Let $X$ be a real and discrete random variable and $m \in \mathbb { R }$. For $T \in \mathbb { R } _ { + } ^ { * }$, we set $V _ { m } ( T ) = \frac { 1 } { 2 T } \int _ { - T } ^ { T } \phi _ { X } ( t ) \mathrm { e } ^ { - \mathrm { i } m t } \mathrm {~d} t$. We assume that $X ( \Omega )$ is finite and we use the notations of question 1: $X ( \Omega ) = \left\{ x _ { 1 } , \ldots , x _ { r } \right\}$ with $a _ { k } = \mathbb { P } \left( X = x _ { k } \right)$. Show that, for all $T \in \mathbb { R } _ { + } ^ { * }$, we have $V _ { m } ( T ) = \sum _ { n = 1 } ^ { r } \operatorname { sinc } \left( T \left( x _ { n } - m \right) \right) \mathbb { P } \left( X = x _ { n } \right)$.
Let $X$ be a real and discrete random variable and $m \in \mathbb { R }$. For $T \in \mathbb { R } _ { + } ^ { * }$, we set $V _ { m } ( T ) = \frac { 1 } { 2 T } \int _ { - T } ^ { T } \phi _ { X } ( t ) \mathrm { e } ^ { - \mathrm { i } m t } \mathrm {~d} t$.\\
We assume that $X ( \Omega )$ is finite and we use the notations of question 1: $X ( \Omega ) = \left\{ x _ { 1 } , \ldots , x _ { r } \right\}$ with $a _ { k } = \mathbb { P } \left( X = x _ { k } \right)$.\\
Show that, for all $T \in \mathbb { R } _ { + } ^ { * }$, we have $V _ { m } ( T ) = \sum _ { n = 1 } ^ { r } \operatorname { sinc } \left( T \left( x _ { n } - m \right) \right) \mathbb { P } \left( X = x _ { n } \right)$.