Let $P \in \mathbb{R}_n[X]$. For $k \in \mathbb{N}$, give the expression of $\tau^k(P)$ as a function of $P$.

Give the matrix $M = \left(M_{i,j}\right)_{1 \leqslant i,j \leqslant n+1}$ of $\tau$ in the basis $\left(P_k\right)_{k \in \llbracket 1, n+1 \rrbracket}$. Express the coefficients $M_{i,j}$ in terms of $i$ and $j$.

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Show that the family $\left(H_k\right)_{k \in \llbracket 0, n \rrbracket}$ is a basis of $\mathbb{R}_n[X]$.

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Let $k \in \mathbb{Z}$. Calculate $H_n(k)$. Distinguish three cases: $k \in \llbracket 0, n-1 \rrbracket$, $k \geqslant n$, and $k < 0$. For the latter case, set $k = -p$.

We consider the function $\varphi$ defined on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}, \quad \varphi(x) = \begin{cases} 1 & \text{if } x \in \left[-\frac{1}{2}, \frac{1}{2}\right] \\ 0 & \text{otherwise} \end{cases}$$
Justify that $\varphi$ belongs to $E_{\mathrm{cpm}}$ and calculate its Fourier transform $\mathcal{F}(\varphi)$.

Let $f$ be a function in $\mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. According to the Fourier inversion formula, we have
$$\forall x \in \mathbb{R}, \quad f(x) = \int_{-1/2}^{1/2} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x\xi} \mathrm{d}\xi$$
Prove that $\mathcal{F}(f)$ is of class $C^{\infty}$ on $\mathbb{R}$ and that $\mathcal{F}(f) \in \mathcal{S}$. Deduce that $f$ is of class $C^{\infty}$ on $\mathbb{R}$.

Let $f$ be a function in $\mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. According to the Fourier inversion formula, we have
$$\forall x \in \mathbb{R}, \quad f(x) = \int_{-1/2}^{1/2} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x\xi} \mathrm{d}\xi$$
Prove that
$$\forall (x, x_{0}) \in \mathbb{R}^{2}, \quad \sum_{n=0}^{+\infty} \frac{(x-x_{0})^{n}}{n!} \int_{-1/2}^{1/2} (2\pi\mathrm{i}\xi)^{n} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x_{0}\xi} \mathrm{d}\xi = f(x)$$

Let $f$ be a function in $\mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. According to the Fourier inversion formula, we have
$$\forall x \in \mathbb{R}, \quad f(x) = \int_{-1/2}^{1/2} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x\xi} \mathrm{d}\xi$$
We assume that $f$ is zero outside a segment $[a, b]$. Show that $f = 0$.

For every natural number $n$, we denote by $S_{n}$ the function defined on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}, \quad S_{n}(x) = \sum_{k=-n}^{n} e^{2\pi\mathrm{i} kx}$$
Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. We consider the function $g$ defined on $[-1,1]$ by
$$\forall x \in ]-1,1[\backslash\{0\}, \quad g(x) = \frac{f(x)-f(0)}{\sin(\pi x)} \quad g(0) = \frac{f'(0)}{\pi} \quad g(1) = g(-1) = -g(0)$$
and the sequence of complex numbers $(c_{n}(f))_{n \in \mathbb{Z}}$ defined by
$$\forall n \in \mathbb{Z}, \quad c_{n}(f) = \int_{-1/2}^{1/2} f(x) e^{-2\pi\mathrm{i} nx} \mathrm{d}x$$
IV.A.1) Show that the function $g$ is of class $C^{1}$ on $]-1,1[\backslash\{0\}$ and continuous on $]-1,1[$.
IV.A.2) Calculate the limit of $g'$ at 0. Deduce that $g$ is of class $C^{1}$ on $]-1,1[$.

For every natural number $n$, we denote by $S_{n}$ the function defined on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}, \quad S_{n}(x) = \sum_{k=-n}^{n} e^{2\pi\mathrm{i} kx}$$
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
$$\forall n \in \mathbb{Z}, \quad c_{n}(f) = \int_{-1/2}^{1/2} f(x) e^{-2\pi\mathrm{i} nx} \mathrm{d}x$$
Let $n \in \mathbb{N}$. Calculate the integral $\int_{-1/2}^{1/2} S_{n}(x) \mathrm{d}x$.

For every natural number $n$, we denote by $S_{n}$ the function defined on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}, \quad S_{n}(x) = \sum_{k=-n}^{n} e^{2\pi\mathrm{i} kx}$$
Prove that
$$\forall n \in \mathbb{N}, \quad \forall x \in \left[-\frac{1}{2}, \frac{1}{2}\right] \backslash\{0\}, \quad S_{n}(x) = \frac{\sin((2n+1)\pi x)}{\sin(\pi x)}$$

For every natural number $n$, we denote by $S_{n}$ the function defined on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}, \quad S_{n}(x) = \sum_{k=-n}^{n} e^{2\pi\mathrm{i} kx}$$
Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. We consider the function $g$ defined on $[-1,1]$ by
$$\forall x \in ]-1,1[\backslash\{0\}, \quad g(x) = \frac{f(x)-f(0)}{\sin(\pi x)} \quad g(0) = \frac{f'(0)}{\pi} \quad g(1) = g(-1) = -g(0)$$
and the sequence of complex numbers $(c_{n}(f))_{n \in \mathbb{Z}}$ defined by
$$\forall n \in \mathbb{Z}, \quad c_{n}(f) = \int_{-1/2}^{1/2} f(x) e^{-2\pi\mathrm{i} nx} \mathrm{d}x$$
Justify that
$$\forall n \in \mathbb{N}^{*}, \quad \sum_{k=-n}^{n} c_{k}(f) = f(0) + \int_{-1/2}^{1/2} g(x) \sin((2n+1)\pi x) \mathrm{d}x$$

Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. We consider the function $g$ defined on $[-1,1]$ by
$$\forall x \in ]-1,1[\backslash\{0\}, \quad g(x) = \frac{f(x)-f(0)}{\sin(\pi x)} \quad g(0) = \frac{f'(0)}{\pi} \quad g(1) = g(-1) = -g(0)$$
We henceforth admit that $g$ is of class $C^{1}$ on $[-1,1]$.
Using integration by parts, show the existence of a real number $C$ such that
$$\forall n \in \mathbb{N}, \quad \left|\int_{-1/2}^{1/2} g(x) \sin((2n+1)\pi x) \mathrm{d}x\right| \leqslant \frac{C}{2n+1}$$

Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. Let $t \in [-1/2, 1/2]$. We consider the function $G_{t}$ defined on $[-1/2, 1/2]$ by
$$\forall x \in \left[-\frac{1}{2}, \frac{1}{2}\right], \quad G_{t}(x) = f'(x+t)\sin(\pi x) - (f(x+t)-f(t))\pi\cos(\pi x)$$
Establish the existence of a real number $D$, independent of $x$ and $t$, such that
$$\forall t \in \left[-\frac{1}{2}, \frac{1}{2}\right], \quad \forall x \in \left[-\frac{1}{2}, \frac{1}{2}\right], \quad |G_{t}(x)| \leqslant D x^{2}$$

Let $f : \mathbb{R}_{+} \rightarrow \mathbb{C}$ be a continuous function and zero outside a segment. We define the function $\mathcal{L}(f)$ (Laplace transform of $f$) on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}, \quad \mathcal{L}(f)(x) = \int_{0}^{+\infty} f(t) e^{-xt} \mathrm{d}t$$
We will admit that $\mathcal{L}(f)$ is of class $C^{\infty}$ on $\mathbb{R}$ and that
$$\forall x \in \mathbb{R}, \quad \forall n \in \mathbb{N}, \quad (\mathcal{L}(f))^{(n)}(x) = (-1)^{n} \int_{0}^{+\infty} f(t) t^{n} e^{-xt} \mathrm{d}t$$
In the remainder of this part, we shall assume that
$$\lim_{n \rightarrow +\infty} \sum_{0 \leqslant k \leqslant \lfloor nx \rfloor} \frac{(n\lambda)^{k}}{k!} e^{-n\lambda} = \frac{1}{2} \quad \text{if } x = \lambda$$
VI.C.1) Let $x \in \mathbb{R}_{+}$. Prove that
$$\lim_{n \rightarrow +\infty} \sum_{0 \leqslant k \leqslant \lfloor nx \rfloor} (-1)^{k} \frac{n^{k}}{k!} (\mathcal{L}(f))^{(k)}(n) = \int_{0}^{x} f(y) \mathrm{d}y$$
VI.C.2) Deduce that the map $\mathcal{L} : f \mapsto \mathcal{L}(f)$ is injective on the set of complex-valued functions, continuous on $\mathbb{R}_{+}$ and zero outside a bounded interval.

Show that $\Sigma_{N}$ is a closed, bounded and convex subset of $\mathbb{R}^{N}$.
Where $\Sigma_{N}$ denotes the set of vectors $p \in \mathbb{R}^{N}$ such that $\sum_{i=1}^{N} p_{i} = 1$ and $p_{i} \geqslant 0$ for all $1 \leqslant i \leqslant N$.

Show that $H_{N}$ is positive, continuous on $\Sigma_{N}$ and calculate the value of $H_{N}(p)$ when $p_{i} = 1/N$ for all $i \in \{1, \ldots, N\}$ (uniform distribution on $\{1, \ldots, N\}$).
Where $H_{N}(p) = \sum_{i=1}^{N} \varphi(p_{i})$ and $\varphi(t) = \begin{cases} 0 & \text{if } t = 0 \\ -t \ln(t) & \text{otherwise.} \end{cases}$

(a) Let $a$ and $b$ be in $[0, +\infty[$ such that $a < b$. Show that there exists $\epsilon \in ]0, b]$ such that $\varphi(a + t) + \varphi(b - t) > \varphi(a) + \varphi(b)$ for all $t > 0$ such that $t \leqslant \epsilon$.
(b) Deduce that $H_{N}$ attains its maximum on $\Sigma_{N}$ at a unique point which you will determine.
Where $\varphi(t) = \begin{cases} 0 & \text{if } t = 0 \\ -t \ln(t) & \text{otherwise,} \end{cases}$ $\Sigma_{N}$ is the set of vectors $p \in \mathbb{R}^{N}$ such that $\sum_{i=1}^{N} p_{i} = 1$ and $p_{i} \geqslant 0$ for all $1 \leqslant i \leqslant N$, and $H_{N}(p) = \sum_{i=1}^{N} \varphi(p_{i})$.

We denote by $\Sigma_{\infty}$ the set of sequences of real numbers $p = (p_{i})_{i \geqslant 1}$ such that $p_{i} \geqslant 0$ for all $i \geqslant 1$ and $\sum_{i=1}^{+\infty} p_{i} = 1$. We denote by $H_{\infty}$ the function on $\Sigma_{\infty}$ defined by $H_{\infty}(p) = \sum_{i=1}^{\infty} \varphi(p_{i})$ taking values in $\mathbb{R}_{+} \cup \{+\infty\}$, where $\varphi(t) = \begin{cases} 0 & \text{if } t = 0 \\ -t \ln(t) & \text{otherwise.} \end{cases}$
(a) We consider $a \in ]0,1[$ and $p_{i} = a(1-a)^{i-1}$ for $i \geqslant 1$. Calculate $H_{\infty}(p)$ and study its variations as a function of $a$.
(b) Show that there exists $p \in \Sigma_{\infty}$ such that $H_{\infty}(p) = +\infty$. (Hint: You may use without proof that the series with general term $n^{-1} \ln(n)^{-\beta}$ for $n \geqslant 2$ converges if and only if $\beta > 1$).

Let $n$ be a strictly positive integer. We consider a family $(X_{k})_{1 \leqslant k \leqslant n}$ of $n$ random variables taking values in $\{1, \ldots, N\}$, pairwise independent and identically distributed, defined on a probability space $(\Omega, \mathscr{A}, \mathbf{P})$. We further assume that $\mathbf{P}(X_{1} = i) = p_{i}$ and that $p_{i} > 0$ for all $i \in \{1, \ldots, N\}$. Show that for all $\epsilon > 0$, we have $\mathbf{P}\left(\left|\frac{1}{n} \ln\left(\prod_{k=1}^{n} p_{X_{k}}\right) + H_{N}(p)\right| \geqslant \epsilon\right)$ tends to 0 as $n$ tends to infinity.
Where $H_{N}(p) = \sum_{i=1}^{N} \varphi(p_{i})$ and $\varphi(t) = \begin{cases} 0 & \text{if } t = 0 \\ -t \ln(t) & \text{otherwise.} \end{cases}$

Let $\mathcal{S}$ be the simplex with vertices $s_0, s_1, \ldots, s_n$, defined by $$\mathcal{S} = \left\{\sum_{i=0}^n t_i s_i \mid \forall i = 0,\ldots,n,\, t_i \geqslant 0,\, \sum_{i=0}^n t_i = 1\right\}.$$ The volume of $\mathcal{S}$ is defined by $\operatorname{Vol}(\mathcal{S}) := \frac{1}{n!}\left|\det(s_1 - s_0, s_2 - s_0, \ldots, s_n - s_0)\right|$.
7a. Show that $\mathcal{S}$ is a compact convex set in $\mathbb{R}^n$.
7b. Show that $\mathring{\mathcal{S}} = \left\{\sum_{i=0}^n t_i s_i \mid \forall i = 0,\ldots,n,\, t_i > 0,\, \sum_{i=0}^n t_i = 1\right\}$. Deduce that if $0 \in \mathring{\mathcal{S}}$, then for all $\lambda \in [0,1[$, $\lambda \mathcal{S} \subset \mathring{\mathcal{S}}$.
7c. For $i = 0, \ldots, n$, we denote $\hat{s}_i = (1, s_i)$ the point of $\mathbb{R}^{n+1}$ whose coordinates are 1 followed by the coordinates of $s_i$. Express $\left|\det(\hat{s}_0, \hat{s}_1, \ldots, \hat{s}_n)\right|$ as a function of $\operatorname{Vol}(\mathcal{S})$. Deduce that the volume of a simplex does not depend on the order of the vertices.

Let $f \in \mathbb{R}^{N}$ and $J_{f} : \Sigma_{N} \rightarrow \mathbb{R}$ defined by $J_{f}(p) = H_{N}(p) + \sum_{i=1}^{N} p_{i} f_{i}$. We denote $$J_{f,*} = \sup\{J_{f}(p) \mid p \in \Sigma_{N}\}$$ the supremum of $J_{f}$ on $\Sigma_{N}$ and $\Sigma_{N}(f) = \{p \in \Sigma_{N} \mid J_{f}(p) = J_{f,*}\}$ the set of $p$ in $\Sigma_{N}$ for which the supremum is attained.
Show that $\Sigma_{N}(f)$ is non-empty.

Let $f \in \mathbb{R}^{N}$ and $J_{f} : \Sigma_{N} \rightarrow \mathbb{R}$ defined by $J_{f}(p) = H_{N}(p) + \sum_{i=1}^{N} p_{i} f_{i}$. We denote $J_{f,*} = \sup\{J_{f}(p) \mid p \in \Sigma_{N}\}$ and $\Sigma_{N}(f) = \{p \in \Sigma_{N} \mid J_{f}(p) = J_{f,*}\}$.
Let $p \in \Sigma_{N}$.
(a) Suppose that $p_{1} = 0$ and $p_{2} > 0$. Show that there exists $p'$ in $\Sigma_{N}$ such that $J_{f}(p') > J_{f}(p)$ (you may look for $p'$ close to $p$).
(b) Deduce that if $p \in \Sigma_{N}(f)$, then $p_{i} > 0$ for all $i \in \{1, \ldots, N\}$.

We are given an enumeration of the set $\Lambda_X = \{y_i \mid i \in \mathbb{N}\}$. Deduce that there exists a sequence of positive reals $\left(q_i\right)_{i \geqslant 0}$ such that for all $x \in \mathbb{R}$, $$f(x) = \sum_{i=0}^{+\infty} q_i g\left(x - y_i\right), \quad \text{and} \quad \sum_{i \in \mathbb{N},\, y_i \in [x-K, x]} q_i = \mathbb{E}(N(x-K, x)).$$

Let $f \in \mathbb{R}^{N}$ and $J_{f} : \Sigma_{N} \rightarrow \mathbb{R}$ defined by $J_{f}(p) = H_{N}(p) + \sum_{i=1}^{N} p_{i} f_{i}$. We denote $J_{f,*} = \sup\{J_{f}(p) \mid p \in \Sigma_{N}\}$ and $\Sigma_{N}(f) = \{p \in \Sigma_{N} \mid J_{f}(p) = J_{f,*}\}$.
Let $p \in \Sigma_{N}$. We now assume that $p_{i} > 0$ for all $i \in \{1, \ldots, N\}$. We denote $E_{0} = \{a \in \mathbb{R}^{N} \mid \sum_{i=1}^{N} a_{i} = 0\}$.
(a) Verify that $E_{0}$ is a vector subspace of $\mathbb{R}^{N}$ and give its dimension. Identify the orthogonal $E_{0}^{\perp}$ of $E_{0}$ for the canonical inner product on $\mathbb{R}^{N}$.
(b) Let $a \in E_{0}$ and $\tilde{p} : \mathbb{R} \rightarrow \mathbb{R}^{N}$ defined by $\tilde{p}(t) = p + ta$. Show that there exists $\epsilon > 0$ such that $\tilde{p}(t) \in \Sigma_{N}$ for all $t \in ]-\epsilon, \epsilon[$. Calculate the derivative of $\tilde{p}$ at 0.
(c) Suppose further that $p \in \Sigma_{N}(f)$. Show that for all $a \in E_{0}$, we have $\sum_{i=1}^{N} a_{i}(f_{i} - \ln(p_{i})) = 0$. Deduce that there exists $c \in \mathbb{R}$ such that $\ln(p_{i}) = f_{i} + c$ for all $i \in \{1, \ldots, N\}$.