In this chapter we will present in detail constructions and properties of some important classes of wavelets. The constructions will follow the general theory established in the previous chapter.

What to look for in a wavelet?

The answer to the question in the title of this section clearly depends on what we want to use the wavelet for. Our approach taken in Chapter 1 and later in Chapters 8 and 9 is to analyze functions from some function space, very often different from L2(ℤ), using wavelets. We will base our answer upon the analysis of arguments given later. This however is only a matter of motivation. Our mathematics will in no way rely on things presented in later chapters.

It is clear from our arguments given in Chapters 8 and 9, and has already been mentioned in chapter 1, that good decay of wavelets plays a crucial role in investigating wavelet expansions of a function. It is obviously also crucial in the following question of clear practical importance but not discussed in any detail in this book. Suppose a function f on ℤ (or on ℤd) is given with supp f ⊂ [0,1] (or some cube Q). How can we recognize it from its wavelet coefficients? Suppose we approximate f by a finite subsum of its wavelet expansion. How will this approximation look outside [0,1]? We will use this type of estimate to estimate ΣB in the proof of the fundamental Proposition 8.8.