For all $k \in \mathbb{N}^*$, we set $g_k(x) = \sqrt{2}\sin(k\pi x)$. We denote by $G = \operatorname{Vect}\left((g_k)_{k \in \mathbb{N}^*}\right)$ and $H = G^\perp$. We admit that, $$H \neq \{0\} \Longrightarrow \exists f \in H \text{ such that } \left\{ \begin{array}{l} \|f\| = 1 \\ \langle T(f), f \rangle = \sup_{h \in H, \|h\|=1} \langle T(h), h \rangle \end{array} \right.$$ Deduce that $H = \{0\}$.

Let $n \in \mathbb{N}^{*}$. Show that $$1 = \sum _ { k = 0 } ^ { n } P \left( S _ { k } = 0 _ { d } \right) P ( R > n - k )$$

From now on, $f$ denotes an infinitely differentiable function from $[0,1]$ to $\mathbb{R}$. We assume that there exists a unique point $x_0 \in [0,1]$ where $f'$ vanishes. We also assume that $f''(x_0) > 0$.
For all $x \in [x_0, 1]$, we define $$h(x) = \sqrt{|f(x) - f(x_0)|}$$ We admit that the bijection $h : [x_0, 1] \rightarrow [0, h(1)]$ admits an inverse application $h^{-1} : [0, h(1)] \rightarrow [x_0, 1]$ that is infinitely differentiable.
We admit the identities: $$\lim_{a \rightarrow +\infty} \int_0^a \sin(x^2) \mathrm{d}x = \lim_{a \rightarrow +\infty} \int_0^a \cos(x^2) \mathrm{d}x = \frac{\sqrt{2\pi}}{4}$$
Show that, as $t \rightarrow +\infty$, $$\int_{x_0}^1 \sin(tf(x)) \mathrm{d}x = \sin\left(tf(x_0) + \frac{\pi}{4}\right) \sqrt{\frac{\pi}{2tf''(x_0)}} + O\left(\frac{1}{t}\right)$$

For all $k \in \mathbb{N}^*$, we set $g_k(x) = \sqrt{2}\sin(k\pi x)$. Show that the family of vectors $(g_k)_{k \in \mathbb{N}^*}$ is orthonormal.

We admit that $(g_k)_{k \in \mathbb{N}^*}$ is a total sequence, where $g_k(x) = \sqrt{2}\sin(k\pi x)$. For all $f \in E$, we set, $$\forall x \in [0,1], \quad \Phi(x) = \sum_{k=1}^{+\infty} \frac{1}{k^2\pi^2} \langle f, g_k \rangle g_k(x)$$ Show that $\Phi$ is continuous.

Let $X$ be a real and discrete random variable and $m \in \mathbb { R }$. For $T \in \mathbb { R } _ { + } ^ { * }$, we set $V _ { m } ( T ) = \frac { 1 } { 2 T } \int _ { - T } ^ { T } \phi _ { X } ( t ) \mathrm { e } ^ { - \mathrm { i } m t } \mathrm {~d} t$. We assume that $X ( \Omega )$ is countable and we use the notations of question 2: $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ with $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. Using the results of Q17--Q19, establish that $V _ { m } ( T ) \xrightarrow [ T \rightarrow + \infty ] { } \mathbb { P } ( X = m )$.

We admit that $(g_k)_{k \in \mathbb{N}^*}$ is a total sequence, where $g_k(x) = \sqrt{2}\sin(k\pi x)$. For all $f \in E$, we set, $$\forall x \in [0,1], \quad \Phi(x) = \sum_{k=1}^{+\infty} \frac{1}{k^2\pi^2} \langle f, g_k \rangle g_k(x)$$ For all $N \in \mathbb{N}$, we set $f_N = \sum_{k=1}^N \langle f, g_k \rangle g_k$. Show that $$\lim_{N \rightarrow +\infty} \left\| T(f_N) - \Phi \right\| = 0$$

We admit that $(g_k)_{k \in \mathbb{N}^*}$ is a total sequence, where $g_k(x) = \sqrt{2}\sin(k\pi x)$. For all $f \in E$, we set, $$\forall x \in [0,1], \quad \Phi(x) = \sum_{k=1}^{+\infty} \frac{1}{k^2\pi^2} \langle f, g_k \rangle g_k(x)$$ For all $N \in \mathbb{N}$, we set $f_N = \sum_{k=1}^N \langle f, g_k \rangle g_k$. Deduce $T(f) = \Phi$.

If $a$ and $b$ are two real numbers, we denote $K _ { a , b }$ the function defined for all real $t$ by $K _ { a , b } ( t ) = \begin{cases} \frac { \mathrm { e } ^ { \mathrm { i } t b } - \mathrm { e } ^ { \mathrm { i } t a } } { 2 \mathrm { i } t } & \text { if } t \neq 0 , \\ \frac { b - a } { 2 } & \text { if } t = 0 . \end{cases}$ Using power series, show that $K _ { a , b }$ is of class $C ^ { \infty }$ on $\mathbb { R }$.

Let $E_1$ denote the vector space of functions $f:[0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$. Show that we define an inner product on $E_1$ by setting $$\forall (f,g) \in (E_1)^2, \quad (f \mid g) = \int_0^1 f'(t) g'(t)\,\mathrm{d}t$$

If $a$ and $b$ are two real numbers, we denote $K _ { a , b }$ the function defined for all real $t$ by $K _ { a , b } ( t ) = \begin{cases} \frac { \mathrm { e } ^ { \mathrm { i } t b } - \mathrm { e } ^ { \mathrm { i } t a } } { 2 \mathrm { i } t } & \text { if } t \neq 0 , \\ \frac { b - a } { 2 } & \text { if } t = 0 . \end{cases}$ Let $N$ be a natural integer and let $F _ { N }$ be the function defined, for all real $x$, by $F _ { N } ( x ) = \int _ { - N } ^ { N } K _ { a , x } ( t ) \mathrm { d } t$. Show that $F _ { N }$ is of class $C ^ { 1 }$ on $\mathbb { R }$ and that, for all real $x , F _ { N } ^ { \prime } ( x ) = N \operatorname { sinc } ( N x )$.

We fix a real vector space $E$ of dimension $n \geq 2$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$ with $\operatorname{dim} \mathcal{V} = \frac{n(n-1)}{2}$, equipped with an inner product $(-\mid-)$. We choose $x$ in $\mathcal{V}^{\bullet} \backslash \{0\}$. We denote by $p$ the generic nilindex of $\mathcal{V}$ (with $p \geq n-1$), and we fix $u \in \mathcal{V}$ such that $x \in \operatorname{Im} u^{p-1}$. We have $\mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$ and $L^{\perp} = \operatorname{Vect}(x) \oplus \mathcal{V} x$.
Suppose that there exists $v_{0}$ in $\mathcal{V}$ such that $v_{0}(x) \neq 0$. Let $v \in \mathcal{V}$. By considering $v + tv_{0}$ for $t$ real, show that $\operatorname{Im} v^{p-1} \subset \operatorname{Vect}(x) \oplus \mathcal{V} x$.

Show that, for every function $f:[0,1] \rightarrow \mathbb{R}$ of class $\mathcal{C}^1$ such that $f(0) = 0$, we have $$\forall x \in [0,1] \quad |f(x)| \leqslant \sqrt{x \int_0^x (f'(t))^2\,\mathrm{d}t}$$

If $a$ and $b$ are two real numbers, we denote $K _ { a , b }$ the function defined for all real $t$ by $K _ { a , b } ( t ) = \begin{cases} \frac { \mathrm { e } ^ { \mathrm { i } t b } - \mathrm { e } ^ { \mathrm { i } t a } } { 2 \mathrm { i } t } & \text { if } t \neq 0 , \\ \frac { b - a } { 2 } & \text { if } t = 0 . \end{cases}$ Show that $\int _ { - N } ^ { N } K _ { a , b } ( t ) \mathrm { d } t = \int _ { N a } ^ { N b } \operatorname { sinc } ( s ) \mathrm { d } s$.

We fix a real vector space $E$ of dimension $n \geq 2$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$ with $\operatorname{dim} \mathcal{V} = \frac{n(n-1)}{2}$, equipped with an inner product $(-\mid-)$. We choose $x$ in $\mathcal{V}^{\bullet} \backslash \{0\}$. We denote by $p$ the generic nilindex of $\mathcal{V}$ (with $p \geq n-1$), and we fix $u \in \mathcal{V}$ such that $x \in \operatorname{Im} u^{p-1}$. We have established that $K(\mathcal{V}) = \operatorname{Vect}(\mathcal{V}^{\bullet}) \subset \operatorname{Vect}(x) \oplus \mathcal{V} x$ (from question 23 applied to all $v \in \mathcal{V}$), and Lemma B states that if $K(\mathcal{V}) \subset \operatorname{Vect}(x) + \mathcal{V} x$ then $v(x) = 0$ for every $v \in \mathcal{V}$.
Conclude the proof of Gerstenhaber's theorem: if $\operatorname{dim} \mathcal{V} = \frac{n(n-1)}{2}$ then there exists a basis of $E$ in which every element of $\mathcal{V}$ is represented by a strictly upper triangular matrix.

Let $E_1$ denote the vector space of functions $f:[0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$. We set, for all $f \in E_1$, $$U(f)(s) = \int_0^1 k_s'(t) f'(t)\,\mathrm{d}t$$ where $k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$ Let $f \in E_1$ of class $\mathcal{C}^2$. Show that $U(f) = -T(f'')$. Deduce that $U(f) = f$.

Let $E_1$ denote the vector space of functions $f:[0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$. We set, for all $f \in E_1$, $$U(f)(s) = \int_0^1 k_s'(t) f'(t)\,\mathrm{d}t$$ Show that $U$ is the identity map on $E_1$.

Let $E_1$ denote the vector space of functions $f:[0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$, equipped with the inner product $$\forall (f,g) \in (E_1)^2, \quad (f \mid g) = \int_0^1 f'(t) g'(t)\,\mathrm{d}t$$ Prove that the pre-Hilbert space $(E_1, (\cdot \mid \cdot))$ is a reproducing kernel Hilbert space and that its reproducing kernel is the application $K$ defined by $K(s,t) = k_s(t)$ where $$k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$

We consider the space $E$ of continuous functions from $[0,1]$ to $\mathbb{R}$, equipped with the inner product defined by $$\langle f, g \rangle = \int_0^1 f(t) g(t)\,\mathrm{d}t$$ Show that $(E, \langle \cdot, \cdot \rangle)$ is not a reproducing kernel Hilbert space.

Let $(a_n)_n \in \mathbb{R}^{\mathbb{N}}$ be a sequence of reals such that the series $\sum (a_n)^2$ is convergent. Show that the radius of convergence of the power series $\sum a_n t^n$ is greater than or equal to 1.

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.

Write in Python language a function \texttt{puiss2k} that takes as argument a square matrix $M$ and a natural integer $k$ and returns the matrix $M^{2^k}$ by performing $k$ matrix products. One may exploit the fact that $M^{2^{k+1}} = M^{2^k} M^{2^k}$.

We are given a real $a > 0$. We consider the space $E_3$ of functions $f:[0,a] \rightarrow \mathbb{R}$, continuous and of class $\mathcal{C}^1$ piecewise on $[0,a]$, and satisfying $f(0) = 0$. We equip $E_3$ with the inner product defined, for $f, g \in E_3$, by $$(f \mid g) = \int_0^a f'(t) g'(t)\,\mathrm{d}t$$ Show that the function $(x,y) \mapsto \min(x,y)$ is a reproducing kernel on $(E_3, (\cdot \mid \cdot))$.