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 \(c_k\)? Construct the associated
wavelet.
\item[(b)] Prove that if the functions \(\{\phi(x-k)\}_k\) form an orthonormal set then the
functions \(\{2^{j/2}W(2^jx-k)\}_k\) form an orthonormal set for each fixed integer \(j\).
\end{enumerate}