Prove that, for any continuous application $f : C(0,1) \rightarrow \mathbb{R}$, the set $\mathcal{D}_f$ admits exactly one element.

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 application $$\phi_{m-2} : \left|\begin{array}{rll} \mathcal{P}_{m-2} & \rightarrow & \mathcal{P} \\ Q & \mapsto & \Delta \tilde{Q} \end{array}\right. \quad \text{where} \quad \tilde{Q}(x,y) = (1 - x^2 - y^2) Q(x,y)$$ is linear and injective and that $\operatorname{Im} \phi_{m-2} \subset \mathcal{P}_{m-2}$.

Let $m$ be an integer greater than or equal to 2. We consider a polynomial $P \in \mathcal{P}_m$.
Deduce that there exists a polynomial $T \in \mathcal{P}_{m-2}$ such that $P + (1 - x^2 - y^2) T$ is a harmonic polynomial.

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 $m$.

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 $C(0,1)$.
Explicitly determine the set $\mathcal{D}_{P_C}$ when the polynomial $P$ is defined by $P(x,y) = x^3$.

Let $P \in \mathcal{P}$. Show that $P$ decomposes uniquely in the form: $$P(x,y) = H(x,y) + (1 - x^2 - y^2) Q(x,y)$$ where $H$ is a harmonic polynomial and $Q \in \mathcal{P}$.

Let $m \in \mathbb{N}$. We denote by $\mathcal{H}_m$ the vector subspace of harmonic polynomials of degree less than or equal to $m$. Determine the dimension of $\mathcal{H}_m$.

Explicitly determine a basis of $\mathcal{H}_3$.

We work on $\mathbb{R}^n$ for a natural integer $n \geqslant 3$. We admit that the Dirichlet problem on the unit ball of $\mathbb{R}^n$, associated with a continuous function defined on the unit sphere $S_n(0,1)$, admits a unique solution. Let $m \in \mathbb{N}^*$.
Determine the dimension of $\mathcal{H}_m$ as a function of $m$ and $n$.

Let $\mathcal{U}$ and $\mathcal{V}$ be two vector subspaces of $\mathbb{R}^{n}$ such that $$\operatorname{dim} \mathcal{U} + \operatorname{dim} \mathcal{V} > n.$$ Show that $\mathcal{U} \cap \mathcal{V}$ is not reduced to $\{0\}$.

Let $M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$. Let $j$ be an integer, $1 \leqslant j \leqslant n$, and $\mathcal{V}$ be a vector subspace of $\mathbb{R}^{n}$ of dimension $j$. Show that $$\inf_{x \in \mathcal{V},\, \|x\|=1} \langle x, Mx \rangle \leqslant m_{j}.$$ (One may use questions $\mathbf{2c}$ and $\mathbf{3a}$, by choosing $\mathcal{U} = \mathcal{W}_{j}$.)

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 $\hat { \lambda } = \left( \lambda _ { 1 } \geqslant \cdots \geqslant \lambda _ { n + 1 } \right) \in \mathbb { R } ^ { n + 1 }$ 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.

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 $s^{\downarrow}(L) \preccurlyeq s^{\downarrow}(M)$.

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

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 $L, M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$ and $\ell = s^{\downarrow}(L)$. Show that $$\max_{1 \leqslant j \leqslant n} \left|\ell_{j} - m_{j}\right| \leqslant \|L - M\|.$$

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

We denote by $\mathcal{S}_{n}^{\dagger}(\mathbb{R})$ the set of $n \times n$ symmetric matrices whose eigenvalues are all simple. Let $M \in \mathcal{S}_{n}^{\dagger}(\mathbb{R})$. Determine a real $r > 0$ such that the open ball of $\mathcal{S}_{n}(\mathbb{R})$ centered at $M$ with radius $r$ is included in $\mathcal{S}_{n}^{\dagger}(\mathbb{R})$. Deduce that $\mathcal{S}_{n}^{\dagger}(\mathbb{R})$ is an open set of $\mathcal{S}_{n}(\mathbb{R})$.

We denote by $\mathcal{S}_{n}^{\dagger}(\mathbb{R})$ the set of $n \times n$ symmetric matrices whose eigenvalues are all simple. Show that the first component $s_{1}^{\downarrow}$ of $s^{\downarrow}$ is of class $\mathscr{C}^{1}$ on $\mathcal{S}_{2}^{\dagger}(\mathbb{R})$, but not on $\mathcal{S}_{2}(\mathbb{R})$. (One may use question 1d.)

In this part, we consider two real symmetric matrices $A, B \in \mathcal{S}_{n}(\mathbb{R})$ and their sum $C = A + B$. We denote by $a = s^{\downarrow}(A)$, $b = s^{\downarrow}(B)$ and $c = s^{\downarrow}(C)$.
Show that $$\sum_{i=1}^{n} c_{i} = \sum_{i=1}^{n} a_{i} + \sum_{i=1}^{n} b_{i}.$$

For $j \geqslant 2$ an integer, the function $\psi _ { j } : \mathbb { R } \rightarrow \mathbb { R }$ is 1-periodic and defined on $] - 1 / 2,1 / 2 ]$ by $\psi _ { j } ( t ) = \max ( 0,1 - j | t | )$. For integers $0 \leqslant k < j$, $\psi _ { j , k } ( t ) = \psi _ { j } \left( t - \frac { k } { j } \right)$. We are given $f \in \mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { 2 } \right)$ and $j \geqslant 2$ an integer, and we set $$S _ { j } ( f ) \left( \theta _ { 1 } , \theta _ { 2 } \right) = \sum _ { k _ { 1 } = 0 } ^ { j - 1 } \sum _ { k _ { 2 } = 0 } ^ { j - 1 } f \left( \frac { k _ { 1 } } { j } , \frac { k _ { 2 } } { j } \right) \psi _ { j , k _ { 1 } } \left( \theta _ { 1 } \right) \psi _ { j , k _ { 2 } } \left( \theta _ { 2 } \right) .$$
Show that $S _ { j } ( f ) \in \mathscr { C } _ { \text {sep} } \left( \mathbb { R } ^ { 2 } \right)$ and coincides with $f$ at the points $\left( \frac { \ell _ { 1 } } { j } , \frac { \ell _ { 2 } } { j } \right)$ for $\left( \ell _ { 1 } , \ell _ { 2 } \right) \in \mathbb { Z } ^ { 2 }$.

Deduce the inversion formula: for every integer $k \in \mathbb{N}$, $$u_k = \sum_{j=0}^{k} (-1)^{k-j} \binom{k}{j} v_j$$

The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ Let $P \in \mathbb{R}_{n-1}[X]$. Show that $$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} P(j) = 0$$

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Show that the family $\left(H_k\right)_{k \in \llbracket 0, n \rrbracket}$ is a basis of $\mathbb{R}_n[X]$.

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Calculate $\delta\left(H_0\right)$ and, for $k \in \llbracket 1, n \rrbracket$, express $\delta\left(H_k\right)$ in terms of $H_{k-1}$.