Generalized Frames: Key to Analysis and Synthesis

Summary

In this chapter we develop a general method of analyzing and reconstructing signals, called the theory of generalized frames. The windowed Fourier transform and the continuous wavelet transform are both special cases. So are their manifold discrete versions, such as those described in the next four chapters. In the discrete case the theory reduces to a well-known construction called (ordinary) frames. The general theory shows that the results obtained in Chapters 2 and 3 are not isolated but are part of a broad structure. One immediate consequence is that certain types of theorems (such as reconstruction formulas, consistency conditions, and least-square approximations) do not have to be proved again and again in different settings; instead, they can be proved once and for all in the setting of generalized frames. Since the field of wavelet analysis is so new, it is important to keep a broad spectrum of options open concerning its possible course of development. The theory of generalized frames provides a tool by which many different wavelet-like analyses can be developed, studied, and compared.