Let $T$ be a trigonometric polynomial of the form $$T(\theta) = a_0 + \sum_{k=1}^{n} \left[ a_k \cos(k\theta) + b_k \sin(k\theta) \right]$$ where $a_0, a_1, b_1, \ldots, a_n, b_n \in \mathbb{R}$.
a) Let $k \in \mathbb{N}^*$. Show that there exists a polynomial function $B_k$ of degree $(k-1)$ such that: $$\forall \theta \in \mathbb{R}, \quad \sin(k\theta) = B_k(\cos(\theta)) \sin(\theta).$$
b) Let $\theta_0 \in \mathbb{R}$. Show that there exists a polynomial function $P \in E_{n-1}$ such that, for all $\theta \in \mathbb{R}$, we have: $$T(\theta_0 + \theta) - T(\theta_0 - \theta) = 2 P(\cos\theta) \sin\theta$$
c) Deduce that: $$\sup_{x \in [-1,1]} |P(x)| \leqslant n \sup_{\theta \in \mathbb{R}} |T(\theta)|.$$
d) Show that: $$\sup_{\theta \in \mathbb{R}} \left| T'(\theta) \right| \leq n \sup_{\theta \in \mathbb{R}} |T(\theta)|.$$

We wish to prove the Cartan-Dieudonné theorem, whose statement is: ``if $f \in O(E,q)$, $f$ is the composition of at most $n$ reflections, where $n = \operatorname{dim}(E)$, with the convention that $\operatorname{Id}_E$ is the composition of 0 reflections.''
We reason by induction, assuming $n > 1$ and that the Cartan-Dieudonné theorem is proved for any vector space of dimension $n-1$.
Conclude when there exists $x \in E$ such that $f(x) = x$ with $q(x) \neq 0$.

We wish to prove the Cartan-Dieudonné theorem, whose statement is: ``if $f \in O(E,q)$, $f$ is the composition of at most $n$ reflections, where $n = \operatorname{dim}(E)$, with the convention that $\operatorname{Id}_E$ is the composition of 0 reflections.''
We reason by induction, assuming $n > 1$ and that the Cartan-Dieudonné theorem is proved for any vector space of dimension $n-1$.
Conclude when there exists $x \in E$ such that $q(x) \neq 0$ and $q(f(x)-x) \neq 0$.

We wish to prove the Cartan-Dieudonné theorem, whose statement is: ``if $f \in O(E,q)$, $f$ is the composition of at most $n$ reflections, where $n = \operatorname{dim}(E)$, with the convention that $\operatorname{Id}_E$ is the composition of 0 reflections.''
We reason by induction, assuming $n > 1$ and that the Cartan-Dieudonné theorem is proved for any vector space of dimension $n-1$.
Conclude in the other cases (i.e., when neither of the conditions in IV.A.2 or IV.A.3 holds).

Let $M$ be a strictly positive real number. We assume that $f$ and $g$ are zero outside the interval $[-M, M]$. Deduce that if $f$ and $g$ are zero outside a bounded interval then the "P-L" inequality $$\int_{-\infty}^{+\infty} h(x)\,dx \geq \left(\int_{-\infty}^{+\infty} f(x)\,dx\right)^{\lambda} \left(\int_{-\infty}^{+\infty} g(x)\,dx\right)^{1-\lambda}$$ is satisfied.

Show that the "P-L" inequality $$\int_{-\infty}^{+\infty} h(x)\,dx \geq \left(\int_{-\infty}^{+\infty} f(x)\,dx\right)^{\lambda} \left(\int_{-\infty}^{+\infty} g(x)\,dx\right)^{1-\lambda}$$ is satisfied (if you choose to use the dominated convergence theorem then carefully verify that its conditions of validity are satisfied).

Let $\lambda \in ]0,1[$ and $f, g, h$ be functions from $\mathbb{R}^{2}$ to $\mathbb{R}_{+}$ that are continuous with bounded support and such that $$\forall X \in \mathbb{R}^{2}, \forall Y \in \mathbb{R}^{2}, \quad h(\lambda X + (1-\lambda) Y) \geq f(X)^{\lambda} g(Y)^{1-\lambda}$$ Show that $$\iint_{\mathbb{R}^{2}} h(x,y)\,dx\,dy \geq \left(\iint_{\mathbb{R}^{2}} f(x,y)\,dx\,dy\right)^{\lambda} \left(\iint_{\mathbb{R}^{2}} g(x,y)\,dx\,dy\right)^{1-\lambda}.$$

We assume that $\frac { \sqrt { \lambda _ { 1 } } } { \sqrt { \lambda _ { 2 } } }$ is not a rational number. We fix an arbitrary element $f \in C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$. For each $k \in \mathbb { N } ^ { * }$ we set: $$\forall ( u , v ) \in \mathbb { R } ^ { 2 } , f _ { k } ( u , v ) = \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } R _ { k } \left( u - \theta _ { 1 } \right) R _ { k } \left( v - \theta _ { 2 } \right) f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 }$$
Let $k \in \mathbb { N } ^ { * }$. Prove that there exist $( 2 k + 1 ) ^ { 2 }$ complex numbers $\left( a _ { j , l } \right) _ { - k \leq j , l \leq k }$ such that for every $( u , v ) \in \mathbb { R } ^ { 2 } : f _ { k } ( u , v ) = \sum _ { - k \leq j , l \leq k } a _ { j , l } e ^ { i u j } e ^ { i v l }$. Justify that the function $f _ { k }$ satisfies the Ergodic Theorem.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$.
Let $g$ be a continuous application from $[0,1]$ to $\mathbb{R}$. We assume that for all $n \in \mathbb{N}$, we have $$\int_0^1 t^n g(t)\,dt = 0.$$
VI.A.1) What can we say about $\displaystyle\int_0^1 P(t)g(t)\,dt$ for $P \in \mathbb{R}[X]$?
VI.A.2) Deduce from this that $g$ is the zero application.

We assume that the conditions of question 4 are satisfied (i.e., $\lambda(i) = \lambda(0) - 2i$ for all $i \in \mathbf{Z}$).
6a. Show that for $f \in V$, the vector space spanned by $H^n(f), n \in \mathbf{N}$, is finite-dimensional.
6b. Deduce that a non-zero subspace of $V$ stable under $H$ contains at least one of the $v_i$.

Deduce that for all $n \in \mathbf{N}$ and all $\ell \in \mathcal{T}_{n+1}$, $(S_{n} f)(\ell 2^{-n-1}) = f(\ell 2^{-n-1})$.

Deduce from question 9 that for all $f$ in $\mathcal{C}_{0}$, $\lim_{n \rightarrow +\infty} \|f - S_{n} f\|_{\infty} = 0$.

We assume that $f$ is the zero application on $C(0,1)$ and that $u$ is an element of $\mathcal{D}_f$. For all $n \in \mathbb{N}$, we define the application $$u_n : \begin{array}{rll} \bar{D}(0,1) & \rightarrow & \mathbb{R} \\ (x,y) & \mapsto & u(x,y) + \dfrac{1}{n}(x^2 + y^2) \end{array}$$
Deduce that, for all $(x,y) \in D(0,1)$, $u_n(x,y) \leqslant 1/n$.

We assume that $f$ is the zero application on $C(0,1)$ and that $u$ is an element of $\mathcal{D}_f$. For all $n \in \mathbb{N}$, we define the application $$u_n : \begin{array}{rll} \bar{D}(0,1) & \rightarrow & \mathbb{R} \\ (x,y) & \mapsto & u(x,y) + \dfrac{1}{n}(x^2 + y^2) \end{array}$$
Show that $u$ is identically zero on $\bar{D}(0,1)$.

Let $m$ be an integer greater than or equal to 2. We consider a polynomial $P \in \mathcal{P}_m$ and we denote by $P_C$ the restriction of $P$ to the circle $C(0,1)$.
Show that the unique element of the set $\mathcal{D}_{P_C}$ is the restriction to $\bar{D}(0,1)$ of a polynomial of degree less than or equal to $m$.

By using the notations of question 3b, deduce that $$\sup_{\mathcal{V} \subset \mathbb{R}^{n},\, \operatorname{dim} \mathcal{V} = j} \inf_{x \in \mathcal{V},\, \|x\|=1} \langle x, Mx \rangle = m_{j}.$$ Is this supremum attained?

Let $\ell$ and $m$ be two $n$-tuples of real numbers. We write $$\ell \preccurlyeq m \quad \text{if and only if, for every integer } j,\, 1 \leqslant j \leqslant n, \quad \ell_{j} \leqslant m_{j}.$$ Let $L, M \in \mathcal{S}_{n}(\mathbb{R})$ such that $(0, \ldots, 0) \preccurlyeq s^{\downarrow}(M - L)$. Show that $s^{\downarrow}(L) \preccurlyeq s^{\downarrow}(M)$.

Conclude that the function $s^{\downarrow} : \mathcal{S}_{n}(\mathbb{R}) \rightarrow \mathbb{R}^{n}$ is continuous.

Let $\hat { \lambda } = \left( \lambda _ { 1 } \geqslant \cdots \geqslant \lambda _ { n + 1 } \right) \in \mathbb { R } ^ { n + 1 }$ and $\widehat { \mu } = \left( \mu _ { 1 } \geqslant \cdots \geqslant \mu _ { n } \right) \in \mathbb { R } ^ { n }$. Let $x \in \mathbb { R }$. Form $$\widehat { \lambda } ^ { \prime } = \left( \lambda _ { 1 } \geqslant \cdots \geqslant \lambda _ { i } \geqslant x > \lambda _ { i + 1 } \geqslant \cdots \geqslant \lambda _ { n + 1 } \right)$$ by choosing the integer $i \in \{ 0 , \ldots , n + 1 \}$ appropriately. If $x > \lambda _ { 1 }$, we thus have $i = 0$, while if $x \leqslant \lambda _ { n + 1 }$, we have $i = n + 1$. Similarly form $$\widehat { \mu } ^ { \prime } = \left( \mu _ { 1 } \geqslant \cdots \geqslant \mu _ { j } \geqslant x > \mu _ { j + 1 } \geqslant \cdots \geqslant \mu _ { n } \right) .$$ Assume that $\widehat { \lambda }$ and $\widehat { \mu }$ are interlaced. Show that $j \leqslant i \leqslant j + 1$. By examining each of the two cases $j = i$ or $i - 1$, show that $\widehat { \lambda } ^ { \prime }$ and $\widehat { \mu } ^ { \prime }$ are interlaced.

Two simplexes $\mathcal{S}$ and $\mathcal{S}'$ in $\mathbb{R}^n$ are called equivalent if there exist an enumeration of the vertices $s_0, s_1, \ldots, s_n$ of $\mathcal{S}$, and $s_0', s_1', \ldots, s_n'$ of $\mathcal{S}'$, and a matrix $A$ in $\mathrm{GL}_n(\mathbb{Z})$ such that $A(s_i - s_0) = s_i' - s_0'$ for all $i = 1, \ldots, n$.
Theorem 2 states: For every strictly positive integer $k$, there exists a strictly positive constant $C(n,k)$ such that for every integer simplex $\mathcal{S}$ in $\mathbb{R}^n$ having exactly $k$ interior integer points, $\operatorname{Vol}(\mathcal{S}) \leqslant C(n,k)$.
Deduce from Theorem 2 that for every strictly positive integer $k$, there exist up to equivalence only finitely many integer simplexes in $\mathbb{R}^n$ having exactly $k$ interior points.

Let $\mathcal{S}$ be an integer simplex of $\mathbb{R}^n$ with vertices $0, s_1, \ldots, s_n$ having exactly $k$ interior integer points and let $x = \sum_{i=1}^n t_i s_i$ be an interior integer point of $\mathcal{S}$.
18a. Show that $\sum_{i=1}^n t_i \leqslant 1 - \alpha(k,n)$. (One may reason by contradiction and construct then $k+1$ distinct integer points interior to $\mathcal{S}$.)
18b. Show that $\frac{\alpha(k,n)}{1-\alpha(k,n)} x \in (\mathcal{S} - x)$.
18c. Deduce that $a(\mathcal{S} - x) \geqslant \frac{\alpha(k,n)}{1-\alpha(k,n)}$.

Conclude the proof of Theorem 2, which states: For every strictly positive integer $k$, there exists a strictly positive constant $C(n,k)$ such that for every integer simplex $\mathcal{S}$ in $\mathbb{R}^n$ having exactly $k$ interior integer points, $\operatorname{Vol}(\mathcal{S}) \leqslant C(n,k)$.

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 $i \in \{1, \ldots, N\}$.

Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$, $q \in \Sigma_{N}$, and for all $\theta \in \mathbb{R}^{d}$, $f(\theta) = M\theta$, $Z(\theta) = \sum_{i=1}^{N} e^{f_{i}(\theta)}$, $$p(\theta) = \left(\frac{e^{f_{1}(\theta)}}{Z(\theta)}, \ldots, \frac{e^{f_{N}(\theta)}}{Z(\theta)}\right) \in \Sigma_{N},$$ and $L(\theta) = \ln(Z(\theta)) - q^{T} M\theta$. We denote $$\Sigma_{N}(\bar{g}, g) = \left\{p \in \Sigma_{N} \mid \sum_{i=1}^{N} p_{i} g_{k}(i) = \bar{g}_{k}, 1 \leqslant k \leqslant d\right\}.$$
Show that if $\theta$ is a critical point of $L$ (that is, a point where the gradient of $L$ vanishes) then $M^{T} p(\theta) = M^{T} q$ and $p(\theta) \in \Sigma_{N}(\bar{g}, g)$.

Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$, $q \in \Sigma_{N}$, and for all $\theta \in \mathbb{R}^{d}$, $f(\theta) = M\theta$, $Z(\theta) = \sum_{i=1}^{N} e^{f_{i}(\theta)}$, $$p(\theta) = \left(\frac{e^{f_{1}(\theta)}}{Z(\theta)}, \ldots, \frac{e^{f_{N}(\theta)}}{Z(\theta)}\right) \in \Sigma_{N},$$ and $L(\theta) = \ln(Z(\theta)) - q^{T} M\theta$. We assume that $\ker \widetilde{M} = \{0\}$ where $\widetilde{M} = (M \mid \mathbf{1})$.
We are interested in this question in the number of points at which the function $L$ attains its minimum.
(a) Show that if $\theta$ and $\theta'$ are two distinct points of $\mathbb{R}^{N}$ such that $L$ has a critical point at $\theta$, then the derivative of $t \rightarrow L(t\theta + (1-t)\theta')$ is strictly increasing on $[0,1]$ and vanishes at $t = 1$.
(b) Deduce that there is at most one critical point for $L$ and conclude on the number of points at which $L$ attains its minimum.