3.4. Frames for the windowed Fourier transform.

The windowed Fourier transform of Chapter 2 can also be discretized. The natural discretization for ?, t in g ^{?, t}( x) = e ^{i ?x} g( x ? t) is ? = m ? ₀, t = nt ₀, where ? ₀, t ₀ > 0 are fixed, and m,n range over ; the discretely labelled family is thus

We can again seek answers to the same questions as in the wavelet case: for which choices of g, ? ₀, t ₀ can a function be characterized by the inner products ? f, g _m,n ?; when is it possible to reconstruct f in a numerically stable way from these inner products; can an efficient algorithm be given to write f as a linear combination of the g _m,n? The answers are again provided by the same abstract framework: stable numerical reconstruction of f from its windowed Fourier co-efficients

is only possible if the g _m,n constitute a frame, i.e., if there exist A > 0, B < ? so that

If the g _m,n constitute a frame, then any function can be written as

where are the vectors in the dual frame; (3.4.1) shows both how to recover f from the ? f, g _m,n ? and how...