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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 1996 • Question 9

Question

Math 6420 • PDE Fall 1996 • Question 9

Suppose \(f(t)\) is a band limited function with Fourier Transform \(F(\mu )=0\) for \(|\mu |>\pi /h\). Define the sinc function \(S_0(t)\) by \begin{equation} S_0(t)=\frac {\sin (\pi t/h)}{\pi t/h}. \end{equation} Show that \begin{equation} f(t)=\sum _{k=-\infty }^{\infty } f(kh)S_0(t-kh). \end{equation}

Raw LaTeX

Suppose \(f(t)\) is a band limited function with Fourier
Transform \(F(\mu)=0\) for \(|\mu|>\pi/h\). Define the sinc function \(S_0(t)\) by
\begin{equation}
  S_0(t)=\frac{\sin(\pi t/h)}{\pi t/h}.
\end{equation}
Show that
\begin{equation}
  f(t)=\sum_{k=-\infty}^{\infty} f(kh)S_0(t-kh).
\end{equation}

Topic Tags

Fourier Transform and Sampling Theory Linear Programming and Optimization Real Integral Evaluation by Complex Methods