The structure of shift–modulation invariant spaces: The rational case

In this paper we study structural properties of shift–modulation invariant (SMI) spaces, also called Gabor subspaces, or Weyl–Heisenberg subspaces, in the case when shift and modulation lattices are rationally dependent. We prove the characterization of SMI spaces in terms of range functions analogous to the well-known description of shift-invariant spaces [C. de Boor, R. DeVore, A. Ron, The structure of finitely generated shift-invariant spaces in L2(Rd), J. Funct. Anal. 119 (1994) 37–78; M. Bownik, The structure of shift-invariant subspaces of L2(Rn), J. Funct. Anal. 177 (2000) 282–309; H. Helson, Lectures on Invariant Subspaces, Academic Press, New York/London, 1964]. We also give a simple characterization of frames and Riesz sequences in terms on their behavior of the fibers of the range function. Next, we prove several orthogonal decomposition results of SMI spaces into simpler blocks, called principal SMI spaces. Then, this is used to characterize operators invariant under both shifts and modulations in terms of families of linear maps acting on the fibers of the range function. We also introduce the fundamental concept of the dimension function for SMI spaces. As a result, this leads to the classification of unitarily equivalent SMI spaces in terms of their dimension functions. Finally, we show several results illustrating our fiberization techniques to characterize dual Gabor frames.