There are three classes of signals that we will be using. The most basic is called L²(R) which contains all functions which have a finite, well-defined integral of the square: . This class is important because it is a generalization of normal Euclidean geometry and because it gives a simple representation of the energy in a signal.

The next most basic class is L¹(R), which requires a finite integral of the absolute value of the function: . This class is important because one may interchange infinite summations and integrations with these functions although not necessarily with L² functions. These classes of function spaces can be generalized to those with and designated L^p.

A more general class of signals than any L^p space contains what are called 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 δ(t) where it is defined by the property: .

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.

We will need the Fourier transform of φ(t) which, if it exists, is defined to be Φ

(6.2)

and the discrete-time Fourier transform (DTFT) 23 of h(n) defined to be

(6.3)

where and n is an integer (n∈Z). If convolution with h(n) is viewed as a digital filter, as defined in Section: Analysis - From Fine Scale to Coarse Scale, then the DTFT of h(n) is the filter's frequency response, 23, 24 which is 2π periodic.

If Φ(ω) exits, the defining recursive equation Equation 6.1 becomes

(6.4)

which after iteration becomes

(6.5)

if and Φ(0) is well defined. This may be a distribution or it may be a smooth function depending on H(ω) and, therefore, h(n)33, 4. This makes sense only if Φ(0) is well defined. Although Equation 6.1 and Equation 6.5 are equivalent term-by-term, the requirement of Φ(0) being well defined and the nature of the limits in the appropriate function spaces may make one preferable over the other. Notice how the zeros of H(ω) determine the zeros of Φ(ω).

There are two matrices that are particularly important to determining the properties of wavelet systems. The first is the refinement matrixM, which is obtained from the basic recursion equation Equation 6.1 by evaluating φ(t) at integers 22, 6, 7, 29, 30. This looks like a convolution matrix with the even (or odd) rows removed. Two particular submatrices that are used later in Section 6.10 to evaluate φ(t) on the dyadic rationals are illustrated for N=6 by

(6.6)

which we write in matrix form as

(6.7)

with M₀ being the 6×6 matrix of the h(n) and being 6×1 vectors of integer samples of φ(t). In other words, the vector with entries φ(k) is the eigenvector of M₀ for an eigenvalue of unity.

The second submatrix is a shifted version illustrated by

(6.8)

with the matrix being denoted M₁. The general refinement matrix M is the infinite matrix of which M₀ and M₁ are partitions. If the matrix H is the convolution matrix for h(n), we can denote the M matrix by [↓2]H to indicate the down-sampled convolution matrix H. Clearly, for φ(t) to be defined on the dyadic rationals, M₀ must have a unity eigenvalue.

A third, less obvious but perhaps more important, matrix is called the transition matrixT and it is built up from the autocorrelation matrix of h(n). The transition matrix is constructed by

(6.9)

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 φ(t) and the associated wavelet system.