In signal processing the marks of great physicist-mathematicians of the early 20th century and beyond are everywhere. Apparently when Gabor authored his seminal paper (cited by 4087 according to google scholar), he was building on the foundations of laid by Weyl, and others besides, in his Theory of Groups and Quantum Mechanics. More can be found in the intro to this book, Gabor Analysis and Algorithms: Theory and Applications, by Hans G. Feichtinger and Thomas Strohmer. I also found this paper that has an interesting introduction, which should be compared with Aldrovani’s discussion (in his book Special Matrices of Mathematical Physics) of Weyl-Heisenberg groups, which says that “The duality between U and V leads to a deep and rich structure standing behind the Quantum Mechanics of simple systems.” U and V are NxN matrices, U is the first-order permutation matrix, and V has the N roots of unity on its diagonal. Aldrovanialso advises (p. 82), “The relation of matrices to Fourier transformations is obscured by our inclination to look at them as sets of rows and columns. … It is actually only a convenient convention, and comes from the preferred use of a special (“canonical”) basis in matrix space. Other bases, in particular that formed by Weyl’s [or Gabor if it’s ‘information’ ..mt] operators, give the prominent role to the diagonals. Matrix product acquires a new aspect and many a Fourier-related property come to the fore.” Furthermore, by this tact, “We shall see how a matrix space can be made into a differential manifold, and how a natural symplectic (phase space) setup turns up.”
A google search turned up this site that got me started down this path of inquiry [link]. I got to that by checking deeper into work by Holger Rauhut, there’s a pdf of his slides with motivation and coverage of topics along these lines, and also this thorough paper Compressive Sensing, which looks like a very nice review with new insight.
It’s very exciting to start to see the connection between these various branches of mathematics, physics, information theory. More to come on those topics…