We consider the function $\psi$ defined on $\mathbb{R}$ by $$\forall x \in \mathbb{R}^{*}, \quad \psi(x) = \frac{\sin(\pi x)}{\pi x} \quad \text{and} \quad \psi(0) = 1$$ I.B.1) Justify that $\psi$ is expandable as a power series. Specify this expansion and its radius of convergence. Deduce that $\psi$ is of class $C^{\infty}$ on $\mathbb{R}$. I.B.2) Prove that $$\forall n \in \mathbb{N}, \quad \int_{n}^{n+1} |\psi(x)| \mathrm{d}x \geqslant \frac{2}{(n+1)\pi^{2}}$$ Deduce that $\psi$ does not belong to $E_{\mathrm{cpm}}$.
Let $f \in \mathcal{S}$. I.D.1) Justify that, for every natural number $n$, the function $x \mapsto x^{n} f(x)$ is integrable on $\mathbb{R}$. I.D.2) Prove that the function $\mathcal{F}(f)$ is of class $C^{\infty}$ on $\mathbb{R}$ and that $$\forall n \in \mathbb{N}, \quad \forall \xi \in \mathbb{R}, \quad (\mathcal{F}(f))^{(n)}(\xi) = (-2\pi\mathrm{i})^{n} \int_{-\infty}^{+\infty} t^{n} f(t) e^{-2\pi\mathrm{i} t\xi} \mathrm{~d}t$$
QI.E
First order differential equations (integrating factor)View
We consider the function $\theta : \mathbb{R} \rightarrow \mathbb{C}$ defined by $\theta(x) = \exp(-\pi x^{2})$, for $x \in \mathbb{R}$. I.E.1) Justify that $\theta$ belongs to $\mathcal{S}$ and that $\mathcal{F}(\theta)$ is a solution of the differential equation $$\forall \xi \in \mathbb{R}, \quad y'(\xi) = -2\pi\xi\, y(\xi)$$ I.E.2) Establish that $\mathcal{F}(\theta) = \theta$. We will admit that $\int_{-\infty}^{+\infty} \theta(x) \mathrm{d}x = 1$.
Let $f \in \mathcal{S}$. We assume that $\mathcal{F}(f)$ is integrable on $\mathbb{R}$. For every positive natural number $n$, we set $$I_{n} = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) \theta\left(\frac{\xi}{n}\right) \mathrm{d}\xi \quad J_{n} = \int_{-\infty}^{+\infty} f\left(\frac{t}{n}\right) \mathcal{F}(\theta)(t) \mathrm{d}t$$ Show that $\lim_{n \rightarrow +\infty} I_{n} = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) \mathrm{d}\xi$.
Let $f \in \mathcal{S}$. We assume that $\mathcal{F}(f)$ is integrable on $\mathbb{R}$. For every positive natural number $n$, we set $$I_{n} = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) \theta\left(\frac{\xi}{n}\right) \mathrm{d}\xi \quad J_{n} = \int_{-\infty}^{+\infty} f\left(\frac{t}{n}\right) \mathcal{F}(\theta)(t) \mathrm{d}t$$ Calculate $\lim_{n \rightarrow +\infty} J_{n}$.
Let $f \in \mathcal{S}$. We assume that $\mathcal{F}(f)$ is integrable on $\mathbb{R}$. For every positive natural number $n$, we set $$I_{n} = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) \theta\left(\frac{\xi}{n}\right) \mathrm{d}\xi \quad J_{n} = \int_{-\infty}^{+\infty} f\left(\frac{t}{n}\right) \mathcal{F}(\theta)(t) \mathrm{d}t$$ Prove that $\forall n \in \mathbb{N}^{*}, I_{n} = J_{n}$. We will admit the Fubini formula: $$\int_{-\infty}^{+\infty} \left(\int_{-\infty}^{+\infty} f(t) \theta\left(\frac{\xi}{n}\right) e^{-2\pi\mathrm{i} t\xi} \mathrm{d}\xi\right) \mathrm{d}t = \int_{-\infty}^{+\infty} \left(\int_{-\infty}^{+\infty} f(t) \theta\left(\frac{\xi}{n}\right) e^{-2\pi\mathrm{i} t\xi} \mathrm{d}t\right) \mathrm{d}\xi$$
Let $f \in \mathcal{S}$. We assume that $\mathcal{F}(f)$ is integrable on $\mathbb{R}$. Prove that $f(0) = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) \mathrm{d}\xi$. Deduce, using the function $h : t \mapsto f(x+t)$, that $$\forall x \in \mathbb{R}, \quad f(x) = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x\xi} \mathrm{d}\xi$$
QII.E
Integration using inverse trig and hyperbolic functionsView
Prove that $\forall x \in \mathbb{R}, \quad \int_{-\infty}^{+\infty} \frac{e^{2\pi\mathrm{i} x\xi}}{1+(2\pi\xi)^{2}} \mathrm{d}\xi = \frac{1}{2} e^{-|x|}$.
Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. The sequence of complex numbers $(c_{n}(f))_{n \in \mathbb{Z}}$ is defined by $$\forall n \in \mathbb{Z}, \quad c_{n}(f) = \int_{-1/2}^{1/2} f(x) e^{-2\pi\mathrm{i} nx} \mathrm{d}x$$ Prove the existence of a real number $E$ such that $$\forall t \in \left[-\frac{1}{2}, \frac{1}{2}\right], \quad \left|f(t) - \sum_{k=-n}^{n} c_{k}(f) e^{2\pi\mathrm{i} kt}\right| \leqslant \frac{E}{2n+1}$$ One may introduce the function $h_{t} : x \mapsto f(x+t)$.
Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. We set $$\forall k \in \mathbb{Z}, \quad \forall x \in \mathbb{R}, \quad \psi_{k}(x) = \psi(x+k)$$ where $\psi(x) = \frac{\sin(\pi x)}{\pi x}$ for $x \neq 0$ and $\psi(0) = 1$. Justify that $\forall n \in \mathbb{N}, \quad (\mathcal{F}(f))^{(n)}\left(\frac{1}{2}\right) = (\mathcal{F}(f))^{(n)}\left(-\frac{1}{2}\right) = 0$.
Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. Let $h$ be the function defined on $\mathbb{R}$, which is 1-periodic and which equals $\mathcal{F}(f)$ on the interval $[-1/2, 1/2]$. Show that $h$ is of class $C^{\infty}$ on $\mathbb{R}$.
Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. Let $h$ be the function defined on $\mathbb{R}$, which is 1-periodic and which equals $\mathcal{F}(f)$ on the interval $[-1/2, 1/2]$. Using the inequality from IV.G, prove the existence of a sequence of complex numbers $(d_{k})_{k \in \mathbb{Z}}$ such that the sequence of functions $\left(x \mapsto \sum_{k=-n}^{n} d_{k} e^{2\pi\mathrm{i} kx}\right)_{n \in \mathbb{N}}$ converges uniformly to $\mathcal{F}(f)$ on $[-1/2, 1/2]$.
Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. We set $$\forall k \in \mathbb{Z}, \quad \forall x \in \mathbb{R}, \quad \psi_{k}(x) = \psi(x+k)$$ where $\psi(x) = \frac{\sin(\pi x)}{\pi x}$ for $x \neq 0$ and $\psi(0) = 1$. Let $(d_{k})_{k \in \mathbb{Z}}$ be the sequence of complex numbers from V.C. Prove that the sequence of functions $\left(\sum_{k=-n}^{n} d_{k} \psi_{k}\right)_{n \in \mathbb{N}}$ converges uniformly to $f$ on $\mathbb{R}$.
Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. We set $$\forall k \in \mathbb{Z}, \quad \forall x \in \mathbb{R}, \quad \psi_{k}(x) = \psi(x+k)$$ where $\psi(x) = \frac{\sin(\pi x)}{\pi x}$ for $x \neq 0$ and $\psi(0) = 1$, and $f = \sum_{k=-\infty}^{+\infty} d_{k} \psi_{k}$ (uniform limit). Establish that $\forall j \in \mathbb{Z},\ f(-j) = d_{j}$.
We consider $(X_{n})_{n \in \mathbb{N}^{*}}$ a sequence of random variables defined on the same probability space $(\Omega, \mathcal{A}, P)$, mutually independent and following the same Poisson distribution with parameter $\lambda > 0$. We set $$\forall n \in \mathbb{N}^{*}, \quad S_{n} = X_{1} + \cdots + X_{n}$$ VI.A.1) By induction, prove that, for every integer $n \in \mathbb{N}$, $S_{n}$ follows the Poisson distribution with parameter $n\lambda$. We will admit that, for every integer $n \in \mathbb{N}^{*}$, the variables $S_{n}$ and $X_{n+1}$ are mutually independent. VI.A.2) Let $\varepsilon \in \mathbb{R}_{+}^{*}$. Prove that $$\forall n \in \mathbb{N}^{*}, \quad P\left(\left|S_{n} - n\lambda\right| \geqslant n\varepsilon\right) \leqslant \frac{\lambda}{n\varepsilon^{2}}$$ VI.A.3) Let $\varepsilon > 0$. Justify the following two inclusions $$\begin{aligned} & \left(S_{n} > n(\lambda+\varepsilon)\right) \subset \left(\left|S_{n}-n\lambda\right| \geqslant n\varepsilon\right) \\ & \left(S_{n} \leqslant n(\lambda-\varepsilon)\right) \subset \left(\left|S_{n}-n\lambda\right| \geqslant n\varepsilon\right) \end{aligned}$$ VI.A.4) In all the following questions, we assume $x \geqslant 0$. Deduce from VI.A.3 that $$\begin{cases} \lim_{n \rightarrow +\infty} P\left(S_{n} \leqslant nx\right) = 0 & \text{if } 0 \leqslant x < \lambda \\ \lim_{n \rightarrow +\infty} P\left(S_{n} \leqslant nx\right) = 1 & \text{if } x > \lambda \end{cases}$$
We consider $(X_{n})_{n \in \mathbb{N}^{*}}$ a sequence of random variables defined on the same probability space $(\Omega, \mathcal{A}, P)$, mutually independent and following the same Poisson distribution with parameter $\lambda > 0$. We set $S_{n} = X_{1} + \cdots + X_{n}$. Using question VI.A, show that $$\lim_{n \rightarrow +\infty} \sum_{0 \leqslant k \leqslant \lfloor nx \rfloor} \frac{(n\lambda)^{k}}{k!} e^{-n\lambda} = \begin{cases} 0 & \text{if } 0 \leqslant x < \lambda \\ 1 & \text{if } x > \lambda \end{cases}$$