Ex. 5.18

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

$$f(x)=\sum _{m,n}{f}_{mn}{\psi }_{mn}(x)$$

where

$$f_{mn}:=\int f(x){\psi }_{mn}(x)dx.$$

The constraint on the moments

$$\int \psi (x)x^{l}dx=0,l=0,...,p-1$$

has important consequences. First it implies

$$\begin{array}{rcl}& & \int {\psi }_{0m}(x)x^{l}dx=\int \psi (x-m)x^{l}dx=\int \psi (y)(y+m{)}^{l}dy\\ & =& \sum _{k=0}^l\frac{l!}{k!(l-k)!}\int \psi (y)y^{k}dy=0,l=0,...,p-1,\end{array}$$

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

$$\begin{array}{rcl}& & \int {\psi }_{10}(x)x^l=\frac{1}{\sqrt{2}}\int \psi (x/2)x^{l}dx\\ & =& 2^{l+1/2}\int \psi (y)y^{l}dy=0,l=0,...,p-1.\end{array}$$

It's easy to proceed inductively to show for all $m$ and $n$ that

$$\int {\psi }_{nm}(x)x^{l}dx=0,l=0,...,p-1.$$

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 $\varphi (x-k)$, which spans the reference space $V_0$.