**Signal Classes**

There are three classes of signals that we will be using. The most basic is
called

The next most basic class is *these* functions although not necessarily with

A more general class of signals than any *distributions*. These are generalized functions which are not
defined by their having “values" but by the value of an “inner product"
with a normal function. An example of a distribution would be the Dirac
delta function

Another detail to keep in mind is that the integrals used in these
definitions are *Lebesque integrals* which are somewhat more general
than the basic Riemann integral. The value of a Lebesque integral is not
affected by values of the function over any countable set of values of its
argument (or, more generally, a set of measure zero). A function defined
as one on the rationals and zero on the irrationals would have a zero
Lebesque integral. As a result of this, properties derived using measure
theory and Lebesque integrals are sometime said to be true “almost
everywhere," meaning they may not be true over a set of measure zero.

Many of these ideas of function spaces, distributions, Lebesque measure, etc. came out of the early study of Fourier series and transforms. It is interesting that they are also important in the theory of wavelets. As with Fourier theory, one can often ignore the signal space classes and can use distributions as if they were functions, but there are some cases where these ideas are crucial. For an introductory reading of this book or of the literature, one can usually skip over the signal space designation or assume Riemann integrals. However, when a contradiction or paradox seems to arise, its resolution will probably require these details.

**Fourier Transforms**

We will need the Fourier transform of

and the discrete-time Fourier transform (DTFT) [23] of

where

If

which after iteration becomes

if

**Refinement and Transition Matrices**

There are two matrices that are particularly important to determining the
properties of wavelet systems. The first is the *refinement matrix*

which we write in matrix form as

with

The second submatrix is a shifted version illustrated by

with the matrix being denoted

A third, less obvious but perhaps more important, matrix is called the
*transition matrix*

This matrix (sometimes called the Lawton matrix) was used by Lawton (who
originally called it the Wavelet-Galerkin matrix) [14] to
derive necessary and sufficient conditions for an orthogonal wavelet basis.
As we will see later in this chapter, its eigenvalues are also important
in determining the properties of