In this, the longest chapter in this book, we discuss various aspects of non-orthonormal, discrete wavelet expansions, together with some parallels with the windowed Fourier transform. The "frames" of the chapter title are sets of non-independent vectors; they can nevertheless be used to write a straightforward and completely explicit expansion for every vector in the space. We will discuss wavelet frames as well as frames for the windowed Fourier transform; in the latter case, the approach can be viewed as "oversampled" with respect to the Nyquist density in time-frequency space.

A lot of the material in this chapter has been taken from Daubechies (1990), updated here and there. A very nicely written review of frames (and of the continuous transforms as well), with some additional original theorems, is Heil and Walnut (1989).

3.1. Discretizing the wavelet transform.

In the continuous wavelet transform, we consider the family

where with a ? 0, and ? is admissible. For convenience, in the discretization we restrict a to positive values only, so that the admissibility condition becomes

(See 2.4.) We would like to restrict a, b to discrete values only. The discretization of the dilation parameter seems natural: we choose a = a ^m ₀, where m ? , and the dilation step a ₀ ? 1 is fixed. For convenience we will assume a ₀ > 1 (although it does not matter, since we take negative as well as positive powers m). For