This appendix contains outline proofs and derivations for the theorems and formulas given in early part of Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients . They are not intended to be complete or formal, but they should be sufficient to understand the ideas behind why a result is true and to give some insight into its interpretation as well as to indicate assumptions and restrictions.

**Proof 1*** The conditions given by (Reference) and (Reference) can be derived by
integrating both sides of*

*and making the change of variables *

*and noting the integral is independent of translation which gives*

*With no further requirements other than *

*and for *

This is the most basic necessary condition for the existence of

**Proof 2*** The conditions in (Reference) and (Reference) are a down-sampled
orthogonality of translates by *

*in (Reference). The basic scaling equation Equation 1 is substituted for
both functions in Equation 5 giving*

*which, after reordering and a change of variable *

*Using the orthogonality in Equation 5 gives our result*

*in (Reference) and (Reference). This result requires the orthogonality
condition Equation 5, *

**Proof 3 (Corollary 2)** *
The result that*

*in (Reference) or, more generally*

*is obtained by breaking Equation 4 for *

*Next we use Equation 8 and sum over *

*which we then split into even and odd sums and reorder to give:*

*Solving Equation 11 and Equation 13 simultaneously gives *

*If the same approach is taken with (Reference) and (Reference) for *

*which, in terms of the partial sums *

*Using the orthogonality condition Equation 13 as was done in Equation 12 and
(Reference) gives*

*Equation Equation 15 and Equation 16 are simultaneously true if and only if
*

**Proof 3*** If the support of *

*where the support of the right hand side of Equation 17 is
*

**Proof 4** *First define the autocorrelation function*

*and the power spectrum*

*which after changing variables, *

*If we look at Equation 18 as being the inverse Fourier transform of
Equation 21 and sample *

*but this integral is the form of an inverse discrete-time Fourier
transform (DTFT) which means*

*If the integer translates of *

*If the scaling function is not normalized*

*which is similar to Parseval's theorem relating the energy in the
frequency domain to the energy in the time domain.*

**Proof 6*** Equation (Reference) states a very interesting property of the frequency
response of an FIR filter with the scaling coefficients as filter
coefficients. This result can be derived in the frequency or time domain.
We will show the frequency domain argument. The scaling equation
Equation 1 becomes (Reference) in the frequency domain. Taking the squared
magnitude of both sides of a scaled version of*

*gives*

*Add *

*which is unity from (Reference). Summing the right side of Equation 28 gives*

*Break this sum into a sum of the even and odd indexed terms.*

*which after using Equation 29 gives*

*which gives (Reference). This requires both the scaling and orthogonal
relations but no specific normalization of *

**Proof 10*** The multiresolution assumptions in (Reference) require the scaling
function and wavelet satisfy (Reference) and (Reference)*

*and orthonormality requires*

*and*

*for all *

*Rearranging and making a change of variables gives*

*Using Equation 35 gives*

*for all *

*Separating Equation 40 into even and odd indices gives*

*which must be true for all integer *

*From the orthonormality of the translates of *

*This can be compactly represented as*

*Assuming the sequences are finite length Equation 45 can
be used to show that*

*where *

*Similarly by convolving both sides of Equation 46 by
*

*Combining Equation 47 and Equation 48 gives the result*

**Proof 11*** We show the integral of the wavelet is zero by integrating both sides of
(Equation 34b) gives*

*But the integral on the right hand side is *

*and, therefore, from Equation 50, the integral of the wavelet is zero.*

*The fact that multiplying in the time domain by *