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This is a very powerful result [link] , [link] . It not only ties the number of zero moments to the regularity but also to the degree ofpolynomials that can be exactly represented by a sum of weighted and shifted scaling functions.
Theorem 21 If $\psi \left(t\right)$ is $K$ -times differentiable and decays fast enough, then the first $K-1$ wavelet moments vanish [link] ; i.e.,
implies
Unfortunately, the converse of this theorem is not true. However, we can relate the differentiability of $\psi \left(t\right)$ to vanishing moments by
Theorem 22 There exists a finite positive integer $L$ such that if ${m}_{1}\left(k\right)=0$ for $0\le k\le K-1$ then
for $L\phantom{\rule{0.166667em}{0ex}}P>K$ .
For example, a three-times differentiable $\psi \left(t\right)$ must have three vanishing moments, but three vanishing moments results in only one-timedifferentiability.
These theorems show the close relationship among the moments of ${h}_{1}\left(n\right)$ , $\psi \left(t\right)$ , the smoothness of $H\left(\omega \right)$ at $\omega =0$ and $\pi $ and to polynomial representation. It also states a loose relationship with thesmoothness of $\phi \left(t\right)$ and $\psi \left(t\right)$ themselves.
Daubechies used the above relationships to show the following important result which constructs orthonormal wavelets with compact support with themaximum number of vanishing moments.
Theorem 23 The discrete-time Fourier transform of $h\left(n\right)$ having $K$ zeros at $\omega =\pi $ of the form
satisfies
if and only if $L\left(\omega \right)={\left|L\left(\omega \right)\right|}^{2}$ can be written
with $K\le N/2$ where
and $R\left(y\right)$ is an odd polynomial chosen so that $P\left(y\right)\ge 0$ for $0\le y\le 1$ .
If $R=0$ , the length $N$ is minimum for a given regularity $K=N/2$ . If $N>2\phantom{\rule{0.166667em}{0ex}}K$ , the second term containing $R$ has terms with higher powers of $y$ whose coefficients can be used for purposes other than regularity.
The proof and a discussion are found in Daubechies [link] , [link] . Recall from [link] that $H\left(\omega \right)$ always has at least one zero at $\omega =\pi $ as a result of $h\left(n\right)$ satisfying the necessary conditions for $\phi \left(t\right)$ to exist and have orthogonal integer translates. We are now placing restrictions on $h\left(n\right)$ to have as high an order zero at $\omega =\pi $ as possible. That accounts for the form of [link] . Requiring orthogonality in [link] gives [link] .
Because the frequency domain requirements in [link] are in terms of the square of the magnitudes of the frequency response, spectralfactorization is used to determine $H\left(\omega \right)$ and therefore $h\left(n\right)$ from ${\left|H\left(\omega \right)\right|}^{2}$ . [link] becomes
If we use the functional notation:
then [link] becomes
Since $M\left(\omega \right)$ and $L\left(\omega \right)$ are even functions of $\omega $ they can be written as polynomials in $cos\left(\omega \right)$ and, using $cos\left(\omega \right)=1-2\phantom{\rule{0.166667em}{0ex}}{sin}^{2}(\omega /2)$ , [link] becomes
which, after a change of variables of $y={sin}^{2}(\omega /2)=1-{cos}^{2}(\omega /2)$ , becomes
where $P\left(y\right)$ is an $(N-K)$ order polynomial which must be positive since it will have to be factored to find $H\left(\omega \right)$ from [link] . This now gives [link] in terms of new variables which are easier to use.
In order that this description supports an orthonormal wavelet basis, we now require that [link] satisfies [link]
which using [link] and [link] becomes
Equations of this form have an explicit solution found by using Bezout's theorem. The details are developed by Daubechies [link] . If all the $(N/2-1)$ degrees of freedom are used to set wavelet moments to zero, we set $K=N/2$ and the solution to [link] is given by
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