Math 6420 • PDE

The dilation equation is
\[
  \phi(x)=\sum_k c_k\phi(2x-k)
\]
and the associated wavelet is
\[
  W(x)=\sum_k(-1)^k c_{1-k}\phi(2x-k).
\]
\begin{enumerate}
\item[(a)] Show that the function
\[
N_2(x)=
\begin{cases}
x, & 0<x<1,\\
2-x, & 1\le x<2,\\
0, & \text{elsewhere}
\end{cases}
\]
satisfies the dilation equation and determine the values of \(c_k\). Sketch the associated wavelet.
Briefly explain whether or not \(N_2(x)\) can act as a scaling function for a multiresolution
analysis?

\item[(b)] Prove that if the functions \(\{\phi(x-k)\}_k\) form an orthonormal set then
\[
  \int_{-\infty}^{\infty}W(x)W(x-m)\,dx
  =
  \delta_{0,m}\int_{-\infty}^{\infty}\phi^2(x)\,dx.
\]

\item[(c)] Prove that
\[
  \int_{-\infty}^{\infty} W(x)\phi(x-m)\,dx=0
\]
for all \(m\).
\end{enumerate}