9. Characterization of Functional Spaces by Means of Wavelets
Society for Industrial and Applied Mathematics eBooks · 1992
The major message of this chapter is that the orthonormal bases we have discussed for the last four chapters also give good (i.e., unconditional) bases for many other spaces than , out-performing the Fourier basis functions in this respect. Almost all the material in this chapter is borrowed from Meyer (1990), but it is here presented in (I believe) a more pedestrian way, accessible to readers with a lower level of mathematical sophistication. (Meyer's book also contains much more on this subject than is explained in this chapter.) In §9.1 I start by reviewing a classic theorem of pure harmonic analysis, the Calderón—Zygmund decomposition. It can be found also in many textbooks (such as Stein (1970)); I include a detailed proof here as an illustration of techniques using different (dyadic) scales, practiced in pure harmonic analysis long before wavelets came along. Together with some other classic theorems, it leads to the proof that wavelets are an unconditional bases for , . Section 9.2 lists the characterizations, by means of wavelets, of other functional spaces, without proof. Also included is a short discussion on the detection of singularities with orthonormal wavelet bases. Section 9.3 treats expansions of -functions by means of wavelets; since has no unconditional bases, wavelets cannot do the impossible, but they still do a better job than Fourier expansions. Finally, §9.4 points out an amusing difference in emphasis between wavelet and Fourier expansions.