Math 6420 • PDE

The \(z\)-transform is defined on sequences in \(\ell^2\) by the
following formulas
\[
  U(z)=\sum_{n=-\infty}^{\infty}u_nz^n,\qquad
  u_n=\frac{1}{2\pi i}\int_C U(z)z^{-n-1}\,dz
\]
where \(C\) is a closed contour enclosing the origin, and integration is counter-clockwise.

If \(z\) is restricted to lie on the unit circle, then \(U(z)=U(e^{i\theta})=f(\theta)\).
\begin{enumerate}
\item[a).] Find the relation between the \(z\) transform and the Fourier series
representation of \(f(\theta)\).
\item[b).] Use the \(z\)-transform to solve the system of equations
\[
  \frac{du_n}{dt}=\frac{u_{n+1}-u_{n-1}}{2h},\qquad (n=0,\pm1,\pm2,\ldots)
\]
with initial condition \(u_n(0)=\delta_{n0}\) (\(h\) is a positive constant).
\end{enumerate}
\end{enumerate}