  We present a comprehensive analysis of the convergence properties of the
frame operators of Weyl-Heisenberg systems and shift-invariant systems, and
relate 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 convergence
of the Walnut series are all different, but that weak and norm convergence are
the same, while there are WH-systems for which the Walnut representation has
none of these convergence properties. We make a detailed study 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 this passes to
the Wexler-Raz dual. We also show that a condition of Tolimieri 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 unconditional convergence of the Walnut
representation - even in a more general setting. Many of the results are
generalized to shift-invariant systems. We give classifications for numerous
important classes of WH-systems including: (1) The WH-systems for which the
frame operator extends to a bounded operator on L^p(R), 1\leq p; (2) The
WH-systems for which the frame operator extends to a bounded operator on the
Wiener amalgam space; (3) The families of frames which have the same frame
operator.

Functional Analysis