In this appendix we develop most of the results on scaling functions, wavelets and scaling and wavelet coefficients presented in (Reference) and elsewhere. For convenience, we repeat (Reference), (Reference), (Reference), and (Reference) here

If normalized

The results in this appendix refer to equations in the text written in bold face fonts.

**Equation (Reference)**
is the normalization of (Reference) and part of the
orthonormal conditions required by Equation 3 for

**Equation (Reference)** If the

Summing both sides over

which after reordering is

Using (Reference), Equation 21, and Equation 24 gives

but

If the scaling function is not normalized to unity, one can show the more general result of (Reference). This is done by noting that a more general form of (Reference) is

if one does not normalize

**Equation (Reference)** follows from summing Equation 3 over

which after reordering gives

and using Equation 10 gives (Reference).

**Equation (Reference)**
is derived by applying the basic recursion equation to its own right hand
side to give

which, with a change of variables of

Applying this

**Equation (Reference)**
is derived by defining the sum

and using the basic recursive equation Equation 1 to give

Interchanging the order of summation gives

but the summation over

This is the linear difference equation

which has as a solution the geometric sequence

If the limit exists, equation Equation 15 divided by

It is stated in (Reference) and shown in Equation 6 that if

which gives (Reference).

Equation Equation 21 shows another remarkable property of

**Equation (Reference)** is the “partitioning of unity" by

**Equation (Reference)** is generalization of (Reference) by noting that
the sum in (Reference) is independent of a shift of the form

for any integers

This gives (Reference) and becomes (Reference) for

The first four relationships for the scaling function hold in a generalized form for the more general defining equation (Reference). Only (Reference) is different. It becomes

for

**Equations (Reference), (Reference), and (Reference)** are the recursive
relationship for the Fourier transform of the scaling function and are
obtained by simply taking the transform (Reference) of both sides of
Equation 1 giving

which after the change of variables

and using (Reference) gives

which is (Reference) and (Reference). Applying this recursively gives the infinite product (Reference) which holds for any normalization.

**Equation (Reference)** states that the sum of the squares of samples of
the Fourier transform of the scaling function is one if the samples are
uniform every

Summing over

but

therefore

which becomes

Because of the orthogonality of integer translates of

**Equations (Reference) and (Reference)** show how the scaling function
determines the equation coefficients. This is derived by multiplying both
sides of Equation 1 by

Using the orthogonality condition Equation 3 gives

which gives (Reference). A similar argument gives (Reference).