Let $f$ be continuously differentiable on $\mathbb { R }$. Let $f _ { n } ( x ) = n \left( f \left( x + \frac { 1 } { n } \right) - f ( x ) \right)$. Then,
(a) $f _ { n }$ converges uniformly on $\mathbb { R }$;
(b) $f _ { n }$ converges on $\mathbb { R }$, but not necessarily uniformly;
(c) $f _ { n }$ converges to the derivative of $f$ uniformly on $[ 0,1 ]$;
(d) there is no guarantee that $f _ { n }$ converges on any open interval.

Let $f _ { n } ( x ) = \frac { 1 } { 1 + x ^ { n } }$. Pick the correct statement(s) from below.
(A) $f _ { n }$ converges uniformly on $[ 0,1 / 2 ]$.
(B) $f _ { n }$ converges uniformly on $[ 0,1 )$.
(C) $f _ { n }$ converges uniformly on $[ 0,2 ]$.
(D) $f _ { n }$ converges pointwise on $[ 0 , \infty )$.

Let $f : [ 0,1 ] \longrightarrow \mathbb { R }$ and $g : \mathbb { R } \longrightarrow \mathbb { R }$ be continuous functions. Assume that $g$ is periodic with period 1. Show that $$\lim _ { n \mapsto \infty } \int _ { 0 } ^ { 1 } f ( x ) g ( n x ) d x = \left( \int _ { 0 } ^ { 1 } f ( x ) d x \right) \left( \int _ { 0 } ^ { 1 } g ( x ) d x \right)$$

We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay. Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$. For every integer $n \in \mathbb{N}$, $F_n(x) = \cos(n \arccos x)$ extended as a polynomial to $\mathbb{R}$.
Show that, for all $x \in [-1,1]$, the series $$\sum_{n \geqslant 0} \alpha_n F_n(x)$$ is convergent.

We denote by $\mathcal{S}$ the set of sequences of real numbers with rapid decay. Let $(\alpha_n)_{n \in \mathbb{N}}$ be a sequence of $\mathcal{S}$. For every integer $n \in \mathbb{N}$, $F_n(x) = \cos(n \arccos x)$ extended as a polynomial to $\mathbb{R}$. We define the function $f$ on the segment $[-1,1]$ by: $$\forall x \in [-1,1], \quad f(x) = \sum_{n=0}^{+\infty} \alpha_n F_n(x).$$
Show that $f$ is of class $C^\infty$ on $[-1,1]$.

For a function $h \in C([-1,1])$, we denote by $\widetilde{h}$ the following $2\pi$-periodic function: $$\tilde{h} : \begin{cases} \mathbb{R} \rightarrow \mathbb{R} \\ \theta \mapsto h(\cos(\theta)) \end{cases}$$
Let $f \in C^\infty([-1,1])$.
Show that the Fourier series of $\widetilde{f}$ converges normally to $\widetilde{f}$.

We denote by $C([-1,1])$ the vector space of continuous functions on $[-1,1]$ with real values, equipped with the infinite norm $\|f\|_\infty = \sup_{x \in [-1,1]} |f(x)|$. For every integer $n \in \mathbb{N}$, $V_n$ denotes the set of restrictions to $[-1,1]$ of polynomial functions of degree at most $n$, and $d(f, V_n) = \inf_{p \in V_n} \|f - p\|_\infty$. For a function $h \in C([-1,1])$, $\widetilde{h}$ denotes the $2\pi$-periodic function $\theta \mapsto h(\cos(\theta))$.
Let $f \in C([-1,1])$. We assume that the sequence $(d(f, V_n))_{n \in \mathbb{N}}$ has rapid decay.
The purpose of this question is to show that the function $f$ is of class $C^\infty$.
a) Let $k \in \mathbb{N}^*$. Show that, for $P \in E_{k-1}$, $a_k(\widetilde{f}) = a_k(\widetilde{f-P})$.
b) Deduce that the sequence $(a_n(\widetilde{f}))_{n \in \mathbb{N}}$ of Fourier coefficients of the function $\widetilde{f}$ has rapid decay.
c) Conclude.

For every natural integer $p \geqslant 1$ and every real number $x$, we set $\widetilde { A } _ { p } ( x ) = A _ { p } \left( \frac { x } { 2 \pi } - \left[ \frac { x } { 2 \pi } \right] \right)$. Study the convergence of the Fourier series of $\widetilde { A } _ { p }$.

We assume that $\frac { \sqrt { \lambda _ { 1 } } } { \sqrt { \lambda _ { 2 } } }$ is not a rational number. We fix an arbitrary element $f \in C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$. For each $k \in \mathbb { N } ^ { * }$ we set: $$\forall ( u , v ) \in \mathbb { R } ^ { 2 } , f _ { k } ( u , v ) = \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } R _ { k } \left( u - \theta _ { 1 } \right) R _ { k } \left( v - \theta _ { 2 } \right) f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 }$$
Let $\epsilon \in ] 0 , \pi [$ and $k \in \mathbb { N } ^ { * }$. By writing $f _ { k } ( u , v ) - f ( u , v )$ as a sum of two terms and applying Question 10, prove that for every $( u , v ) \in \mathbb { R } ^ { 2 }$: $$\left| f _ { k } ( u , v ) - f ( u , v ) \right| \leq 2 \epsilon \left( \left\| \frac { \partial f } { \partial \theta _ { 1 } } \right\| + \left\| \frac { \partial f } { \partial \theta _ { 2 } } \right\| \right) + 8 \pi \| f \| d _ { k } ( \epsilon )$$

We assume that $\frac { \sqrt { \lambda _ { 1 } } } { \sqrt { \lambda _ { 2 } } }$ is not a rational number. We fix an arbitrary element $f \in C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$. For each $k \in \mathbb { N } ^ { * }$ we set: $$\forall ( u , v ) \in \mathbb { R } ^ { 2 } , f _ { k } ( u , v ) = \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } R _ { k } \left( u - \theta _ { 1 } \right) R _ { k } \left( v - \theta _ { 2 } \right) f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 }$$
Prove the Ergodic Theorem for the function $f$. (One may set $M = 2 \left( \left\| \frac { \partial f } { \partial \theta _ { 1 } } \right\| + \left\| \frac { \partial f } { \partial \theta _ { 2 } } \right\| \right) + 8 \pi \| f \|$. For given $\epsilon > 0$, one may choose $k \in \mathbb { N } ^ { * }$ such that $d _ { k } ( \epsilon ) < \epsilon$. Next, one may apply Question 14 to $f _ { k }$ and consider $T _ { 0 } > 0$ such that for every $T \geq T _ { 0 }$: $$\left| \frac { 1 } { T } \int _ { 0 } ^ { T } f _ { k } \circ \theta ( t ) d t - ( 2 \pi ) ^ { - 2 } \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } f _ { k } \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 } \right| < \epsilon .)$$

Let $a , b \in ] 0,2 \pi [$ such that $a < b$. We denote by $\phi _ { a , b } : \mathbb { R } \rightarrow \mathbb { R }$ the continuous $2 \pi$-periodic function defined as follows. The function $\phi _ { a , b }$ is zero on $[ 0 , a ]$ and $[ b , 2 \pi ]$. For every $t \in [ a , b ] , \phi _ { a , b } ( t ) = \sin ^ { 2 } \left( \frac { \pi } { b - a } ( t - a ) \right)$.
Recall that every non-empty open set of $] - 1,1 [ ^ { 2 }$ contains a rectangle of the form $] \cos b , \cos a [ \times ] \cos d , \cos c [$ where $0 < a < b < \pi$ and $0 < c < d < \pi$.
Consider the solution $x ( t ) = \sum _ { i = 1 } ^ { 2 } \cos \left( t \sqrt { \lambda _ { i } } + \varphi _ { i } \right) e _ { i }$ of (1) obtained by taking $c _ { 1 } = c _ { 2 } = 1$ in (2). Let $\Omega$ be a non-empty open set of $\left\{ u e _ { 1 } + v e _ { 2 } \mid u , v \in ] - 1,1 [ \right\}$. Prove that there exists $t \in [ 0 , + \infty [$ such that $x ( t ) \in \Omega$. (One may reason by contradiction and justify the existence of a function of the type $\left( \theta _ { 1 } , \theta _ { 2 } \right) \mapsto \phi _ { a , b } \left( \theta _ { 1 } \right) \phi _ { c , d } \left( \theta _ { 2 } \right) = \Phi \left( \theta _ { 1 } , \theta _ { 2 } \right)$ such that $\Phi ( \theta ( t ) )$ is zero for all $t \in [ 0 , + \infty [)$.

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We extend to $C^0([0;1], \mathbb{R})$ the inner product $\langle \cdot, \cdot \rangle$ by setting $$\forall f, g \in C^0([0;1], \mathbb{R}), \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, dt$$ and we denote by $\|\cdot\|$ the associated norm. For each $n \in \mathbb{N}$, $\Pi_n$ denotes the unique polynomial in $\mathbb{R}_n[X]$ minimizing $\|Q - f\|$ over $\mathbb{R}_n[X]$.
Show that the sequence $\left(\left\|\Pi_n - f\right\|\right)_{n \in \mathbb{N}}$ is decreasing and converges to 0.

For any function $h$ in $C(\mathbb{R})$ and any real $\alpha$, we define the function $T_{\alpha}(h)$ by setting $T_{\alpha}(h)(x) = h(x - \alpha)$ for all $x \in \mathbb{R}$. We assume that $f$ and $g$ belong to $L^{2}(\mathbb{R})$. Show that a function $h$ is uniformly continuous on $\mathbb{R}$ if and only if $\lim_{\alpha \rightarrow 0} \|T_{\alpha}(h) - h\|_{\infty} = 0$.

For any function $h$ in $C(\mathbb{R})$ and any real $\alpha$, we define the function $T_{\alpha}(h)$ by setting $T_{\alpha}(h)(x) = h(x - \alpha)$ for all $x \in \mathbb{R}$. We assume that $f$ and $g$ belong to $L^{2}(\mathbb{R})$. Deduce that $f * g$ is uniformly continuous on $\mathbb{R}$ in the case where $f$ has compact support.

For any function $h$ in $C(\mathbb{R})$ and any real $\alpha$, we define the function $T_{\alpha}(h)$ by setting $T_{\alpha}(h)(x) = h(x - \alpha)$ for all $x \in \mathbb{R}$. We assume that $f$ and $g$ belong to $L^{2}(\mathbb{R})$. Show that $f * g$ is uniformly continuous on $\mathbb{R}$ in the general case.

Assume that $f \in L^{1}(\mathbb{R})$ and $g \in C_{b}(\mathbb{R})$. a) Show that $f * g$ is continuous. b) Show that if $g$ is uniformly continuous on $\mathbb{R}$, then $f * g$ is uniformly continuous on $\mathbb{R}$.

Let $f \in C_{b}(\mathbb{R})$ and let $(\delta_{n})$ be a sequence of functions forming an approximate identity. Show that the sequence $\left(f * \delta_{n}\right)_{n \in \mathbb{N}}$ converges pointwise to $f$ on $\mathbb{R}$.

Let $f \in C_{b}(\mathbb{R})$ and let $(\delta_{n})$ be a sequence of functions forming an approximate identity. Show that if $f$ has compact support, then the sequence $\left(f * \delta_{n}\right)_{n \in \mathbb{N}}$ converges uniformly to $f$ on $\mathbb{R}$.

For every natural number $n$, we denote by $h_{n}$ the function defined on $[-1,1]$ by $$h_{n}(t) = \frac{\left(1 - t^{2}\right)^{n}}{\lambda_{n}}$$ and zero outside $[-1,1]$, where the real number $\lambda_{n}$ is given by the formula $$\lambda_{n} = \int_{-1}^{1} \left(1 - t^{2}\right)^{n} \mathrm{~d}t$$ a) Show that the sequence of functions $\left(h_{n}\right)_{n \in \mathbb{N}}$ is an approximate identity. b) Show that if $f$ is a continuous function with support included in $\left[-\frac{1}{2}, \frac{1}{2}\right]$, then $f * h_{n}$ is a polynomial function on $\left[-\frac{1}{2}, \frac{1}{2}\right]$ and zero outside the interval $\left[-\frac{3}{2}, \frac{3}{2}\right]$. c) Deduce a proof of Weierstrass's theorem: every complex continuous function on a closed interval of $\mathbb{R}$ is the uniform limit on this interval of a sequence of polynomial functions.

For any function $f \in L^{1}(\mathbb{R})$, the Fourier transform of $f$ is the function $\hat{f}$ defined by $$\forall x \in \mathbb{R} \quad \hat{f}(x) = \int_{\mathbb{R}} f(t) \mathrm{e}^{-\mathrm{i}xt} \mathrm{~d}t$$ For any function $f \in L^{1}(\mathbb{R})$, show that $\hat{f}$ belongs to $C_{b}(\mathbb{R})$.

We define, for every non-zero natural number $n$, the function $k_{n}$ by $$\begin{cases} k_{n}(x) = 1 - \frac{|x|}{n} & \text{if } |x| \leqslant n \\ k_{n}(x) = 0 & \text{otherwise} \end{cases}$$ We admit that $\int_{\mathbb{R}} \varphi = \pi$ where $\varphi(x) = \left(\frac{\sin x}{x}\right)^2$ for $x\neq 0$ and $\varphi(0)=1$. We set $K_{n} = \frac{1}{2\pi} \hat{k}_{n}$. Show that the sequence of functions $\left(K_{n}\right)_{n \geqslant 1}$ is an approximate identity.

Show that if $E$ is not empty, then $Lf$ is continuous on $E$, where for $x \in E^{\prime}$, $$Lf(x) = \int_0^{+\infty} f(t)e^{-\lambda(t)x}\,dt.$$

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{\sin t}{t}$ for all $t \in \mathbb{R}^{+*}$, $f$ being extended by continuity at 0.
We denote for $n \in \mathbb{N}$ and $x \geqslant 0$, $$f_n(x) = \int_{n\pi}^{(n+1)\pi} \frac{\sin t}{t} e^{-xt}\,dt.$$ Show that $\sum_{n \geqslant 0} f_n$ converges uniformly on $[0, +\infty[$.

Justify the equality
$$\forall t \in \mathbb { R } \quad G _ { x } ( t ) = e ^ { i x \sin t } = \sum _ { n = - \infty } ^ { + \infty } \varphi _ { n } ( x ) e ^ { i n t }$$
What can be said about the convergence of the Fourier series of $G _ { x }$ ?

Show that $\varphi _ { n }$ is of class $\mathcal { C } ^ { \infty }$ on $\mathbb { R }$.