We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that only finitely many $p_i$ are strictly positive. We set $$g_0(x) = \begin{cases} \mathbb{P}(X > x) & \text{if } x \geqslant 0 \\ 0 & \text{if } x < 0 \end{cases}$$ and $Lg_0$ denotes the unique bounded solution of (E) with support in $\mathbb{R}^+$ for $g = g_0$. Show that $Lg_0(x) = 1$ for $x \geqslant 0$ and $Lg_0(x) = 0$ for $x < 0$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$ equipped with a scalar product $(.|.)$. Let $T, M$ be two endomorphisms of $E$ satisfying (H1)–(H4). We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$. Let $G$ be the set of elements $u \in E$ satisfying (a) $u \in \operatorname{Im}(T)$ and (b) $\forall v \in E, S(u,v) = 0$.
We assume that $G$ satisfies hypothesis (H5): $\operatorname{dim}(G) = 2m-2$.
Show that if $(w_1, w_2)$ is a characterizing pair of $G$ then $(T(w_1), T(w_2))$ constitutes a basis of $G^\perp$.

We equip $\mathbb{R}_{n}[X]$ with the inner product defined by $$\langle P, Q \rangle = \int_{-1}^{1} P(x)Q(x)\,dx$$ and the associated norm $\|P\|_{2} = \sqrt{\langle P, P \rangle}$.
For $j \in \mathbb{N}$, we define the polynomial $$P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$$ By convention, $P_{0} = 1$.
(a) What is the degree of $P_{j}$?
(b) Show that $P_{j}$ is an even or odd polynomial, depending on the value of $j$.
(c) Show that $P_{j}(1) = 1$ and $P_{j}(-1) = (-1)^{j}$.

We equip $\mathbb{R}_{n}[X]$ with the inner product defined by $$\langle P, Q \rangle = \int_{-1}^{1} P(x)Q(x)\,dx$$ and the associated norm $\|P\|_{2} = \sqrt{\langle P, P \rangle}$.
For $j \in \mathbb{N}$, we define the polynomial $$P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$$ By convention, $P_{0} = 1$.
By means of integration by parts, show that the family $\left(P_{j}\right)_{0 \leqslant j \leqslant n}$ is orthogonal in $\mathbb{R}_{n}[X]$.

We assume that $k \geqslant 0$ and that $C$ is a matrix in $\Delta_{k+1}$. We set $P = I_n + C$. We consider the endomorphism $\varphi$ of $\mathcal{M}_n(\mathbb{R})$ defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), \varphi: M \mapsto P^{-1}MP$.
Let $i \in \llbracket 0, k \rrbracket$ and $M \in \Delta_i$. Show that there exists $M'$ in $H_{k+1}$ such that $\varphi(M) = M + M'$.

We assume that $k \geqslant 0$ and that $C$ is a matrix in $\Delta_{k+1}$. We set $P = I_n + C$. We consider the endomorphism $\varphi$ of $\mathcal{M}_n(\mathbb{R})$ defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), \varphi: M \mapsto P^{-1}MP$. The matrix $N$ being the matrix defined in III.A.4, show that there exists $N'$ in $H_{k+1}$ such that $$\varphi(N) = N + NC - CN + N'$$

We assume that $k \geqslant 0$ and that $C$ is a matrix in $\Delta_{k+1}$. We set $P = I_n + C$. We consider the endomorphism $\varphi$ of $\mathcal{M}_n(\mathbb{R})$ defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), \varphi: M \mapsto P^{-1}MP$. Let $T$ be an upper triangular matrix. We set $A = N + T$, $B = \varphi(A)$. Show that $B \in H_{-1}$ and that $$\begin{cases} \forall i \in \llbracket -1, k-1 \rrbracket, \quad B^{(i)} = A^{(i)} \\ B^{(k)} = A^{(k)} + NC - CN \end{cases}$$

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$). Let $f_1$ and $f_2$ be two functions continuous on $\bar{U}$, of class $\mathcal{C}^2$ and harmonic on $U$. Show that if the functions $f_1$ and $f_2$ are equal on $\partial U$, then $f_1$ and $f_2$ are equal on $U$.

Let $n$ be a non-zero natural number and $\Phi_n : \{0,1\}^n \rightarrow \llbracket 0, 2^n - 1 \rrbracket$, $(x_j)_{j \in \llbracket 1,n \rrbracket} \mapsto \sum_{j=1}^{n} x_j 2^{n-j}$.
Specify $\operatorname{Im} \Phi_n$ as a function of $A_n$, where $A_n = \left\{\sum_{j=1}^{n} x_j 2^{n-j}, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n\right\}$.

We are given $x _ { 0 } \in \mathbb { R } ^ { N }$. We consider the finite real sequences $\left( \alpha _ { k } \right)$ and $\left( \beta _ { k } \right)$, as well as the finite sequences $\left( \tilde { x } _ { k } \right) , \left( \tilde { r } _ { k } \right)$ and $\left( \tilde { p } _ { k } \right)$ of elements of $\mathbb { R } ^ { N }$, constructed according to the following recurrence relations, for $k \in \{ 0 , \ldots , m - 1 \}$, $$\begin{aligned} \alpha _ { k } & = \frac { \left\| \tilde { r } _ { k } \right\| ^ { 2 } } { \left\langle A \tilde { p } _ { k } , \tilde { p } _ { k } \right\rangle } \\ \tilde { x } _ { k + 1 } & = \tilde { x } _ { k } + \alpha _ { k } \tilde { p } _ { k } \\ \tilde { r } _ { k + 1 } & = \tilde { r } _ { k } - \alpha _ { k } A \tilde { p } _ { k } \\ \beta _ { k } & = \frac { \left\| \tilde { r } _ { k + 1 } \right\| ^ { 2 } } { \left\| \tilde { r } _ { k } \right\| ^ { 2 } } \\ \tilde { p } _ { k + 1 } & = \tilde { r } _ { k + 1 } + \beta _ { k } \tilde { p } _ { k } \end{aligned}$$ with $\tilde { x } _ { 0 } = x _ { 0 } , \tilde { r } _ { 0 } = b - A x _ { 0 }$ and $\tilde { p } _ { 0 } = \tilde { r } _ { 0 }$.
Show that the following properties are satisfied:
(i) For all $k \in \{ 0 , \ldots , m - 1 \}$, for all $i \in \{ 0 , \ldots , k - 1 \}$, we have $$\left\langle \tilde { r } _ { i } , \tilde { r } _ { k } \right\rangle = 0 , \left\langle \tilde { p } _ { i } , \tilde { r } _ { k } \right\rangle = 0 , \left\langle \tilde { p } _ { i } , A \tilde { p } _ { k } \right\rangle = 0$$
(ii) For all $k \in \{ 0 , \ldots , m \} , \tilde { x } _ { k }$ is identified with $x _ { k }$, the minimizer of $J$ on $x _ { 0 } + H _ { k }$ defined in question 13.
(iii) For all $k \in \{ 0 , \ldots , m \} , \tilde { r } _ { k }$ is identified with $r _ { k } = b - A x _ { k }$.
(iv) The family $\left( \tilde { p } _ { 0 } , \ldots , \tilde { p } _ { k } \right)$ is a basis of $H _ { k + 1 }$, for all $k \in \{ 0 , \ldots , m - 1 \}$.

Show that, for all $n \in \mathbb{N}$, $\left\|T_n'\right\|_{L^\infty([-1,1])} = n^2$.
One may begin by establishing that, for all $n \in \mathbb{N}$ and $\theta \in \mathbb{R}$, $|\sin(n\theta)| \leqslant n|\sin\theta|$.
The sequence of polynomials $\left(T_n\right)_{n \in \mathbb{N}}$ is defined by $T_0 = 1, T_1 = X$ and $\forall n \in \mathbb{N}, T_{n+2} = 2X T_{n+1} - T_n$.

Let $n$ be a non-zero natural number. Let $A \in \mathbb{C}_{2n}[X]$, split with simple roots, and $(\alpha_1, \ldots, \alpha_{2n})$ its roots. Show that $$\forall B \in \mathbb{C}_{2n-1}[X], \quad B(X) = \sum_{k=1}^{2n} B(\alpha_k) \frac{A(X)}{(X - \alpha_k) A'(\alpha_k)}$$

Justify that, for every natural integer $k$, $p _ { 1 } ^ { ( k ) } + \cdots + p _ { n } ^ { ( k ) } = 1$.

For each complex number $w$, we denote by $\operatorname{Re}(w)$ the real part of $w$. Show that, for all $z \in \stackrel{\circ}{\mathbb{D}}$: $$\ln|1-z| = -\operatorname{Re}\left(\sum_{n=1}^{\infty} \frac{z^n}{n}\right)$$ To do this, one may write $z = re^{i\theta}$ with $0 \leq r < 1$ and $\theta \in \mathbb{R}$, then study the function: $$\begin{aligned} F : [0,1[ &\rightarrow \mathbb{R} \\ \rho &\mapsto \ln\left|1 - \rho e^{i\theta}\right| \end{aligned}$$

We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$ and denote by $x_1 \leq \ldots \leq x_{n+m}$ the roots of $P$ counted with multiplicity. Show that: $$Q = \prod_{k=m+1}^{n+m}\left(X - x_k\right) \quad \text{and} \quad R = \prod_{k=1}^{m}\left(X - x_k\right)$$

We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$. By contradiction, show that $|P(-1)| = \|P\|_I$.
To do this, one may choose a real number $\epsilon > 0$, introduce the segment $I_\epsilon = [-1-\epsilon, 1]$ and bound the quantity: $$\frac{\|Q\|_{I_\epsilon} \|R\|_{I_\epsilon}}{\|P\|_{I_\epsilon}}$$ using question 4.37.

We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$. Show that $P$ satisfies the differential equation: $$\|P\|_I^2 - P^2 = \frac{1}{(n+m)^2}\left(1 - X^2\right)P'^2$$

We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$. By considering the function: $$\begin{aligned} f : \mathbb{R} &\rightarrow \mathbb{R} \\ y &\mapsto P(\cos y), \end{aligned}$$ verify that for all $x \in [-1,1]$, $$P(x) = \|P\|_I \cos\left((n+m)\operatorname{Arccos} x\right).$$

Let $H$ be the matrix of the inner product $\phi(P,Q) = \int_0^1 P(t)Q(t)\,\mathrm{d}t$ in the canonical basis of $\mathbb{R}_{n-1}[X]$, with general term $h_{i,j} = \phi(X^i, X^j)$. Let $U \in \mathcal{M}_{n,1}(\mathbb{R})$. Express the product $U^\top H U$ using $\phi$ and the coefficients of $U$.

Let $n \in \mathbf { N }$. Show that for all real $t > 0$,
$$p _ { n } = \frac { e ^ { n t } } { 2 \pi } \int _ { - \pi } ^ { \pi } e ^ { - i n \theta } P \left( e ^ { - t + i \theta } \right) \mathrm { d } \theta$$
so that
$$p _ { n } = \frac { e ^ { n t } P \left( e ^ { - t } \right) } { 2 \pi } \int _ { - \pi } ^ { \pi } e ^ { - i n \theta } \frac { P \left( e ^ { - t + i \theta } \right) } { P \left( e ^ { - t } \right) } \mathrm { d } \theta$$

The function $q$ associates to any real $x$ the real number $q ( x ) = x - \lfloor x \rfloor - \frac { 1 } { 2 }$.
Show, for all real $t > 0$, the identity
$$\int _ { 1 } ^ { + \infty } \frac { t q ( u ) } { e ^ { t u } - 1 } \mathrm {~d} u = - \frac { 1 } { 2 } \ln \left( 1 - e ^ { - t } \right) - \ln P \left( e ^ { - t } \right) - \int _ { 1 } ^ { + \infty } \ln \left( 1 - e ^ { - t u } \right) \mathrm { d } u$$

Show, for all real $t > 0$, the identity $$\int_{1}^{+\infty} \frac{tq(u)}{e^{tu}-1} \mathrm{~d}u = -\frac{1}{2}\ln(1-e^{-t}) - \ln P(e^{-t}) - \int_{1}^{+\infty} \ln(1-e^{-tu}) \mathrm{d}u$$

Let $A$ and $B$ be two polynomials in $\mathbf{R}[X]$ whose coefficients are all strictly positive. Prove that the coefficients of the product $AB$ are also strictly positive.

Let $V$ and $V^{\prime}$ be two subspaces of $E$ of dimension $p$. Let $u = (u_1, \ldots, u_p)$ and $u^{\prime} = (u_1^{\prime}, \ldots, u_p^{\prime})$ be the orthonormal families constructed in question (1).
(a) Show that there exist $0 \leqslant \theta_1 \leqslant \cdots \leqslant \theta_p \leqslant \pi/2$ such that $\cos(\theta_k) = \langle u_k, u_k^{\prime}\rangle$ for all $k \in \llbracket 1, p \rrbracket$.
(b) Calculate the value of $\operatorname{det}(\operatorname{Gram}(u, u^{\prime}))$ as a function of the $\cos(\theta_k)$.
(c) Deduce that $\operatorname{det}(\operatorname{Gram}(u, u^{\prime})) \leqslant 1$. What can be said about $V$ and $V^{\prime}$ in the case of equality?

Let $e = (e_1, \ldots, e_d)$ be an orthonormal basis of $E$, $p$ an integer such that $1 \leqslant p \leqslant d$ and $\mathcal{I}_p = \{\alpha = (i_1, \ldots, i_p) \in \mathbb{N}^p \mid 1 \leqslant i_1 < \cdots < i_p \leqslant d\}$. For all $\alpha = (i_1, \ldots, i_p) \in \mathcal{I}_p$, we denote $e_{\alpha} = (e_{i_1}, \ldots, e_{i_p}) \in E^p$ and for all $\omega$ and $\omega^{\prime}$ elements of $\mathscr{A}_p(E, \mathbb{R})$ $$\langle\omega, \omega^{\prime}\rangle = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \omega^{\prime}(e_{\alpha}).$$
We consider $u, v \in E^p$. Show that $$\Omega_p(u)(v) = \langle\Omega_p(u), \Omega_p(v)\rangle.$$