Chapter 3
Signal Representation and Frames
A
pervasive and useful idea in mathematics is that functions (signals) may
be decomposed into elementary atomic functions. Suppose that fn
is such a collection of building blocks or atoms from which we may
construct functions. Any such constructed function f has
the form
for some constants cn which we may choose. By specifying
both an underlying atomic set fn and an associated
scalar sequence cn, we may provide a description of the
function f which has both practical and analytic value. This
is true with the caveat that the atomic set is chosen with some care. In
particular, the interpretation of the equality in Equation (*) is not straight-forward
for arbitrary atomic sets. In addition, there are deep and interesting
convergence issues concerning the right hand side of (*) when the atomic
set has an infinite number of members. In a practical sense, we may argue
that choosing atomic sets which lead to fundamental analytical problems
such as these are bad choices and should be avoided. Such problems may
be circumvented by placing some modest requirements on the atomic set;
namely, that it form a frame for a large enough space of interest.
This chapter describes a general theory for the discrete representation
of analog signals using mathematical frames. The presentation starts with
familiar cases in which the underlying atomic functions form orthonormal
bases and progresses through Riesz bases to the general overcomplete case.
Always, functions of interest are assumed to come from a general Hilbert
space H of interest.
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