Weyl-Heisenberg frames, translation invariant systems and the Walnut representation

We present a comprehensive analysis of the convergence properties of the fr ame operators of Weyl-Heisenberg systems and shift-invariant systems, and r elate these to the convergence of the Walnut representation. We give a deep analysis of necessary conditions and sufficient conditions for convergence of the frame operator. We show that symmetric, norm and unconditional conv ergence of the Walnut series are all different, but that weak and norm conv ergence are the same, while there are WH-systems for which the Walnut repre sentation has none of these convergence properties. We make a detailed stud y of the CC-condition (a sufficient condition for WH-systems to have finite upper frame bounds) and show that (for ab rational) a uniform version of t his passes to the Wexler-Raz dual. We also show that a condition of Tolimie ri and Orr implies the uniform CC-condition. We obtain stronger results in the case when (g, a, b) is a WH-system and ab is rational. For example, if ab is rational, then the CC-condition becomes equivalent to the uncondition al convergence of the Walnut representation-even in a more general setting. Many of the results are generalized to shift-invariant systems. We give cl assifications for numerous important classes of WH-systems including: ii) T he WH-systems for which the frame operator extends to a bounded operator on L-p(R), for all 1 less than or equal to p less than or equal to infinity; (2) The WH-systems for which the Frame operator extends to a bounded operat or on the Wiener amalgam spacer (3) The families of frames which have the s ame frame operator. (C) 2001 Academic Press.