3. Discrete Wavelet Transforms: Frames
Society for Industrial and Applied Mathematics eBooks · 1992
In this, the longest chapter in this book, we discuss various aspects of nonorthonormal, 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).