grandes-ecoles 2018 Q4

grandes-ecoles · France · centrale-maths1__pc Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions

Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, continuous and integrable on $\mathbb{R}$. Show that, for any real $\xi$, the function $\left\lvert\, \begin{aligned} & \mathbb{R} \rightarrow \mathbb{C} \\ & x \mapsto f(x) \exp(-\mathrm{i} 2\pi \xi x) \end{aligned}\right.$ is integrable on $\mathbb{R}$.

Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, continuous and integrable on $\mathbb{R}$. Show that, for any real $\xi$, the function $\left\lvert\, \begin{aligned} & \mathbb{R} \rightarrow \mathbb{C} \\ & x \mapsto f(x) \exp(-\mathrm{i} 2\pi \xi x) \end{aligned}\right.$ is integrable on $\mathbb{R}$.

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