Show that there exists a real sequence $\left( a _ { n } \right) _ { n \in \mathbb { N } }$ having the following property: for every integer $p \in \mathbb { N } ^ { * }$, for every non-degenerate interval $I$ and for every complex function $f$ of class $C ^ { \infty }$ on $I$, the function $g$ defined on $I$ by $g = a _ { 0 } f + a _ { 1 } f ^ { \prime } + \cdots + a _ { p - 1 } f ^ { ( p - 1 ) }$ satisfies $$g ^ { \prime } + \frac { 1 } { 2 ! } g ^ { \prime \prime } + \frac { 1 } { 3 ! } g ^ { ( 3 ) } + \cdots + \frac { 1 } { p ! } g ^ { ( p ) } = f ^ { \prime } + \sum _ { l = 1 } ^ { p - 1 } b _ { l , p } f ^ { ( p + l ) }$$ where the $b _ { l , p }$ are coefficients independent of $f$ which we do not seek to calculate.

Show that $a _ { 0 } = 1$ and that for every $p \geqslant 1 , a _ { p } = - \sum _ { i = 2 } ^ { p + 1 } \frac { a _ { p + 1 - i } } { i ! }$. Deduce that $\left| a _ { p } \right| \leqslant 1$ for every natural integer $p$. Determine $a _ { 1 }$ and $a _ { 2 }$.

a) For every $z \in \mathbb { C }$ such that $| z | < 1$, justify that the series $\sum _ { p \in \mathbb { N } } a _ { p } z ^ { p }$ is convergent.
We denote by $\varphi ( z )$ its sum: $\varphi ( z ) = \sum _ { p = 0 } ^ { \infty } a _ { p } z ^ { p }$.
b) For $z \in \mathbb { C }$ such that $| z | < 1$, calculate the product $\left( e ^ { z } - 1 \right) \varphi ( z )$. Deduce that for every $z \in \mathbb { C } ^ { * }$ satisfying $| z | < 1$, we have $\varphi ( z ) = \frac { z } { e ^ { z } - 1 }$.
c) Show that $a _ { 2 k + 1 } = 0$ for every integer $k \geqslant 1$. Calculate $a _ { 4 }$.

We define a sequence of polynomials $\left( A _ { n } \right) _ { n \in \mathbb { N } }$ satisfying $A _ { 0 } = 1 , A _ { n + 1 } ^ { \prime } = A _ { n }$ and $\int _ { 0 } ^ { 1 } A _ { n + 1 } ( t ) d t = 0$ for every $n \in \mathbb { N }$.
a) Show that the series $\sum _ { n } A _ { n } ( t ) z ^ { n }$ converges for every real $t \in [ - 1,1 ]$ and every complex $z$ satisfying $| z | < 1$.
Under these conditions, we set $f ( t , z ) = \sum _ { n = 0 } ^ { + \infty } A _ { n } ( t ) z ^ { n }$.
b) Let $z \in \mathbb { C }$ such that $| z | < 1$. Show that the function $t \mapsto f ( t , z )$ is differentiable on $[ 0,1 ]$ and express its derivative in terms of $f ( t , z )$. Deduce that if $| z | < 1$ and $z \neq 0$, $$\sum _ { n = 0 } ^ { + \infty } A _ { n } ( t ) z ^ { n } = \frac { z e ^ { t z } } { e ^ { z } - 1 }$$
c) Show that if $z \in \mathbb { C }$ and $| z | < 2 \pi$, we have $\frac { z e ^ { z / 2 } } { e ^ { z } - 1 } + \frac { z } { e ^ { z } - 1 } = 2 \frac { z / 2 } { e ^ { z / 2 } - 1 }$.
Deduce that for every natural integer $n , A _ { n } \left( \frac { 1 } { 2 } \right) = \left( \frac { 1 } { 2 ^ { n - 1 } } - 1 \right) a _ { n }$.

For every natural integer $p \geqslant 1$ and every real number $x$, we set $\widetilde { A } _ { p } ( x ) = A _ { p } \left( \frac { x } { 2 \pi } - \left[ \frac { x } { 2 \pi } \right] \right)$. Show that $\widetilde { A } _ { p }$ is $2 \pi$-periodic and piecewise continuous.

The function $h$ is defined on $\mathbb{R}$ by $$h(u) = u - [u] - 1/2$$ Show that the function $H$ defined on $\mathbb{R}$ by: $$H(x) = \int_{0}^{x} h(t) \, dt$$ is continuous, of class $\mathcal{C}^{1}$ piecewise, and periodic with period 1.

The function $h$ is defined on $\mathbb{R}$ by $$h(u) = u - [u] - 1/2$$ Let $\varphi$ be the application defined for all $x > 0$ by: $$\varphi(x) = \int_{0}^{+\infty} \frac{h(u)}{u+x} du$$ By repeating the integration by parts from question IV.C, prove that the application $\varphi$ is of class $\mathcal{C}^{1}$ on $\mathbb{R}_{+}^{*}$ and that for all $x > 0$, $$\varphi^{\prime}(x) = -\int_{0}^{+\infty} \frac{h(u)}{(u+x)^{2}} du$$

Using the identity $$\ln \Gamma(x+1) = \left(x+\frac{1}{2}\right)\ln x - x + \ln\sqrt{2\pi} - \int_{0}^{+\infty} \frac{h(u)}{u+x} du$$ show that for all strictly positive real $x$, $$\frac{\Gamma^{\prime}(x+1)}{\Gamma(x+1)} = \ln x + \frac{1}{2x} + \int_{0}^{+\infty} \frac{h(u)}{(u+x)^{2}} du$$

Let $f, g \in C(\mathbb{R})$ be such that $f * g(x)$ is defined for every real $x$. Show that $f * g = g * f$.

Show that if $f$ and $g$ have compact support, then $f * g$ 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 $L^{2}(\mathbb{R})$. For any real $\alpha$, show that $T_{\alpha}(f * g) = \left(T_{\alpha}(f)\right) * g$.

Let $k$ be a non-zero natural number. Assume that $g$ is of class $C^{k}$ on $\mathbb{R}$ and that all its derivative functions, up to order $k$, are bounded on $\mathbb{R}$. Show that $f * g$ is of class $C^{k}$ on $\mathbb{R}$ and specify its derivative function of order $k$.

To any function $g$ in $C(\mathbb{R})$, we associate the linear form $\varphi_{g}$ on $L^{1}(\mathbb{R})$ defined by $$\varphi_{g}(f) = \int_{\mathbb{R}} f(t) g(-t) \mathrm{d}t$$ Let $(g_{1}, \ldots, g_{p})$ be a family of elements of $C_{b}(\mathbb{R})$. Show that the family $(g_{1}, \ldots, g_{p})$ is free if and only if the family $(\varphi_{g_{1}}, \ldots, \varphi_{g_{p}})$ is free.

Let $E$ be an infinite-dimensional vector space and $\left(f_{n}\right)_{n \in \mathbb{N}}$ a family of linear forms on $E$. We denote $$K = \bigcap_{n \in \mathbb{N}} \operatorname{Ker}\left(f_{n}\right)$$ Show that the codimension of $K$ in $E$ is equal to the rank of the family $\left(f_{n}\right)_{n \in \mathbb{N}}$ in the dual space $E^{*}$ (begin with the case where this rank is finite).

We assume that $g \in C_{b}(\mathbb{R})$. We consider the vector subspace $$N_{g} = \left\{f \in L^{1}(\mathbb{R}) \mid f * g = 0\right\}$$ and the vector space $V_{g} = \operatorname{Vect}\left(T_{\alpha}(g)\right)_{\alpha \in \mathbb{R}}$ where $T_{\alpha}(g)(x) = g(x-\alpha)$. Show that the codimension of $N_{g}$ in $L^{1}(\mathbb{R})$ is equal to the dimension of $V_{g}$.

a) Let $\beta \in \mathbb{R}$ and let $g$ be the function defined by $g(t) = \mathrm{e}^{\mathrm{i}\beta t}$ for all $t \in \mathbb{R}$. Determine the codimension of $N_{g}$ in $L^{1}(\mathbb{R})$. b) Let $n$ be a natural number. Show that there exists a function $g$ in $C_{b}(\mathbb{R})$ such that $N_{g}$ has codimension $n$ in $L^{1}(\mathbb{R})$.

Let $g \in C_{b}(\mathbb{R})$. We assume that $N_{g}$ has finite codimension $n$ in $L^{1}(\mathbb{R})$, and that $V_{g} = \operatorname{Vect}\left(T_{\alpha}(g)\right)_{\alpha \in \mathbb{R}}$ has dimension $n$. Show that there exist real numbers $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ and functions $m_{1}, \ldots, m_{n}$ of a real variable such that, for every real $\alpha$, $$T_{\alpha}(g) = \sum_{i=1}^{n} m_{i}(\alpha) T_{\alpha_{i}}(g)$$

Let $F$ be a finite-dimensional subspace of $C(\mathbb{R})$, with dimension denoted $p$. For any function $f \in C(\mathbb{R})$ and for any real $x$, we denote $e_{x}(f) = f(x)$. a) Show that there exist real numbers $a_{1}, \ldots, a_{p}$ such that $(e_{a_{1}}, \ldots, e_{a_{p}})$ is a basis of the dual space $F^{*}$. b) If $\left(f_{1}, \ldots, f_{p}\right)$ is a family of elements of $F$, show that $\operatorname{Det}\left(f_{i}\left(a_{j}\right)\right)_{1 \leqslant i,j \leqslant p}$ is non-zero if and only if $\left(f_{1}, \ldots, f_{p}\right)$ is a basis of $F$.

Let $g \in C_{b}(\mathbb{R})$ with $N_{g}$ of finite codimension $n$ in $L^{1}(\mathbb{R})$, and let $\alpha_{1}, \ldots, \alpha_{n}$, $m_{1}, \ldots, m_{n}$ be as in III.C.1 such that $T_{\alpha}(g) = \sum_{i=1}^{n} m_{i}(\alpha) T_{\alpha_{i}}(g)$ for every real $\alpha$. By applying question III.C.2) to $V_{g}$, show that if $g$ is of class $C^{k}$ then the functions $m_{1}, \ldots, m_{n}$ are of class $C^{k}$.

Let $g \in C_{b}(\mathbb{R})$ with $N_{g}$ of finite codimension in $L^{1}(\mathbb{R})$. The functions $h_{r}$ are those from question I.D.3, defined on $[-1,1]$ by $h_{r}(t) = \frac{(1-t^2)^r}{\lambda_r}$ and zero outside $[-1,1]$. Show that for every non-zero natural number $r$, $V_{h_{r} * g}$ is finite-dimensional.

Let $g \in C_{b}(\mathbb{R})$ with $N_{g}$ of finite codimension in $L^{1}(\mathbb{R})$. The functions $h_{r}$ are those from question I.D.3. Show that for $r$ sufficiently large the dimension of $V_{h_{r} * g}$ is equal to that of $V_{g}$.

Let $g \in C_{b}(\mathbb{R})$ with $N_{g}$ of finite codimension $n$ in $L^{1}(\mathbb{R})$, and let $m_{1}, \ldots, m_{n}$ be as in III.C.1. Deduce that the functions $m_{1}, \ldots, m_{n}$ are of class $C^{\infty}$.

Determine the set of functions $g \in C_{b}(\mathbb{R})$ such that $N_{g}$ has finite codimension in $L^{1}(\mathbb{R})$.

Show that if $E$ is not empty, then $E$ is an unbounded interval of $\mathbb{R}$.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$.
Show that if $E$ is not empty and if $\alpha$ is its infimum (we agree that $\alpha = -\infty$ if $E = \mathbb{R}$), then $Lf$ is of class $C^{\infty}$ on $]\alpha, +\infty[$ and express its successive derivatives using an integral.