We denote by $\mathcal{E}$ the set of continuous maps $g$ from $[0,1]$ to $\mathbf{C}$ such that $g(0)=-1$ and $g(1)=1$.
Determine the unique element $f_0$ of $\mathcal{E}$ which is affine.

We denote by $\mathcal{E}$ the set of continuous maps $g$ from $[0,1]$ to $\mathbf{C}$ such that $g(0)=-1$ and $g(1)=1$. We denote by $\phi_0$ and $\phi_1$ the maps from $\mathbf{C}$ to $\mathbf{C}$ defined by: $\phi_0(z) = \frac{1+\mathrm{i}}{2}\bar{z} + \frac{-1+\mathrm{i}}{2}$ and $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$. If $g\in\mathcal{E}$, we denote by $Tg$ the map from $[0,1]$ to $\mathbf{C}$ defined by: $$Tg(x) = \phi_0(g(2x)) \text{ if } x\in\left[0,\frac{1}{2}\right] \text{ and } Tg(x) = \phi_1(g(2x-1)) \text{ if } x\in\left]\frac{1}{2},1\right]$$
Show that $Tg\in\mathcal{E}$ for all $g\in\mathcal{E}$.

We denote by $\mathcal{E}$ the set of continuous maps $g$ from $[0,1]$ to $\mathbf{C}$ such that $g(0)=-1$ and $g(1)=1$. The norm of uniform convergence on the $\mathbf{C}$-vector space of continuous maps from $[0,1]$ to $\mathbf{C}$ is denoted $\|\cdot\|_\infty$. We denote by $\phi_0$ and $\phi_1$ the maps from $\mathbf{C}$ to $\mathbf{C}$ defined by: $\phi_0(z) = \frac{1+\mathrm{i}}{2}\bar{z} + \frac{-1+\mathrm{i}}{2}$ and $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$. If $g\in\mathcal{E}$, we denote by $Tg$ the map from $[0,1]$ to $\mathbf{C}$ defined by: $$Tg(x) = \phi_0(g(2x)) \text{ if } x\in\left[0,\frac{1}{2}\right] \text{ and } Tg(x) = \phi_1(g(2x-1)) \text{ if } x\in\left]\frac{1}{2},1\right]$$
Let $g_1$ and $g_2$ be two elements of $\mathcal{E}$. Prove that: $$\|Tg_2 - Tg_1\|_\infty = \frac{1}{\sqrt{2}}\|g_2 - g_1\|_\infty$$

We denote by $\mathcal{E}$ the set of continuous maps $g$ from $[0,1]$ to $\mathbf{C}$ such that $g(0)=-1$ and $g(1)=1$. The norm of uniform convergence on the $\mathbf{C}$-vector space of continuous maps from $[0,1]$ to $\mathbf{C}$ is denoted $\|\cdot\|_\infty$. We denote by $\phi_0$ and $\phi_1$ the maps from $\mathbf{C}$ to $\mathbf{C}$ defined by: $\phi_0(z) = \frac{1+\mathrm{i}}{2}\bar{z} + \frac{-1+\mathrm{i}}{2}$ and $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$. If $g\in\mathcal{E}$, we denote by $Tg$ the map from $[0,1]$ to $\mathbf{C}$ defined by: $$Tg(x) = \phi_0(g(2x)) \text{ if } x\in\left[0,\frac{1}{2}\right] \text{ and } Tg(x) = \phi_1(g(2x-1)) \text{ if } x\in\left]\frac{1}{2},1\right]$$ We now define a sequence $(f_n)_{n\in\mathbf{N}}$ of elements of $\mathcal{E}$ by choosing $f_0$ affine (i.e. $f_0(x) = -1 + 2x$) and $f_{n+1} = Tf_n$ for every natural number $n$.
a) Prove that the sequence $(f_n)$ converges uniformly on $[0,1]$ to a function $f\in\mathcal{E}$.
b) Prove that $Tf = f$.
c) Prove that, for all $x\in[0,1]$, $f(x) = -\overline{f(1-x)}$ and give a geometric interpretation of this relation.

The integer part of the real number $x$ is denoted $[x]$. For every real $x$ and every non-zero natural number $n$: $r_n(x) = [2^n x] - 2[2^{n-1}x]$. We denote by $\mathbf{Z}\left[\frac{1}{2}\right]$ the set of rationals of the form $\frac{k}{2^n}$ where $k\in\mathbf{Z}$ and $n\in\mathbf{N}$. The map $f\in\mathcal{E}$ satisfies $Tf = f$.
Conversely, let $x\in[0,1[$.
a) Establish that, for every non-zero natural number $n$, $r_n(x)\in\{0,1\}$.
b) Show that, for every non-zero natural number $N$ and every real $x\in[0,1[$: $$\frac{[2^N x]}{2^N} = \sum_{n=1}^{N}\frac{r_n(x)}{2^n} \quad \text{then} \quad x = \sum_{n=1}^{\infty}\frac{r_n(x)}{2^n}.$$
c) Show that if, moreover, $x\in\mathbf{Z}\left[\frac{1}{2}\right]$ then there exists $N\in\mathbf{N}$ such that $r_n(x) = 0$ for every natural number $n > N$.
d) Calculate $f\left(\frac{1}{2}\right)$ and $f\left(\frac{1}{4}\right)$. Recognize $\phi_0\circ\phi_0$ and deduce $f\left(\frac{1}{2^k}\right)$ for all $k\in\mathbf{N}$.

For every integer $n \in \mathbb{N}$, we set $F_n(x) = \cos(n \arccos x)$.
Write a function \texttt{chebyshev} that takes as argument an integer $n$ and returns the display of the expression $F_n(x)$. Use the programming language associated with the usual computer algebra software.

We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay. Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$. For every integer $n \in \mathbb{N}$, $F_n(x) = \cos(n \arccos x)$ extended as a polynomial to $\mathbb{R}$. We define the function $f$ on the segment $[-1,1]$ by: $$\forall x \in [-1,1], \quad f(x) = \sum_{n=0}^{+\infty} \alpha_n F_n(x).$$
Show that $f$ is of class $C^\infty$ on $[-1,1]$.

We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay. Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$. For every integer $n \in \mathbb{N}$, $F_n(x) = \cos(n \arccos x)$ extended as a polynomial to $\mathbb{R}$. We define the function $f$ on the segment $[-1,1]$ by: $$\forall x \in [-1,1], \quad f(x) = \sum_{n=0}^{+\infty} \alpha_n F_n(x).$$ For every integer $n \in \mathbb{N}$, $V_n$ denotes the set of restrictions to $[-1,1]$ of polynomial functions of degree at most $n$, and $d(f, V_n) = \inf_{p \in V_n} \|f - p\|_\infty$.
Show that the sequence $(d(f, V_n))_{n \in \mathbb{N}}$ has rapid decay.

For a function $h \in C([-1,1])$, we denote by $\widetilde{h}$ the following $2\pi$-periodic function: $$\tilde{h} : \begin{cases} \mathbb{R} \rightarrow \mathbb{R} \\ \theta \mapsto h(\cos(\theta)) \end{cases}$$ The Fourier coefficients of $\widetilde{h}$ are given by: $$a_0(\widetilde{h}) = \frac{1}{2\pi} \int_{-\pi}^{\pi} \widetilde{h}(t)\, dt, \quad a_n(\widetilde{h}) = \frac{1}{\pi} \int_{-\pi}^{\pi} \widetilde{h}(t) \cos(nt)\, dt, \quad b_n(\widetilde{h}) = \frac{1}{\pi} \int_{-\pi}^{\pi} \widetilde{h}(t) \sin(nt)\, dt.$$
Let $f \in C^\infty([-1,1])$.
Show that the sequence $(a_n(\widetilde{f}))_{n \in \mathbb{N}}$ has rapid decay. What is the value of $b_n(\widetilde{f})$?

For a function $h \in C([-1,1])$, we denote by $\widetilde{h}$ the following $2\pi$-periodic function: $$\tilde{h} : \begin{cases} \mathbb{R} \rightarrow \mathbb{R} \\ \theta \mapsto h(\cos(\theta)) \end{cases}$$
Let $f \in C^\infty([-1,1])$.
Show that the Fourier series of $\widetilde{f}$ converges normally to $\widetilde{f}$.

For a function $h \in C([-1,1])$, we denote by $\widetilde{h}$ the following $2\pi$-periodic function: $$\tilde{h} : \begin{cases} \mathbb{R} \rightarrow \mathbb{R} \\ \theta \mapsto h(\cos(\theta)) \end{cases}$$ For every integer $n \in \mathbb{N}$, $F_n(x) = \cos(n \arccos x)$ extended as a polynomial to $\mathbb{R}$.
Let $f \in C^\infty([-1,1])$.
Show that there exists a sequence $(\alpha_n(f))_{n \in \mathbb{N}}$ with rapid decay such that $$f(x) = \sum_{n=0}^{+\infty} \alpha_n(f) F_n(x)$$ for all $x \in [-1,1]$. Give an expression for $\alpha_n(f)$ in terms of $f$ and $n$.

We denote by $C([-1,1])$ the vector space of continuous functions on $[-1,1]$ with real values, equipped with the infinite norm $\|f\|_\infty = \sup_{x \in [-1,1]} |f(x)|$. For every integer $n \in \mathbb{N}$, $V_n$ denotes the set of restrictions to $[-1,1]$ of polynomial functions of degree at most $n$, and $d(f, V_n) = \inf_{p \in V_n} \|f - p\|_\infty$.
Let $f \in C([-1,1])$. We assume that the sequence $(d(f, V_n))_{n \in \mathbb{N}}$ has rapid decay.
Show that we can construct a sequence $(p_n)_{n \in \mathbb{N}}$ of polynomial functions such that:

• for every integer $n$, $\deg(p_n) \leqslant n$;
• $(\|f - p_n\|_\infty)_{n \in \mathbb{N}}$ has rapid decay.

We denote by $C([-1,1])$ the vector space of continuous functions on $[-1,1]$ with real values, equipped with the infinite norm $\|f\|_\infty = \sup_{x \in [-1,1]} |f(x)|$. For every integer $n \in \mathbb{N}$, $V_n$ denotes the set of restrictions to $[-1,1]$ of polynomial functions of degree at most $n$, and $d(f, V_n) = \inf_{p \in V_n} \|f - p\|_\infty$. For a function $h \in C([-1,1])$, $\widetilde{h}$ denotes the $2\pi$-periodic function $\theta \mapsto h(\cos(\theta))$.
Let $f \in C([-1,1])$. We assume that the sequence $(d(f, V_n))_{n \in \mathbb{N}}$ has rapid decay.
The purpose of this question is to show that the function $f$ is of class $C^\infty$.
a) Let $k \in \mathbb{N}^*$. Show that, for $P \in E_{k-1}$, $a_k(\widetilde{f}) = a_k(\widetilde{f-P})$.
b) Deduce that the sequence $(a_n(\widetilde{f}))_{n \in \mathbb{N}}$ of Fourier coefficients of the function $\widetilde{f}$ has rapid decay.
c) Conclude.

We denote $\mathscr{C}$ the image in $\mathbb{R}^2$ of the application $$\gamma : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}^2, t \mapsto (\cos t, 2\sin t)$$
III.A.1) Represent $\mathscr{C}$. III.A.2) Specify the following topological properties of $\mathscr{C}$. a) Is it an open set of $\mathbb{R}^2$? b) A closed set? c) A bounded set? d) A compact? e) A convex set?

We recall that $\mathscr{C}$ was defined as the image of the application $$\gamma : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}^2, t \mapsto (\cos t, 2\sin t)$$
In this question, we seek a complex parametrization of $\mathscr{C}$, of the form $$z : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{C}, t \mapsto \rho(t) \mathrm{e}^{\mathrm{i}\theta(t)}$$ where $\rho$ and $\theta$ are two continuous functions from $\left[\frac{\pi}{4}, \frac{7\pi}{4}\right]$ to $\mathbb{R}$, the function $\rho$ taking strictly positive values.
III.B.1) Calculate $\rho(t)$ for all $t \in \left[\frac{\pi}{4}, \frac{7\pi}{4}\right]$. III.B.2) Represent on the calculator the parametrized arc $$\mathscr{G} : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{C}, t \mapsto \rho(t) \mathrm{e}^{\mathrm{i}t}$$ and reproduce the curve roughly on the paper. What letter does this curve evoke? III.B.3) From the expression of $\gamma(t)$, calculate $\tan\theta(t)$. III.B.4) a) Represent the function $t \mapsto \arctan(2\tan t)$ on the part of the interval $\left[\frac{\pi}{4}, \frac{7\pi}{4}\right]$ on which this function is defined. b) Modify this function to determine the continuous function $\theta$ sought. The result will be verified by representing with the aid of the calculator the parametrized curve $z$. III.B.5) Indicate a sequence of Maple or Mathematica instructions allowing one to obtain this plot.

We define the applications: $$\alpha : \mathbb{N}^* \times \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}, (n, t) \mapsto \frac{\pi}{4} + \frac{3\pi}{2n} \mathrm{E}\left(\frac{2n}{3\pi}\left(t - \frac{\pi}{4}\right)\right)$$ $$\omega : \mathbb{N}^* \times \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}, (n, t) \mapsto \cos^2\left(\frac{2n}{3}\left(t - \frac{\pi}{4}\right)\right)$$ where $\mathrm{E}(x)$ denotes the integer part of the real number $x$.
III.C.1) Briefly study $\alpha$ and $\omega$, then represent on the same graph the two functions $t \mapsto \alpha(10, t)$ and $t \mapsto \omega(10, t)$. III.C.2) Represent the function $\psi : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}, t \mapsto \frac{1}{4}\sin\left(\frac{2}{3}\left(t - \frac{\pi}{4}\right)\right)$. III.C.3) We define the function: $$w : \mathbb{N}^* \times \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{C}, (n, t) \mapsto \rho(t)\left(1 + \psi(t)\omega(n, t)\right) \mathrm{e}^{\mathrm{i}\theta(\alpha(n, t))}$$ Identify which of the four graphics represents the function $t \mapsto w(40, t)$, and explain why. III.C.4) Write a sequence of Maple or Mathematica instructions allowing one to create the sequence of the first 100 curves (one may create an animation).

We propose to calculate the area $\mathscr{A}$ of the domain $\mathscr{H}$ of $\mathbb{R}^2$ containing all the points $w(n, t)$ when $n$ ranges over $\mathbb{N}^*$ and $t$ ranges over $I = \left[\frac{\pi}{4}, \frac{7\pi}{4}\right]$. This domain is bounded by two parametrized arcs defined by $$z : I \rightarrow \mathbb{C}, t \mapsto \rho(t) \mathrm{e}^{\mathrm{i}\theta(t)} = \sqrt{1 + 3\sin^2 t}\, \mathrm{e}^{\mathrm{i}\left(\arctan(2\tan t) + \pi \mathrm{E}\left(\frac{t}{\pi} + \frac{1}{2}\right)\right)}$$ $$v : I \rightarrow \mathbb{C}, t \mapsto \sqrt{1 + 3\sin^2 t}\left(1 + \frac{1}{4}\sin\left(\frac{2}{3}\left(t - \frac{\pi}{4}\right)\right)\right) \mathrm{e}^{\mathrm{i}\left(\arctan(2\tan t) + \pi \mathrm{E}\left(\frac{t}{\pi} + \frac{1}{2}\right)\right)}$$
III.D.1) Recall the statement of the Green-Riemann theorem. Explain how this theorem translates in the case of an area calculation. III.D.2) Recall the formula giving the scalar product of two complex numbers. Deduce the expression of the scalar product $\langle u \circ v(t), v'(t) \rangle$, when $u$ and $v$ are the applications $u : \mathbb{C} \rightarrow \mathbb{C}, z \mapsto \mathrm{i}z$ and $v : t \mapsto \sigma(t) \mathrm{e}^{\mathrm{i}\mu(t)}$, where $\sigma$ and $\mu$ are two functions defined on an interval $J$ of $\mathbb{R}$, with real values and of class $C^1$. III.D.3) If $d(t) = \arctan(2\tan(t))$, simplify $\frac{1}{2}\left(1 + 3\sin^2 t\right) d'(t)$. III.D.4) Deduce from the previous questions an expression of $\mathscr{A}$ in the form of an integral. Simplify this integral using the identity obtained in III.D.3). Finally, calculate $\mathscr{A}$.

For every element $x$ of $E$, we denote by $h(x)$ the application from $E$ to $E$ such that $\forall y \in E, h(x)(y) = \varphi(x,y)$.
a) Show that, for all $x$ in $E$, $h(x)$ is an element of the dual of $E$, denoted $E^{*}$.
b) Show that $h$ is a linear application from $E$ to $E^{*}$.

If $A$ is a subset of $E$, we denote $A^{\perp\varphi} = \{ x \in E \mid \forall a \in A,\ \varphi(x,a) = 0 \}$. Show that $A^{\perp\varphi}$ is a vector subspace of $E$.

We say that $\varphi$ is non-degenerate if and only if $E^{\perp\varphi} = \{0\}$.
Show that $\varphi$ is non-degenerate if and only if $h$ is an isomorphism.

Let $e = (e_1, \ldots, e_n)$ be a basis of $E$. We denote by $e^* = (e_1^*, \ldots, e_n^*)$ the dual basis of $e$.
a) Show that the matrix of $h$ in the bases $e$ and $e^*$ is: $$\operatorname{mat}(h, e, e^*) = \left(\varphi(e_i, e_j)\right)_{\substack{1 \leq i \leq n \\ 1 \leq j \leq n}}$$ This latter matrix will also be called the matrix of $\varphi$ in the basis $e$ and denoted $\operatorname{mat}(\varphi, e)$.
b) Let $(x,y) \in E^2$. We denote by $X$ and $Y$ the column matrices whose coefficients are the components of $x$ and $y$ in the basis $e$. Show that $\varphi(x,y) = {}^t X \Omega Y$ where $\Omega = \operatorname{mat}(\varphi, e)$ and where ${}^t X$ denotes the row matrix obtained by transposing $X$.

Let $q \in Q(E)$.
Show that there exists a unique symmetric bilinear form on $E$, denoted $\varphi$, such that $q = q_\varphi$.

Let $q$ be a quadratic form on $E$. Let $E'$ be a second $\mathbb{K}$-vector space of dimension $n$, and let $q'$ be a quadratic form on $E'$.
We call an isometry from $(E,q)$ to $(E',q')$ any isomorphism $f$ from $E$ to $E'$ satisfying: for all $x \in E$, $q'(f(x)) = q(x)$. We will say that $(E,q)$ and $(E',q')$ are isometric if and only if there exists an isometry from $(E,q)$ to $(E',q')$.
Show that $(E,q)$ and $(E',q')$ are isometric if and only if there exists a basis $e$ of $E$ and a basis $e'$ of $E'$ such that $\operatorname{mat}(q,e) = \operatorname{mat}(q',e')$.

Let $p \in \mathbb{N}^*$. We denote by $c = (c_1, \ldots, c_{2p})$ the canonical basis of $\mathbb{K}^{2p}$. $$\text{For all } x = \sum_{i=1}^{2p} x_i c_i \in \mathbb{K}^{2p}, \text{ we set } q_p(x) = 2\sum_{i=1}^{p} x_i x_{i+p}.$$
a) Show that $q_p$ is a quadratic form on $\mathbb{K}^{2p}$ and compute $\operatorname{mat}(q_p, c)$.
b) We call an Artin space (or artinian space) of dimension $2p$ any pair $(F,q)$, where $F$ is a $\mathbb{K}$-vector space of dimension $2p$, and where $q$ is a quadratic form on $F$ such that $(F,q)$ and $(\mathbb{K}^{2p}, q_p)$ are isometric. Show that in this case, $q$ is non-degenerate. When $p=1$, we say that $(F,q)$ is an artinian plane.
c) We assume that $\mathbb{K} = \mathbb{C}$ and for all $$x = \sum_{k=1}^{2p} x_k c_k \in \mathbb{C}^{2p}, \text{ we set } q(x) = \sum_{k=1}^{2p} x_k^2.$$ Show that $(\mathbb{C}^{2p}, q)$ is an Artin space.
d) We assume that $\mathbb{K} = \mathbb{R}$ and for all $$x = \sum_{i=1}^{2p} x_i c_i \in \mathbb{R}^{2p}, \text{ we set } q'(x) = \sum_{i=1}^{p} x_i^2 - \sum_{i=p+1}^{2p} x_i^2.$$ Show that $(\mathbb{R}^{2p}, q')$ is an Artin space.
e) If $(F,q)$ is an Artin space of dimension $2p$, show that there exists a vector subspace $G$ of $F$ of dimension $p$ such that the restriction of $q$ to $G$ is identically zero.

For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Let $e = (e_1, \ldots, e_n)$ be a basis of $E$. We still denote by $e^* = (e_1^*, \ldots, e_n^*)$ the dual basis of $e$. Let $p \in \{1, \ldots, n\}$. We denote by $F$ the space spanned by $e_1, \ldots, e_p$.
a) Show that $F^\perp$ is the preimage under $h$ of $\operatorname{Vect}(e_{p+1}^*, \ldots, e_n^*)$, where $h$ is defined in I.A.1.
b) Show that $\operatorname{dim}(F) + \operatorname{dim}(F^\perp) = n$.
c) Show that $(F^\perp)^\perp = F$.