Signal Representation and Frames

Abstract

A pervasive and useful idea in mathematics is that functions (signals) may be decomposed into elementary atomic functions. Suppose that {φ _n } is such a collection of building blocks or atoms from which functions may be constructed. Any such constructed function f has the form

$$ f(t) = \sum {{{c}_{n}}{{\phi }_{n}}(t)} $$

(3.1)

for some constants c _n that may be chosen. By specifying both an underlying atomic set {φ _n } and an associated scalar sequence {c _n }, one may provide a description of the function f that 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 (3.1) is not straightforward for arbitrary atomic sets. In addition, there are deep and interesting convergence issues concerning the right-hand side of (3.1) when the atomic set has an infinite number of members. In a practical sense, it may be argued that choosing atomic sets that 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.