Let $\mathcal{A}$ be an open bounded non-empty subset of $\mathbb{R}^{2}$. We denote by $C(\mathcal{A})$ the set of continuous functions $f$ from $\mathbb{R}^{2}$ to $[0,1]$ such that $\forall (x,y) \in \mathbb{R}^{2} \setminus \mathcal{A},\, f(x,y) = 0$ (in other words $f$ is zero outside $\mathcal{A}$). Show that the supremum $$\sup_{f \in C(\mathcal{A})} \iint_{\mathbb{R}^{2}} f(x,y)\,dx\,dy$$ exists and defines a real number denoted $V(\mathcal{A})$.

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.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$.
Show that the application which associates to $f$ the function $Lf$ is injective.

We denote $\mathcal{I} = \{(j, k) \in \mathbf{N}^{2} \mid j \in \mathbf{N} \text{ and } 0 \leq k < 2^{j}\}$; for $j \in \mathbf{N}$, $\mathcal{T}_{j} = \{k \in \mathbf{N} \mid 0 \leq k < 2^{j}\}$. For all $(j, k) \in \mathcal{I}$, $\theta_{j,k} : [0,1] \rightarrow [0,1]$ is defined by $$\theta_{j,k}(x) = \left\{ \begin{array}{l} 1 - |2^{j+1} x - 2k - 1| \quad \text{if } x \in [k 2^{-j}, (k+1) 2^{-j}] \\ 0 \text{ otherwise} \end{array} \right.$$ Show that for all $(j, k) \in \mathcal{I}$, the function $\theta_{j,k}$ is continuous, affine on each interval of the form $[\ell 2^{-n}, (\ell+1) 2^{-n}]$ where $n > j$ and $\ell \in \mathcal{T}_{n}$.

We denote $\mathcal{I} = \{(j, k) \in \mathbf{N}^{2} \mid j \in \mathbf{N} \text{ and } 0 \leq k < 2^{j}\}$; for $j \in \mathbf{N}$, $\mathcal{T}_{j} = \{k \in \mathbf{N} \mid 0 \leq k < 2^{j}\}$. For all $(j, k) \in \mathcal{I}$, $\theta_{j,k} : [0,1] \rightarrow [0,1]$ is defined by $$\theta_{j,k}(x) = \left\{ \begin{array}{l} 1 - |2^{j+1} x - 2k - 1| \quad \text{if } x \in [k 2^{-j}, (k+1) 2^{-j}] \\ 0 \text{ otherwise} \end{array} \right.$$ Prove that for all $(j, k) \in \mathcal{I}$ and $(x, y) \in [0,1]^{2}$, we have $$|\theta_{j,k}(x) - \theta_{j,k}(y)| \leq 2^{j+1} |x - y|$$

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 $v_i$.

In the rest of the second part, $f$ is an element of $\mathcal{C}_{0}$. For all $n \in \mathbf{N}$, let $S_{n} f$ be the function of $\mathcal{C}_{0}$ defined by $$S_{n} f = \sum_{j=0}^{n} \sum_{k \in \mathcal{T}_{j}} c_{j,k}(f) \theta_{j,k}$$ where, for all $(j, k) \in \mathcal{I}$, $$c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2} .$$ Show that $\lim_{j \rightarrow +\infty} \max_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| = 0$.

We denote $\mathcal{I} = \{(j, k) \in \mathbf{N}^{2} \mid j \in \mathbf{N} \text{ and } 0 \leq k < 2^{j}\}$. For all $(j, k) \in \mathcal{I}$, $$c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2} .$$ For all $(j, k) \in \mathcal{I}$, $(i, \ell) \in \mathcal{I}$, calculate $c_{j,k}(\theta_{i,\ell})$.

We denote $\mathcal{I} = \{(j, k) \in \mathbf{N}^{2} \mid j \in \mathbf{N} \text{ and } 0 \leq k < 2^{j}\}$; for $j \in \mathbf{N}$, $\mathcal{T}_{j} = \{k \in \mathbf{N} \mid 0 \leq k < 2^{j}\}$. Let $a_{j,k}$ be a family of real numbers indexed by $(j, k) \in \mathcal{I}$. We denote $b_{j} = \max_{k \in \mathcal{T}_{j}} |a_{j,k}|$, and we suppose that the series $\sum b_{j}$ is convergent.
For all $j \in \mathbf{N}$, let $f_{j}^{a}$ be the function defined by $$f_{j}^{a}(x) = \sum_{k \in \mathcal{T}_{j}} a_{j,k} \theta_{j,k}(x)$$ Show that the series $\sum f_{j}^{a}$ is uniformly convergent on $[0,1]$ towards a function denoted $f^{a}$, which belongs to $\mathcal{C}_{0}$ and which satisfies, for all $(j, k) \in \mathcal{I}$, $c_{j,k}(f^{a}) = a_{j,k}$.

For all $n \in \mathbf{N}$, let $S_{n} f$ be the function of $\mathcal{C}_{0}$ defined by $$S_{n} f = \sum_{j=0}^{n} \sum_{k \in \mathcal{T}_{j}} c_{j,k}(f) \theta_{j,k}$$ where, for all $(j, k) \in \mathcal{I}$, $$c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2} .$$ Suppose $f$ is of class $\mathcal{C}^{1}$. Show that there exists a constant $M \geq 0$ such that for all $(j, k) \in \mathcal{I}$, $|c_{j,k}(f)| \leq M 2^{-j}$.
Deduce that the sequence of functions $S_{n} f$ is uniformly convergent on $[0,1]$ when $n$ tends to $\infty$.

For all $(j, k) \in \mathcal{I}$, $$c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2} .$$ Suppose $f$ is of class $\mathcal{C}^{2}$. Show that there exists a constant $M' \geq 0$ such that for all $(j, k) \in \mathcal{I}$, $|c_{j,k}(f)| \leq M' 4^{-j}$.

For all $n \in \mathbf{N}$, let $S_{n} f$ be the function of $\mathcal{C}_{0}$ defined by $$S_{n} f = \sum_{j=0}^{n} \sum_{k \in \mathcal{T}_{j}} c_{j,k}(f) \theta_{j,k}$$ Show that for all $n \in \mathbf{N}$ and all $\ell \in \mathcal{T}_{n+1}$, the function $S_{n} f$ is affine on the interval $[\ell 2^{-n-1}, (\ell+1) 2^{-n-1}]$.

For all $n \in \mathbf{N}$, let $S_{n} f$ be the function of $\mathcal{C}_{0}$ defined by $$S_{n} f = \sum_{j=0}^{n} \sum_{k \in \mathcal{T}_{j}} c_{j,k}(f) \theta_{j,k}$$ where, for all $(j, k) \in \mathcal{I}$, $$c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2} .$$ Let $n \in \mathbf{N}$. Suppose that for all $\ell \in \mathcal{T}_{n}$, $(S_{n-1} f)(\ell 2^{-n}) = f(\ell 2^{-n})$. Show that we also have that for all $\ell \in \mathcal{T}_{n+1}$, $(S_{n} f)(\ell 2^{-n-1}) = f(\ell 2^{-n-1})$.
One may distinguish cases according to the parity of $\ell$.

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 $\mathcal{C}_{0}$, $\lim_{n \rightarrow +\infty} \|f - S_{n} f\|_{\infty} = 0$.

Let $n \in \mathbf{N}$. Show that $S_{n}$ is a projector on $\mathcal{C}_{0}$, whose subordinate norm (to $\|\cdot\|_{\infty}$) equals 1.

Let $s \in ]0,1[$. Show that if $a, b \geq 0$, then $a^{s} + b^{s} \leq 2^{1-s}(a+b)^{s}$.

Let $s \in ]0,1[$. Show that if $f \in \Gamma^{s}(x_{0}) \cap \mathcal{C}_{0}$, then there exists a real number $c_{1} > 0$, such that for all $(j, k) \in \mathcal{I}$, we have $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s} .$$ Recall that $c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2}$.

We return to the case where $\mathbb { K }$ is an arbitrary field of characteristic zero. If $V$ and $V ^ { \prime }$ are two $\mathbb { K }$-vector spaces of finite dimension, $q \in \mathcal { Q } ( V )$ and $q ^ { \prime } \in \mathcal { Q } \left( V ^ { \prime } \right)$ are two non-degenerate quadratic forms, the orthogonal sum $q \perp q ^ { \prime }$ of $q$ and $q ^ { \prime }$ is the quadratic form on $V \times V ^ { \prime }$ defined by $$q \perp q ^ { \prime } \left( x , x ^ { \prime } \right) = q ( x ) + q ^ { \prime } \left( x ^ { \prime } \right)$$ for all $x \in V$ and all $x ^ { \prime } \in V ^ { \prime }$.
Let $V , V ^ { \prime }$ and $V ^ { \prime \prime }$ be three $\mathbb { K }$-vector spaces of finite dimension and $\left( q , q ^ { \prime } , q ^ { \prime \prime } \right) \in \mathcal { Q } ( V ) \times \mathcal { Q } \left( V ^ { \prime } \right) \times \mathcal { Q } \left( V ^ { \prime \prime } \right)$.
(a) Show that $q \perp q ^ { \prime } \in \mathcal { Q } \left( V \times V ^ { \prime } \right)$ and then that $\left( q \perp q ^ { \prime } \right) \perp q ^ { \prime \prime } \cong q \perp \left( q ^ { \prime } \perp q ^ { \prime \prime } \right)$.
(b) Show that if $q ^ { \prime } \cong q ^ { \prime \prime }$ then $q \perp q ^ { \prime } \cong q \perp q ^ { \prime \prime }$.
(c) Prove that if $V = V ^ { \prime } \oplus V ^ { \prime \prime }$ and $\tilde { q } ( x , y ) = 0$ for all $x \in V ^ { \prime }$ and all $y \in V ^ { \prime \prime }$, then $q \cong q ^ { \prime } \perp q ^ { \prime \prime }$ where $q ^ { \prime }$ is the restriction of $q$ to $V ^ { \prime }$ and $q ^ { \prime \prime }$ is the restriction of $q$ to $V ^ { \prime \prime }$.

Let $\Omega$ be a non-empty open set of $\mathbb{R}^2$ and $P$ a polynomial of two variables, such that $P(x,y) = 0$ for all $(x,y) \in \Omega$.
a) Show that for all $(x,y) \in \Omega$, the open set $\Omega$ contains a subset of the form $I \times J$, where $I$ and $J$ are non-empty open intervals of $\mathbb{R}$ containing $x$ and $y$ respectively.
The use of a drawing will be appreciated; however, this drawing will not constitute a proof.
b) Deduce that $P$ is the zero polynomial.
One may reduce to studying polynomial functions of one variable.

Does this result hold if the set $\Omega$ has infinitely many elements but is not assumed to be open?

Let $f : C(0,1) \rightarrow \mathbb{R}$ be a continuous application. We define: $$u(x,y) = \begin{cases} \mathrm{N}_f(x,y) & \text{if } (x,y) \in D(0,1) \\ f(x,y) & \text{if } (x,y) \in C(0,1) \end{cases}$$ on $\bar{D}(0,1)$.
Prove that $u$ is an application continuous at every point of $C(0,1)$. What can be concluded about the application $u$?

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 $\bar{D}(0,1)$.