Let $W$ denote the inverse of the bijection $f|_{[-1,+\infty[}$, where $f(x) = xe^x$. For which values of the real parameter $\alpha$ is the function $x \mapsto x ^ { \alpha } W ( x )$ integrable on $\left. ]0,1 \right]$?

Let $\lambda > 0$ be fixed. We consider the space $\mathcal{C}(\mathbf{R}, \mathbf{R})$ of continuous functions from $\mathbf{R}$ to $\mathbf{R}$. We denote by $\mathcal{E}$ the vector subspace of $\mathcal{C}(\mathbf{R}, \mathbf{R})$ defined by $$\mathcal{E} = \left\{ f \in \mathcal{C}(\mathbf{R}, \mathbf{R}) \mid \exists (a, A) \in \left(\mathbf{R}_*^+\right)^2 \text{ such that } \forall y \in \mathbf{R},\ |f(y)| \leq A \exp\left(-y^2/a\right) \right\}$$
For all $(f, g) \in \mathcal{E}^2$, show that $fg$ is integrable on $\mathbf{R}$.

Let $W$ denote the inverse of the bijection $f|_{[-1,+\infty[}$, where $f(x) = xe^x$. For which values of the real parameter $\alpha$ is the function $x \mapsto x ^ { \alpha } W ( x )$ integrable on $[ 1 , + \infty [$?

Let $F$ be the vector subspace of $E$ formed of polynomial functions. For $k \in \mathbb{N}$, we denote by $p_k$ the function defined by $p_k(x) = x^k$. For all $k \in \mathbb{N}$, calculate $T(p_k)$. Deduce that $F$ is stable under $T$.

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $s \in [0,1]$, we define the function $k_s$ by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t) \, \mathrm{d}t$$ Let $F$ be the vector subspace of $E$ formed by polynomial functions. For $k \in \mathbb{N}$, we denote by $p_k$ the function defined by $p_k(x) = x^k$. For all $k \in \mathbb{N}$, calculate $T(p_k)$. Deduce that $F$ is stable under $T$.

For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t)\,\mathrm{d}t$$ where $k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$ Let $f \in E$. Calculate $T(f)(0)$ and $T(f)(1)$.

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $s \in [0,1]$, we define the function $k_s$ by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t) \, \mathrm{d}t$$ Let $f \in E$. Calculate $T(f)(0)$ and $T(f)(1)$.

Show that $$\lim_{a \rightarrow +\infty} \int_0^a |\sin(x^2)| \mathrm{d}x = +\infty$$

Show that $$\lim _ { a \rightarrow + \infty } \int _ { 0 } ^ { a } \left| \sin \left( x ^ { 2 } \right) \right| \mathrm { d } x = + \infty$$

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 $$\forall (f,g) \in (E_1)^2, \quad (f \mid g) = \int_0^1 f'(t) g'(t)\,\mathrm{d}t$$

In this part, $E_1$ denotes 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 $$\forall (f,g) \in (E_1)^2, \quad (f \mid g) = \int_0^1 f'(t) g'(t) \, \mathrm{d}t$$

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

In this part, $E_1$ denotes 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 denote by $N$ the norm associated with the inner product $(f \mid g) = \int_0^1 f'(t) g'(t) \, \mathrm{d}t$. Show that, for every function $f : [0,1] \rightarrow \mathbb{R}$ of class $\mathcal{C}^1$ such that $f(0) = 0$, we have $$\forall x \in [0,1] \quad |f(x)| \leqslant \sqrt{x \int_0^x (f'(t))^2 \, \mathrm{d}t}$$

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 $U(f) = f$.

In this part, $E_1$ denotes 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 denote by $N$ the norm associated with the inner product $(f \mid g) = \int_0^1 f'(t) g'(t) \, \mathrm{d}t$. For all $s \in [0,1]$, the function $k_s$ is defined by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ We set, for all $f \in E_1$, $$U(f)(s) = \int_0^1 k_s'(t) f'(t) \, \mathrm{d}t$$ Let $f \in E_1$ of class $\mathcal{C}^2$. Show that $U(f) = -T(f'')$. Deduce that $U(f) = f$.

Show that the integral $\int _ { 0 } ^ { + \infty } \operatorname { sinc } ( s ) \mathrm { d } s$ is convergent.

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 $E_1$.

In this part, $E_1$ denotes 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 denote by $N$ the norm associated with the inner product $(f \mid g) = \int_0^1 f'(t) g'(t) \, \mathrm{d}t$. For all $s \in [0,1]$, the function $k_s$ is defined by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ 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 $E_1$.

We admit that $\int _ { 0 } ^ { + \infty } \operatorname { sinc } ( s ) \mathrm { d } s = \frac { \pi } { 2 }$. If $a$ and $b$ are two real numbers, we denote $K _ { a , b }$ the function defined for all real $t$ by $K _ { a , b } ( t ) = \begin{cases} \frac { \mathrm { e } ^ { \mathrm { i } t b } - \mathrm { e } ^ { \mathrm { i } t a } } { 2 \mathrm { i } t } & \text { if } t \neq 0 , \\ \frac { b - a } { 2 } & \text { if } t = 0 . \end{cases}$ Deduce the existence and value of $\lim _ { N \rightarrow + \infty } \int _ { - N } ^ { N } K _ { a , b } ( t ) \mathrm { d } t$ in the case where $a < b$.

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 $(E_3, (\cdot \mid \cdot))$.

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 $(E_3, (\cdot \mid \cdot))$.

Let $E$ be the set of continuous functions $f$ from $I$ to $\mathbb{R}$ such that $f^2 w$ is integrable on $I$.
Show that, for all functions $f$ and $g$ in $E$, the product $fgw$ is integrable on $I$. You may use the inequality $\forall (a,b) \in \mathbb{R}^2, |ab| \leqslant \frac{1}{2}(a^2 + b^2)$, after justifying it.

Let $E$ be the set of continuous functions $f$ from $I$ to $\mathbb{R}$ such that $f^2 w$ is integrable on $I$.
Show that $E$ is an $\mathbb{R}$-vector space.

For every natural integer $k$ we set $$m_{k} = \frac{1}{2\pi} \int_{-2}^{2} x^{k} \sqrt{4 - x^{2}} \, \mathrm{d}x$$ For $k \in \mathbb{N}$, what is the value of $m_{2k+1}$?

For every natural integer $k$ we set $$m_{k} = \frac{1}{2\pi} \int_{-2}^{2} x^{k} \sqrt{4 - x^{2}} \, \mathrm{d}x$$ For $k \in \mathbb{N}$, what is the value of $m_{2k+1}$?