We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$ and, for all $n \in \mathbb{N}$, we denote by $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$.
The previous question shows that the following application $\varphi$ is well defined: $$\varphi : \left\{ \begin{aligned} E \times E & \rightarrow \mathbb{R} \\ (f, g) & \mapsto \int_{-1}^{1} \frac{1}{\sqrt{1 - x^2}} f(x) g(x)\, dx \end{aligned} \right.$$
Show that $\varphi$ defines an inner product on $E$.

We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$, for all $n \in \mathbb{N}$, $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$, and the space $E$ is equipped with the inner product $(\cdot|\cdot)$ defined by $\varphi(f,g) = \int_{-1}^{1} \frac{1}{\sqrt{1-x^2}} f(x) g(x)\, dx$.
a) Show that there exists a sequence of polynomial functions $(p_n)_{n \in \mathbb{N}}$ such that, for all $n \in \mathbb{N}$, $p_n$ has degree $n$ and leading coefficient 1, and that, for all $n \in \mathbb{N}^*$, $p_n$ is orthogonal to all elements of $E_{n-1}$.
b) Show that there exists a unique family $(Q_n)_{n \in \mathbb{N}}$ of polynomial functions satisfying the following conditions:

• [i)] the family $(Q_n)_{n \in \mathbb{N}}$ is orthogonal for the inner product $(\cdot|\cdot)$;
• [ii)] for all $n \in \mathbb{N}$, $Q_n$ has degree $n$ and leading coefficient 1.

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 $E^{*}$.

We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$ Moreover, we set: $$\forall (P, Q) \in (\mathbb{R}[X])^2, \quad \langle P, Q \rangle = \int_0^1 P(t) Q(t) \, dt$$
Determine a family of polynomials $\left(K_n\right)_{n \in \mathbb{N}}$ satisfying the following two conditions:
i. for all $n \in \mathbb{N}$, the degree of $K_n$ equals $n$ and its leading coefficient is strictly positive;
ii. for all $N \in \mathbb{N}$, $\left(K_n\right)_{0 \leqslant n \leqslant N}$ is an orthonormal basis of $\mathbb{R}_N[X]$ for the inner product $\langle \cdot, \cdot \rangle$.
Justify the uniqueness of such a family.

We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product $$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$ We define for any natural integer $n$ the polynomial $R_n$ as follows $$R_n(X) = (X^2 - 1)^n$$ and we now set $$L_n(X) = \frac{1}{2^n n!} R_n^{(n)}(X)$$
Let $n \in \mathbb{N}$.
(a) What is the degree of the polynomial $L_n$? Express $M(L_n)$ in terms of $L_n$.
(b) Show that if $n \geq 1$ then $$\forall P \in \mathbb{R}_{n-1}[X], \quad (L_n \mid P) = 0$$ (c) Show that for every integer $k$ such that $0 \leq k \leq n$, we have $$L_n^{(k)}(1) = \frac{(n+k)!}{(n-k)!} \frac{1}{k! \, 2^k}$$ (d) Show that for every natural integer $k$, we have $$S\left(L_n, L_n^{(2k+1)}\right) = 2 L_n^{(2k+1)}(1)$$

We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product $$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$ The endomorphisms $T(P) = P'$ and $M(P) = P^*$ are defined as before. The polynomials $L_n$ are defined by $L_n(X) = \frac{1}{2^n n!} R_n^{(n)}(X)$ where $R_n(X) = (X^2-1)^n$.
Show that the pair $(L_{2m}, L_{2m-1})$ is a characterizing pair of $G$.

We consider the set $E_2$ of functions from $]-1,1[$ to $\mathbb{R}$ of the form $$t \mapsto \sum_{n=0}^{+\infty} a_n t^n$$ where $(a_n)_n \in \mathbb{R}^{\mathbb{N}}$ and $\sum (a_n)^2$ is convergent. For $f, g \in E_2$, we set $$\langle f, g \rangle = \sum_{n=0}^{+\infty} a_n b_n \quad \text{where } f: t \mapsto \sum_{n=0}^{+\infty} a_n t^n \text{ and } g: t \mapsto \sum_{n=0}^{+\infty} b_n t^n.$$ Show that $E_2$ equipped with $\langle \cdot, \cdot \rangle$ is a real pre-Hilbert space.

We consider the set $E_2$ of functions from $]-1,1[$ to $\mathbb{R}$ of the form $$t \mapsto \sum_{n=0}^{+\infty} a_n t^n$$ where $(a_n)_n \in \mathbb{R}^{\mathbb{N}}$ and $\sum (a_n)^2$ is convergent. For $f, g \in E_2$, we set $$\langle f, g \rangle = \sum_{n=0}^{+\infty} a_n b_n \quad \text{where } f: t \mapsto \sum_{n=0}^{+\infty} a_n t^n \text{ and } g: t \mapsto \sum_{n=0}^{+\infty} b_n t^n.$$ Let $x \in ]-1,1[$. Determine $g_x \in E_2$ such that, for all $f \in E_2$, $$f(x) = \langle g_x, f \rangle$$

We consider the set $E_2$ of functions from $]-1,1[$ to $\mathbb{R}$ of the form $$t \mapsto \sum_{n=0}^{+\infty} a_n t^n$$ where $(a_n)_n \in \mathbb{R}^{\mathbb{N}}$ and $\sum (a_n)^2$ is convergent. For $f, g \in E_2$, we set $$\langle f, g \rangle = \sum_{n=0}^{+\infty} a_n b_n \quad \text{where } f: t \mapsto \sum_{n=0}^{+\infty} a_n t^n \text{ and } g: t \mapsto \sum_{n=0}^{+\infty} b_n t^n.$$ Deduce that $E_2$ is a reproducing kernel Hilbert space and specify its kernel.

We consider the set $E_2$ of functions from $]-1,1[$ to $\mathbb{R}$ of the form $$t \mapsto \sum_{n=0}^{+\infty} a_n t^n$$ where $(a_n)_n \in \mathbb{R}^{\mathbb{N}}$ and $\sum (a_n)^2$ is convergent. For $f, g \in E_2$, we set $$\langle f, g \rangle = \sum_{n=0}^{+\infty} a_n b_n \quad \text{where } f : t \mapsto \sum_{n=0}^{+\infty} a_n t^n \text{ and } g : t \mapsto \sum_{n=0}^{+\infty} b_n t^n.$$ Show that $E_2$ equipped with $\langle \cdot, \cdot \rangle$ is a real pre-Hilbert space.

We consider the set $E_2$ of functions from $]-1,1[$ to $\mathbb{R}$ of the form $$t \mapsto \sum_{n=0}^{+\infty} a_n t^n$$ where $(a_n)_n \in \mathbb{R}^{\mathbb{N}}$ and $\sum (a_n)^2$ is convergent. For $f, g \in E_2$, we set $$\langle f, g \rangle = \sum_{n=0}^{+\infty} a_n b_n \quad \text{where } f : t \mapsto \sum_{n=0}^{+\infty} a_n t^n \text{ and } g : t \mapsto \sum_{n=0}^{+\infty} b_n t^n.$$ Let $x \in ]-1,1[$. Determine $g_x \in E_2$ such that, for all $f \in E_2$, $$f(x) = \langle g_x, f \rangle$$

We consider the set $E_2$ of functions from $]-1,1[$ to $\mathbb{R}$ of the form $$t \mapsto \sum_{n=0}^{+\infty} a_n t^n$$ where $(a_n)_n \in \mathbb{R}^{\mathbb{N}}$ and $\sum (a_n)^2$ is convergent. For $f, g \in E_2$, we set $$\langle f, g \rangle = \sum_{n=0}^{+\infty} a_n b_n \quad \text{where } f : t \mapsto \sum_{n=0}^{+\infty} a_n t^n \text{ and } g : t \mapsto \sum_{n=0}^{+\infty} b_n t^n.$$ Deduce that $E_2$ is a reproducing kernel Hilbert space and specify its kernel.

Let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system in $\mathbb { R } [ X ]$ equipped with an inner product $( \cdot \mid \cdot )$. Let $n \in \mathbb { N }$ and $P \in \mathbb { R } [ X ]$ such that $\operatorname { deg } P < n$. Show that $\left( V _ { n } \mid P \right) = 0$.

Let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system in $\mathbb { R } [ X ]$ equipped with an inner product $( \cdot \mid \cdot )$. Let $\left( W _ { n } \right) _ { n \in \mathbb { N } }$ be another orthogonal system. Show that $\forall n \in \mathbb { N } , W _ { n } = V _ { n }$.

Let $\left( U _ { n } \right) _ { n \in \mathbb { N } }$ be the sequence of polynomials defined by $U _ { 0 } = 1 , U _ { 1 } = X - 1$ and $\forall n \in \mathbb { N } , U _ { n + 2 } = ( X - 2 ) U _ { n + 1 } - U _ { n }$, with the inner product $$( P \mid Q ) = \frac { 2 } { \pi } \int _ { 0 } ^ { 1 } P ( 4 x ) Q ( 4 x ) \frac { \sqrt { 1 - x } } { \sqrt { x } } \mathrm { ~d} x .$$ Deduce that $\left( U _ { n } \right) _ { n \in \mathbb { N } }$ is an orthogonal system and that, for all $n \in \mathbb { N } , \left\| U _ { n } \right\| = 1$. To calculate the value of $( U _ { m } \mid U _ { n } )$, one may perform the change of variable $x = \cos ^ { 2 } \theta$.

We assume that, for every integer $k \in \mathbb{N}$, the function $x \mapsto x^k w(x)$ is integrable on $I$. Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of orthogonal polynomials associated with the weight $w$ (monic, $\deg(p_n) = n$, and $\langle p_i, p_j \rangle = 0$ for $i \neq j$).
Let $n \in \mathbb{N}^*$. We denote by $x_1, \ldots, x_k$ the distinct roots of $p_n$ that are in $\mathring{I}$ and $m_1, \ldots, m_k$ their respective multiplicities. We consider the polynomial $$q(X) = \prod_{i=1}^k (X - x_i)^{\varepsilon_i}, \quad \text{with } \varepsilon_i = \begin{cases} 1 & \text{if } m_i \text{ is odd} \\ 0 & \text{if } m_i \text{ is even.} \end{cases}$$
By studying $\langle p_n, q \rangle$, show that $p_n$ has $n$ distinct roots in $\mathring{I}$.

In this subsection, $I = ]{-1,1}[$ and $w(x) = \frac{1}{\sqrt{1-x^2}}$. For every integer $n \in \mathbb{N}$, let $Q_n$ denote the polynomial of $\mathbb{R}[X]$ that coincides with $x \mapsto \cos(n \arccos(x))$ on $[-1,1]$. Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of orthogonal polynomials associated with the weight $w$.
Show that $$\begin{cases} p_0 = Q_0 \\ \forall n \in \mathbb{N}^*, \quad p_n = \dfrac{1}{2^{n-1}} Q_n \end{cases}$$

In this subsection, $I = ]{-1,1}[$ and $w(x) = \frac{1}{\sqrt{1-x^2}}$. The orthogonal polynomials associated with $w$ satisfy $p_n = \frac{1}{2^{n-1}} Q_n$ for $n \geqslant 1$, where $Q_n(x) = \cos(n \arccos(x))$.
For $n \in \mathbb{N}$, explicitly determine the points $(x_j)_{0 \leqslant j \leqslant n}$ of $I$ such that the quadrature formula $I_n(f) = \sum_{j=0}^n \lambda_j f(x_j)$ has maximal order.

We equip $\mathscr{M}_{d}(\mathbb{R})$ with the topology associated with the norm $\|M\| = \sqrt{\langle M, M \rangle}$.

• [(a)] Show that the map $f : \mathscr{M}_{d}(\mathbb{R}) \rightarrow \mathscr{M}_{d}(\mathbb{R})$ defined by $f(M) = M^{T}M$ is continuous.
• [(b)] Show that $\mathrm{SO}_{d}(\mathbb{R})$ is a closed bounded subset of $\mathscr{M}_{d}(\mathbb{R})$.