Our aim in this Chapter is to obtain multivariable generalizations of one-variable wavelets. This can be done in many different ways. The most natural way to pass from one variable to several is to use tensors, i.e. functions of the form f(x1, …, xd) = f1(x1) · … · fd(xd). This idea we can employ at two different levels: for wavelets and for scaling functions. We will present this in Section 5.1. In Section 5.2 we will present a genuinely multivariate theory of multiresolution analyses on ℤd, together with some examples. Actually we will present our theory in such generality that even for d = 1 we will get a more general theory than presented so far. The fundamental difference between the above three aproaches is the way we generalize the one-dimensional dyadic dilations Jsf(x) = f(2sx). Tensoring at the level of wavelets corresponds to dilations

Tensoring at the level of the scaling function corresponds to dilations

Our more general approach uses dilations of the form

where A is a suitable linear transformation of ℤd. The last two approaches force us to use instead of one wavelet a finite ‘wavelet set’. Our translations will always be the same as before: for h ∈ ℤd we define

To generate wavelets we will use h ∈ Zd.

In Section 5.2 we will show how to construct wavelet sets from multiresolution analysis in our most general framework. In our last Section 5.3 we will construct many examples of multiresolution analyses and in particular we will give the construction of smooth, fast decaying wavelets on ℤd.

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.

In this chapter we will generalize the Laplacian on Euclidean space to an operator on differential forms on a Riemannian manifold. By a Riemannian manifold, we roughly mean a manifold equipped with a method for measuring lengths of tangent vectors, and hence of curves. Throughout this text, we will concentrate on studying the heat flow associated to these Laplacians. The main result of this chapter, the Hodge theorem, states that the long time behavior of the heat flow is controlled by the topology of the manifold.

In §1.1, the basic examples of heat flow on the one dimensional manifolds S1 and R are studied. The heat flow on the circle already contains the basic features of heat flow on a compact manifold, although the circle is too simple topologically and geometrically to really reveal the information contained in the heat flow. In contrast, heat flow on R is more difficult to study, which indicates why we will restrict attention to compact manifolds. In §1.2, we introduce the notion of a Riemannian metric on a manifold, define the spaces of L2 functions and forms on a manifold with a Riemannian metric, and introduce the Laplacian associated to the metric. The Hodge theorem is proved in §1.3 by heat equation methods. The kernel of the Laplacian on forms is isomorphic to the de Rham cohomology groups, and hence is a topological invariant. The de Rham cohomology groups are discussed in §1.4, and the isomorphism between the kernel of the Laplacian and de Rham cohomology is shown in §1.5.