Let $f ( t )$ be a periodic function of period $T , f ( t + T ) = f ( t ) ( T > 0 )$, and be expanded in the complex Fourier series as follows:
$$f ( t ) = \sum _ { n = - \infty } ^ { \infty } F _ { n } \exp \left( - i \omega _ { n } t \right)$$
Here, $i$ is the imaginary unit, and $t$ is a real number. Answer the following questions.
I. Express $\omega _ { n }$ using $T$ and $n$.
II. Let $M$ be a positive-integer constant and $\delta ( t )$ be the delta function, and define
$$\hat { f } ( t ) = \sum _ { m = 0 } ^ { M - 1 } f ( t ) \delta ( t - m \Delta t ) , \quad \Delta t = \frac { T } { M }$$
Express
$$\lim _ { \varepsilon \rightarrow + 0 } \int _ { - \varepsilon } ^ { T - \varepsilon } \hat { f } ( t ) \exp \left( i \omega _ { k } t \right) d t$$
using $F _ { n } ( n = - \infty , \cdots , - 1,0,1 , \cdots , \infty )$. Here $k$ is an arbitrary integer.
III. $\Delta t$ is given in Question II. Express $F _ { j } ( j = 0,1,2 , \cdots , M - 1 )$ using
$$f ( 0 ) , \quad f ( \Delta t ) , \quad f ( 2 \Delta t ) , \cdots , \quad f ( ( M - 1 ) \Delta t )$$
when $F _ { n } = 0 ( n < 0$ or $n \geq M )$.
IV. Calculate $F _ { j }$ in Question III when $f ( l \Delta t ) = ( - 1 ) ^ { l } \quad ( l = 0,1,2 , \cdots , M - 1 )$.