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.