\section*{Problem 5}
I. We consider a continuous and absolutely integrable function $f ( t )$ of a real variable $t$ and denote the Fourier transform of the function $f ( t )$ as $\mathcal { F } \{ f ( t ) \}$. We define a function $F ( \omega )$ by the following formula:

$$F ( \omega ) = \mathcal { F } \{ f ( t ) \} = \int _ { - \infty } ^ { \infty } f ( t ) \exp ( - i \omega t ) \mathrm { d } t$$

where $\omega$ is a real variable and $i$ is the imaginary unit.

\begin{enumerate}
  \item We define $g ( t ) = f ( a t )$ for a constant $a$ satisfying $a > 0$ and
$$G ( \omega ) = \mathcal { F } \{ g ( t ) \} = \mathcal { F } \{ f ( a t ) \}$$
Express $G ( \omega )$ using the function $F$.
  \item When $f ( t ) = \exp \left( - t ^ { 2 } \right)$ and $a = 2$, sketch the graph of the Fourier transformed functions $F ( \omega )$ and $G ( \omega )$ defined by Equation (2) as a function of $\omega$ to show the difference between them.
  \item We define $h ( t ) = f ( t ) \exp ( - i b t )$ for a constant $b$ satisfying $b > 0$ and
$$H ( \omega ) = \mathcal { F } \{ h ( t ) \} = \mathcal { F } \{ f ( t ) \exp ( - i b t ) \}$$
Express $H ( \omega )$ using the function $F$.
  \item When $f ( t ) = \exp \left( - t ^ { 2 } \right)$ and $b = 2$, sketch the graph of the Fourier transformed functions $F ( \omega )$ and $H ( \omega )$ defined by Equation (3) as a function of $\omega$ to show the difference between them.
\end{enumerate}

II. Let $N$ be a positive integer. We define a discrete Fourier transform $D _ { 1 } , \cdots , D _ { N }$ by the following formula:

$$D _ { m } = \frac { 1 } { \sqrt { N } } \sum _ { n = 1 } ^ { N } c _ { n } \exp \left( - i \frac { 2 \pi } { N } n m \right)$$

for a complex sequence $c _ { 1 } , \cdots , c _ { N }$. Here, $m$ is an integer satisfying $1 \leq m \leq N$.

\begin{enumerate}
  \item Calculate $S \left( n , n ^ { \prime } \right)$ :
$$S \left( n , n ^ { \prime } \right) = \frac { 1 } { N } \sum _ { m = 1 } ^ { N } \exp \left\{ i \frac { 2 \pi } { N } \left( n - n ^ { \prime } \right) m \right\}$$
Here, $n$ is an integer satisfying $1 \leq n \leq N$, and $n ^ { \prime }$ is an integer satisfying $1 \leq n ^ { \prime } \leq N$.
  \item Let $U _ { m n }$ be a complex number satisfying
$$D _ { m } = \sum _ { n = 1 } ^ { N } U _ { m n } c _ { n }$$
Show that the matrix $\mathbf { U } = \left[ U _ { m n } \right] _ { 1 \leq m \leq N , 1 \leq n \leq N }$ is a unitary matrix.
  \item Derive an equation for the inverse discrete Fourier transform $c _ { n }$ from $D _ { 1 } , \cdots , D _ { N }$. Here, $n$ is an integer satisfying $1 \leq n \leq N$.
  \item For any complex value $z , \bar { z }$ is the complex conjugate of $z$. We define $Q$ by
$$Q = \sum _ { n = 1 } ^ { N } \left( \overline { c _ { n } } c _ { n + 1 } + \overline { c _ { n + 1 } } c _ { n } \right)$$
Express $Q$ in terms of $D _ { m }$ and $\overline { D _ { m } }$. Here, we impose the condition $c _ { N + 1 } = c _ { 1 }$.
\end{enumerate}