Ex. 5.18

The wavelet function \(\psi(x)\) of the symmlet-\(p\) wavelet basis has vanishing moments up to order \(p\). Show that this implies that polynomials of order \(p\) are represented exactly in \(V_0\), defined on page 176.

Soln. 5.18

The expansion of a function \(f(x)\) in symmlet-\(p\) wavelet basis has the form

\[\begin{equation} f(x) = \sum_{m,n}f_{mn}\psi_{mn}(x)\non \end{equation}\]

where

\[\begin{equation} f_{mn} := \int f(x)\psi_{mn}(x)dx.\non \end{equation}\]

The constraint on the moments

\[\begin{equation} \int \psi(x)x^ldx = 0, \ \ l=0,...,p-1\non \end{equation}\]

has important consequences. First it implies

\[\begin{eqnarray} &&\int \psi_{0m}(x)x^ldx = \int \psi(x-m)x^ldx = \int \psi(y)(y+m)^ldy\non\\ &=&\sum_{k=0}^l\frac{l!}{k!(l-k)!}\int \psi(y)y^kdy=0, \ \ l=0,...,p-1,\non \end{eqnarray}\]

which means that the first \(p-1\) moments of the unit translates of the mother wavelet function vanish. Second, changing scales gives

\[\begin{eqnarray} &&\int \psi_{10}(x)x^l = \frac{1}{\sqrt{2}}\int \psi(x/2)x^ldx\non\\ &=&2^{l+1/2}\int \psi(y)y^ldy=0,\ \ l=0,...,p-1.\non \end{eqnarray}\]

It's easy to proceed inductively to show for all \(m\) and \(n\) that

\[\begin{equation} \int \psi_{nm}(x)x^ldx = 0,\ \ l=0,...,p-1.\non \end{equation}\]

This means that every symmlet-\(p\) wavelet basis function is orthogonal to all polynomials of degree less than \(p\). Therefore we see the polynomials of degrees less than \(p\) can be represented exactly by the finite linear combination of the scaling functions \(\phi(x-k)\), which spans the reference space \(V_0\).