weil} [Acta Math. 111, 143-211 (1964; Zbl 0203.03305); OEuvres scientifiques. Collected papers. Vol. III (1964-1978), pp. 1-69 (1980; Zbl 0428.01014)] it is well known that the real Heisenberg nilpotent Lie group \(G\) can be periodized. The periodization construction generates the compact Heisenberg nilmanifold which forms a principal circle bundle over the two-dimensional compact torus group \(\mathbb{T}^2\). The periodization of the square integrable irreducible unitary linear Schrödinger representation of \(G\) leads to the Weil-Brezin mapping [\textit{W. Schempp}, ``Harmonic analysis on the Heisenberg nilpotent Lie group, with applications to signal theory'' (1986; Zbl 0632.43001)], which is called Zak transform in solid state physics. Weil was of course working in the adelic formalism of modern number theory, so he considered symplectic groups and Heisenberg groups with coefficients belonging to a general local field. When the field is \(\mathbb{R}\), the real numbers, and \(\mathbb{T}= \mathbb{R}/\mathbb{Z}\), both groups are Lie groups, and so have associated Lie algebras. The commutation relations of the real Heisenberg Lie algebra \(\text{Lie}(G)\) are, as its name suggests, essential Heisenberg's canonical commutation relations (CCR) which form the foundation of quantum mechanics. Furthermore, the Weil-Brezin map permits to prove the Whittaker-Shannon sampling theorem of signal theory via the Poisson summation formula [\textit{R. P. Boas jun.}, Tôhoku Math. J., II. Ser. 24, 121-125 (1972; Zbl 0238.42009)]. The paper under review does not refer to Weil's classic Acta paper quoted above. It studies a dyadic ground field and establishes that the Weil-Brezin map defines bounded operators in the spaces \(L^p\) for exponents \(p= 1\) and \(p= 2\). The aim is a study of the Weil-Brezin map and its application to Wiener spaces and dyadic wavelets defined on binary fields.
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 43A65
Mathematics Subject Classification ID: 94A12
Mathematics Subject Classification ID: 22E27
zbMATH DE Number: 1398580
zbMATH Keywords: Heisenberg nilpotent Lie group
zbMATH Keywords: periodization
zbMATH Keywords: Zak transform
zbMATH Keywords: Weil-Brezin map
zbMATH Keywords: Wiener spaces
zbMATH Keywords: dyadic wavelets
zbMATH Keywords: binary fields
author: Ferenc Schipp
author: William R. Wade
reviewed by: Walter Schempp
MaRDI profile type: Publication
full work available at URL: jath.1999.3403
OpenAlex ID: W2024754499
cites work: Sampling multipliers and the Poisson summation formula
cites work: The wavelet transform, time-frequency localization and signal analysis
cites work: The Generalized Walsh Functions
cites work: Q5552484
cites work: Bargmann transform, Zak transform, and coherent states
cites work: Von Neumann algebras associated with pairs of lattices in Lie groups
cites work: Fourier Analysis on Local Fields. (MN-15)
cites work: Time-frequency representations
cites work: Q5566639
DOI: 10.1006/JATH.1999.3403
description: scientific article; zbMATH DE number 1398580