5 - Multivariable wavelets

Summary

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, …, x_d) = f₁(x₁) · … · f_d(x_d). 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 ℤ^d, 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 J_sf(x) = f(2^sx). 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 ℤ^d. 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 ∈ ℤ^d we define

To generate wavelets we will use h ∈ Z^d.

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 ℤ^d.