  In this dissertation I establish that a broad class of Banach *-algebras of
infinite integral operators, defined by the property that the kernels of the
elements of the algebras possess subexponential off-diagonal decay, is inverse
closed in the Banach space of bounded linear operators on the Hilbert space of
square-integrable functions. I also show that the algebras under consideration
are symmetric. In the second part of this dissertation, I present the results
of the IEEE Transactions on Communications paper written jointly by Thomas
Strohmer and me. We develop a comprehensive framework for the design of
orthogonal frequency-division multiplexing (OFDM) systems, using techniques
from Gabor frame theory, sphere-packing theory, representation theory, and the
Heisenberg group.
