Importance of Zak Transforms for Harmonic Analysis

In engineering and applied mathematics, Zak transforms have been effectively used for over 50 years in various applied settings. As Gelfand observed in a 1950 paper, the variable coefficient Fourier series ideas articulated in Andre Weil’s famous book on integration lead to an exceedingly elementary proof of the Plancherel Theorem for LCA groups. The transform for functions on R appearing in Zak’s seminal 1967 paper is actually a special case of the LCA group transforms earlier introduced by Weil; Zak states this explicitly in his 1967 paper but the mathematical community nonetheless chose to name the transform for him. In brief, the properties of Zak transforms are simply reflections of elementary Fourier series properties and the Plancherel Theorem for non-compact LCA groups is an immediate consequence of the fact that Fourier transforms are averages of Zak transforms. It is remarkable that only a small handful of mathematicians know this proof and that all textbooks continue to give much harder and less transparent proofs for even the case of the group R. Generalized Zak transforms arise naturally as intertwining operators for various representations of Abelian groups and allow formulation of many appealing theorems. Remark: The results discussed below represent joint work by E. Hernandez, H. Sikic, G. Weiss, and the author. 1. The Abelian Group Plancherel Theorem 1.1. Overview. In textbooks on real analysis, one can find a variety of proofs of the Plancherel Theorem for R, n ∈ N. All are lengthy, non-elementary, and technical, e.g.: use of complex analysis to compute Fourier transforms of Gaussian functions followed by use of approximate identities defined by Gaussians to extend the Fourier transform from L(R) ∩ L(R) to a unitary operator on L(R); alternative use of the Hermite function orthonormal basis for L(R) and the computation that Hermite functions are eigenfunctions of the Fourier transform with eigenvalues which are fourth roots of 1; reversion to the 19th century interpretation of Fourier transforms as Riemann sum limits of Fourier series and justification of this approach by somewhat delicate dominated convergence arguments.