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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 completeor formal, but they should be sufficient to understand the ideas behind why a result is true and to give some insight into its interpretation aswell as to indicate assumptions and restrictions.
Proof 1 The conditions given by [link] and [link] can be derived by integrating both sides of
and making the change of variables $y=Mx$
and noting the integral is independent of translation which gives
With no further requirements other than $\phi \in {L}^{1}$ to allow the sum and integral interchange and $\int \phi \left(x\right)\phantom{\rule{0.166667em}{0ex}}dx\ne 0$ , this gives [link] as
and for $M=2$ gives [link] . Note this does not assume orthogonality nor any specific normalization of $\phi \left(t\right)$ and does not even assume $M$ is an integer.
This is the most basic necessary condition for the existence of $\phi \left(t\right)$ and it has the fewest assumptions or restrictions.
Proof 2 The conditions in [link] and [link] are a down-sampled orthogonality of translates by $M$ of the coefficients which results from the orthogonality of translates of the scaling function given by
in [link] . The basic scaling equation [link] is substituted for both functions in [link] giving
which, after reordering and a change of variable $y=M\phantom{\rule{0.166667em}{0ex}}x$ , gives
Using the orthogonality in [link] gives our result
in [link] and [link] . This result requires the orthogonality condition [link] , $M$ must be an integer, and any non-zero normalization $E$ may be used.
Proof 3 (Corollary 2) The result that
in [link] or, more generally
is obtained by breaking [link] for $M=2$ into the sum of the even and odd coefficients.
Next we use [link] and sum over $n$ to give
which we then split into even and odd sums and reorder to give:
Solving [link] and [link] simultaneously gives ${K}_{0}={K}_{1}=1/\sqrt{2}$ and our result [link] or [link] for $M=2$ .
If the same approach is taken with [link] and [link] for $M=3$ , we have
which, in terms of the partial sums ${K}_{i}$ , is
Using the orthogonality condition [link] as was done in [link] and [link] gives
Equation [link] and [link] are simultaneously true if and only if ${K}_{0}={K}_{1}={K}_{2}=1/\sqrt{3}$ . This process is valid for any integer $M$ and any non-zero normalization.
Proof 3 If the support of $\phi \left(x\right)$ is $[0,N-1]$ , from the basic recursion equation with support of $h\left(n\right)$ assumed as $[{N}_{1},{N}_{2}]$ we have
where the support of the right hand side of [link] is $[{N}_{1}/2,(N-1+{N}_{2})/2)$ . Since the support of both sides of [link] must be the same, the limits on the sum, or, the limits on the indices of the non zero $h\left(n\right)$ are such that ${N}_{1}=0$ and ${N}_{2}=N$ , therefore, the support of $h\left(n\right)$ is $[0,N-1]$ .
Proof 4 First define the autocorrelation function
and the power spectrum
which after changing variables, $y=x-t$ , and reordering operations gives
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