Does the sequence of functions $(\varphi_n)_{n \geqslant 1}$ defined by $$\forall n \in \mathbb{N}^{\star}, \quad \varphi_n : \begin{aligned} \mathbb{R} &\rightarrow \mathbb{R} \\ t &\mapsto \mathbb{E}\left(\cos\left(t X_n\right)\right) \end{aligned}$$ converge uniformly on $\mathbb{R}$?

Let $n$ be a non-zero natural number. We set $$\forall n \in \mathbb{N}^{\star}, \quad X_n = \sum_{k=1}^{n} \frac{\varepsilon_k}{2^k}$$ where $(\varepsilon_n)_{n \geqslant 1}$ is a sequence of independent random variables taking values in $\{-1,1\}$ with $\mathbb{P}(\varepsilon_n = 1) = \mathbb{P}(\varepsilon_n = -1) = 1/2$ for all $n \geqslant 1$, and $$\forall n \in \mathbb{N}^{\star}, \quad \varphi_n : \begin{aligned} \mathbb{R} &\rightarrow \mathbb{R} \\ t &\mapsto \mathbb{E}(\cos(t X_n)) \end{aligned}$$
Does the sequence of functions $(\varphi_n)_{n \geqslant 1}$ converge uniformly on $\mathbb{R}$?

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 $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t)\,\mathrm{d}t$$ where $k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$ Show that $T$ is a continuous endomorphism of $E$.

Let $(E, \langle \cdot, \cdot \rangle)$ be a reproducing kernel Hilbert space on an interval $I$, with reproducing kernel $K$. For all $(x,y) \in I^2$, we set $k_x(y) = K(x,y)$. Suppose that $K$ is continuous on $I \times I$. Prove that all functions in $E$ are continuous.

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, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t) \, \mathrm{d}t$$ Show that $T$ is a continuous endomorphism of $E$.

Let $(E, \langle \cdot, \cdot \rangle)$ be a reproducing kernel Hilbert space on an interval $I$, with reproducing kernel $K$. For all $(x,y) \in I^2$, we set $k_x(y) = K(x,y)$. Suppose that $K$ is continuous on $I \times I$. Prove that all functions in $E$ are continuous.

Let $X$ be a real and discrete random variable and $m \in \mathbb { R }$. For $n \in \mathbb { N }$ and $h \in \mathbb { R } _ { + } ^ { * }$, we set $g _ { n } ( h ) = \operatorname { sinc } \left( \frac { x _ { n } - m } { h } \right) \mathbb { P } \left( X = x _ { n } \right)$. Show that the function $g _ { n }$ extends to a function $\tilde { g } _ { n }$ defined and continuous on $\mathbb { R } ^ { + }$.

In this subsection, $I = ]{-1,1}[$ and $w(x) = \frac{1}{\sqrt{1-x^2}}$.
Show that, for every integer $k \in \mathbb{N}$, the function $x \mapsto x^k w(x)$ is integrable on $I$.

For $k \in \mathbf { N } ^ { * }$ and $t \in \mathbf { R } _ { + }$, we set
$$u _ { k } ( t ) = \int _ { k/2 } ^ { ( k + 1 ) / 2 } \frac { t q ( u ) } { e ^ { t u } - 1 } \mathrm {~d} u \quad \text { if } t > 0 , \quad \text { and } \quad u _ { k } ( t ) = \int _ { k/2 } ^ { ( k + 1 ) / 2 } \frac { q ( u ) } { u } \mathrm {~d} u \quad \text { if } t = 0$$
where $q ( x ) = x - \lfloor x \rfloor - \frac { 1 } { 2 }$.
Show that $u _ { k }$ is continuous on $\mathbf { R } _ { + }$ for all $k \in \mathbf { N } ^ { * }$.

Deduce that
$$\int _ { - \pi } ^ { \pi } e ^ { - i \frac { \pi ^ { 2 } \theta } { 6 t ^ { 2 } } } \frac { P \left( e ^ { - t } e ^ { i \theta } \right) } { P \left( e ^ { - t } \right) } \mathrm { d } \theta = O \left( t ^ { 3 / 2 } \right) \quad \text { when } t \text { tends to } 0 ^ { + }$$

For $k \in \mathbf{N}^*$ and $t \in \mathbf{R}_+$, we set $$u_k(t) = \int_{k/2}^{(k+1)/2} \frac{tq(u)}{e^{tu}-1} \mathrm{du} \text{ if } t > 0 \text{, and } u_k(t) = \int_{k/2}^{(k+1)/2} \frac{q(u)}{u} \mathrm{du} \text{ if } t = 0.$$ Show that $u_k$ is continuous on $\mathbf{R}$, for all $k \in \mathbf{N}^*$.

Deduce that $$\int_{-\pi}^{\pi} e^{-i\frac{\theta^2}{6t^2}} \frac{P(e^{-t}e^{i\theta})}{P(e^{-t})} \mathrm{d}\theta = O(t^{3/2}) \text{ when } t \text{ tends to } 0^+.$$

Show that, if $f$ and $g$ are two functions in $E$, then the integral $\int _ { 0 } ^ { + \infty } f ( t ) g ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ is absolutely convergent, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges.

Deduce that $E$ is a vector subspace of the vector space $\mathcal { C } \left( \mathbb { R } _ { + } ^ { * } , \mathbb { R } \right)$ of continuous functions on $\mathbb { R } _ { + } ^ { * }$ with values in $\mathbb { R }$, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges.

The norm $\| \cdot \|$ associated with the inner product $\langle f \mid g \rangle = \int _ { 0 } ^ { + \infty } f ( t ) g ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ is defined for all functions $f \in E$ by $$\| f \| = \left( \int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t \right) ^ { 1 / 2 }$$ Show that $\lim _ { x \rightarrow 0 } \left\| k _ { x } \right\| = 0$. We recall that, for all $x > 0 , k _ { x } ( t ) = \mathrm { e } ^ { \min ( x , t ) } - 1$.

To each function $f \in E$, we associate the function $U ( f )$ defined for all $x > 0$ by $U ( f ) ( x ) = \int _ { 0 } ^ { + \infty } \left( \mathrm { e } ^ { \min ( x , t ) } - 1 \right) f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. It has been shown that $| U ( f ) ( x ) | \leqslant 4 \| f \| \frac { \sqrt { x } \mathrm { e } ^ { x / 2 } } { 1 + x }$ for all $x > 0$. Deduce that $$\| U ( f ) \| \leqslant 4 \| f \|.$$

To each function $f \in E$, we associate the function $U ( f )$ defined for all $x > 0$ by $U ( f ) ( x ) = \int _ { 0 } ^ { + \infty } \left( \mathrm { e } ^ { \min ( x , t ) } - 1 \right) f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Show that $U$ is injective.

We fix two functions $f$ and $g$ in $E$. For $x > 0$, we set $F ( x ) = - U ( f ) ^ { \prime } ( x ) \mathrm { e } ^ { - x }$. Show the existence and calculate the values of the limits at $0$ and at $+ \infty$ of the function $t \mapsto F ( t ) U ( g ) ( t )$.

For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
Show that $T_{\mu}$ is a linear map, which sends the space $\mathcal{C}_{c}(\mathbb{R})$ into itself, and that for all $\varphi \in \mathcal{C}_{c}(\mathbb{R})$ we have $\|T_{\mu}\varphi\|_{\infty} \leqslant \|\varphi\|_{\infty}$.

Show that $C^{0}(\mathbf{R}) \cap CL(\mathbf{R}) \subset L^{1}(\varphi)$.
We admit throughout the rest of the problem that $\int_{-\infty}^{+\infty} \varphi(t) \mathrm{d}t = 1$.

Let $t \in \mathbf{R}_{+}$. Verify that the function $P_{t}(f)$ is well defined for $f \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$ and verify that $P_{t}$ is linear on $C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$.
Recall that $\forall x \in \mathbf{R}, \quad P_{t}(f)(x) = \int_{-\infty}^{+\infty} f\left(\mathrm{e}^{-t} x + \sqrt{1 - \mathrm{e}^{-2t}} y\right) \varphi(y) \mathrm{d}y.$

Let $t \in \mathbf{R}_{+}$. Show that if $f \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$, then $P_{t}(f) \in C^{0}(\mathbf{R})$. Also show that $P_{t}(f)$ is bounded in absolute value by a polynomial function in $|x|$ independent of $t$. Deduce that $P_{t}(f) \in L^{1}(\varphi)$.

Show that $C^{0}(\mathbf{R}) \cap CL(\mathbf{R}) \subset L^{1}(\varphi)$, where $\varphi(x) = \frac{1}{\sqrt{2\pi}}\mathrm{e}^{-x^2/2}$ and $L^1(\varphi) = \{f \in C^0(\mathbf{R}),\, f\varphi \text{ integrable on } \mathbf{R}\}$.

Let $t \in \mathbf{R}_+$. Verify that the function $P_t(f)$ is well defined for $f \in C^0(\mathbf{R}) \cap CL(\mathbf{R})$ and verify that $P_t$ is linear on $C^0(\mathbf{R}) \cap CL(\mathbf{R})$, where $$\forall x \in \mathbf{R}, \quad P_t(f)(x) = \int_{-\infty}^{+\infty} f\!\left(\mathrm{e}^{-t}x + \sqrt{1-\mathrm{e}^{-2t}}\,y\right)\varphi(y)\,\mathrm{d}y.$$

Let $t \in \mathbf{R}_+$. Show that if $f \in C^0(\mathbf{R}) \cap CL(\mathbf{R})$, then $P_t(f) \in C^0(\mathbf{R})$. Also show that $P_t(f)$ is bounded in absolute value by a polynomial function in $|x|$ independent of $t$. Deduce that $P_t(f) \in L^1(\varphi)$.