Zak transforms on binary fields
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.
scientific article; zbMATH DE number 1398580