In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$.
Show that $G_0'$ establishes a continuous bijection from $[u_0; +\infty[$ to $\mathbb{R}_+$. Deduce that the function $u : h \longmapsto u_h$ is continuous on $\mathbb{R}_+$ and differentiable on $\mathbb{R}_+^*$.

In this subsection, we still assume that $J_n = J_n^{(\mathrm{C})}$. Moreover, we assume that $\beta = 1$ and $h = 0$.
We are given $x \in \mathbb{R}$ and $\varepsilon > 0$. Let $k$ be a non-zero natural integer such that $k \geqslant \frac{2}{\varepsilon Z_\infty}$. We define the function $$f_k : u \in \mathbb{R} \longmapsto \begin{cases} 1 & \text{if } u \leqslant x \\ 1 - k(u-x) & \text{if } x < u \leqslant x + \frac{1}{k} \\ 0 & \text{otherwise} \end{cases}$$
Show that $f_k$ is $k$-Lipschitz on $\mathbb{R}$.

Let $\mathcal{F}_n$ be the $\mathbb{R}$-vector space of functions $f : \mathbb{R}^n \rightarrow \mathbb{R}$. For all $X \subset \mathbb{R}^n$, we denote $\mathbb{1}_X$ the indicator function of $X$. Let $\mathcal{U}_n$ be the vector subspace of $\mathcal{F}_n$ generated by the functions $\mathbb{1}_P$ where $P$ is a polytope of $\mathbb{R}^n$.
Let $f \in \mathcal{U}_1$. Prove that for all $x \in \mathbb{R}$ the limit of $f(y)$ as $y$ tends to $x$ while satisfying $y > x$, denoted $\lim_{y \rightarrow x^+} f(y)$, exists and that there exist finitely many reals $x \in \mathbb{R}$ such that $f(x) \neq \lim_{y \rightarrow x^+} f(y)$.

Let $\mathcal{F}_n$ be the $\mathbb{R}$-vector space of functions $f : \mathbb{R}^n \rightarrow \mathbb{R}$. For all $X \subset \mathbb{R}^n$, we denote $\mathbb{1}_X$ the indicator function of $X$. Let $\mathcal{U}_n$ be the vector subspace of $\mathcal{F}_n$ generated by the functions $\mathbb{1}_P$ where $P$ is a polytope of $\mathbb{R}^n$.
Prove that the following definition allows us to define a linear form $\chi_n : \mathcal{U}_n \rightarrow \mathbb{R}$. We define $\chi_1(f)$ by the sum $\chi_1(f) = \sum_{x \in \mathbb{R}} \left(f(x) - \lim_{y \rightarrow x^+} f(y)\right)$, then for $f \in \mathcal{U}_n$ with $n > 1$, we set $$\chi_n(f) = \chi_1(g) \text{ with } g \text{ defined by } g(z) = \chi_{n-1}(f_z) \text{ for } z \in \mathbb{R}.$$ We will show at the same time the formula $\chi_n(\mathbb{1}_P) = 1$ for every polytope $P$ of $\mathbb{R}^n$ and we will justify that $\chi_n$ is independent of the coordinate system, namely that for every invertible linear map $A : \mathbb{R}^n \rightarrow \mathbb{R}^n$ we have $\chi_n(f \circ A) = \chi_n(f)$ for all $f \in \mathcal{U}_n$.

Let $n \geq 1$ be an integer. We denote by $\mathbb{C}[[\mathbb{Z}^n]]$ the $\mathbb{C}$-vector space of functions $f : \mathbb{Z}^n \rightarrow \mathbb{C}$, $\mathcal{R}$ the $\mathbb{C}$-vector space of rational elements, $\mathcal{T}$ the $\mathbb{C}$-vector space of torsion elements, and $\mathbb{C}(\mathbb{Z}^n)$ the field of fractions of $\mathbb{C}[\mathbb{Z}^n]$.
We define a $\mathbb{C}$-linear map $\mathrm{I} : \mathcal{R} \rightarrow \mathbb{C}(\mathbb{Z}^n)$ as follows. If $f \in \mathcal{R}$ satisfies $Qf = P$ with $P, Q \in \mathbb{C}[\mathbb{Z}^n]$, we set $\mathrm{I}(f) = \frac{P}{Q}$. Show that $\mathrm{I}$ is well defined, and that it is a linear map with kernel $\mathcal{T}$ satisfying $\mathrm{I}(Pf) = P\,\mathrm{I}(f)$ for all $f \in \mathcal{R}$ and $P \in \mathbb{C}[\mathbb{Z}^n]$.

We consider a convex function $f \in \mathcal{C}(\mathbb{R})$, admitting a minimizer $x_* \in \mathbb{R}$, and $\tau > 0$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $x_{n+1} := p_f(x_n)$. Show that $\left|p_f(x) - p_f(y)\right| \leq |x-y|$ for all $x, y \in \mathbb{R}$. Deduce that the sequence $\left(\left|x_n - x_*\right|\right)_{n \in \mathbb{N}}$ is decreasing.

19. Show that the polynomials $T _ { n }$ are functions of positive type in dimension 2 .
Hint: you may use the exponential form of the cosine. We shall admit, in the rest of the problem, that for every integer $n \geqslant 0$ and every integer $N \geqslant 4$, the polynomial $a _ { n } ^ { \left( \frac { N } { 2 } - 1 \right) }$ is of positive type in dimension $N$. For an integer $N \geqslant 2$, we say that a polynomial $P \in \mathbb { R } [ X ]$ is $N$-conductive if, for every absolutely monotone function $f$ from $[ - 1,1 ]$ to $\mathbb { R }$, the polynomial $H ( f , P )$ is a function of positive type in dimension $N$.