Math 6420 • PDE

Suppose that the real scaling function \(\phi\) satisfies the dilation equation
\[
  \phi(x)=\sum_k c_k\phi(2x-k).
\]
Also assume that \(\{\phi(x-k),\, k\in\mathbb{Z}\}\) forms an orthonormal set with respect to
the inner product \(\langle u,v\rangle=\int_{-\infty}^{\infty}u(x)v(x)\,dx\). Define the
corresponding wavelet function according to
\[
  \psi(x)=\sum_k(-1)^k c_{1-k}\phi(2x-k).
\]
\begin{enumerate}
\item[(a)] Prove that
\[
  \sum_{k\in\mathbb{Z}} c_kc_{k-2p}=2\delta_{0,p}.
\]
\item[(b)] Show that \(\{2^{j/2}\psi(2^jx-k),\, k\in\mathbb{Z}\}\) forms an orthonormal set.
\item[(c)] Briefly explain signal decomposition and reconstruction within the framework of
multiresolution analysis.
\end{enumerate}