For every function $f$ belonging to $\mathcal{F}_{sr}$ and every non-zero natural number $n$, we set $$\left(f * \rho_n\right)(x) = \int_{\mathbb{R}} f(t) \rho_n(x-t) \mathrm{d}t$$
Let $I$ be the function that equals 1 on the interval $[-1,1]$, and 0 elsewhere. For $n \in \mathbb{N}^*$, we set $I_n(x) = I * \rho_n(x)$. a) For $n \in \mathbb{N}^*$ and $x \in \mathbb{R}$, express $I_n(x)$ in terms of $\varphi$. b) For $n \in \mathbb{N}^*$, show that $I_n$ belongs to $\mathcal{D}$ and study its variations. c) Sketch the graphs of $I_2$ and $I_3$. d) Show that the sequence of functions $(I_n)$ converges pointwise to a function $J$ which we shall determine. Show that $J$ and $I$ are equal except on a finite set of points. e) Does the sequence of functions $(I_n)$ converge uniformly to $J$?

We say that a real function $f$ of class $\mathcal{C}^{\infty}$ on $\mathbb{R}$ has rapid decay if $$\forall (n,m) \in \mathbb{N}^2, \lim_{x \rightarrow +\infty} x^m f^{(n)}(x) = \lim_{x \rightarrow -\infty} x^m f^{(n)}(x) = 0$$ We denote $\mathcal{S}$ the set of functions from $\mathbb{R}$ to $\mathbb{R}$ of class $\mathcal{C}^{\infty}$ with rapid decay.
Show that $\mathcal{S}$ is a vector space over $\mathbb{R}$.

We say that a real function $f$ of class $\mathcal{C}^{\infty}$ on $\mathbb{R}$ has rapid decay if $$\forall (n,m) \in \mathbb{N}^2, \lim_{x \rightarrow +\infty} x^m f^{(n)}(x) = \lim_{x \rightarrow -\infty} x^m f^{(n)}(x) = 0$$ We denote $\mathcal{S}$ the set of functions from $\mathbb{R}$ to $\mathbb{R}$ of class $\mathcal{C}^{\infty}$ with rapid decay.
Show that if $f$ is in $\mathcal{S}$ then $f^{(p)}$ is in $\mathcal{S}$ for every natural number $p$.

We call a distribution on $\mathcal{D}$ any linear map $T : \mathcal{D} \rightarrow \mathbb{R}$ which satisfies $$\forall \varphi \in \mathcal{D}, \forall (\varphi_n)_{n \in \mathbb{N}} \in \mathcal{D}^{\mathbb{N}} \quad \varphi_n \xrightarrow{\mathcal{D}} \varphi \Longrightarrow T(\varphi_n) \rightarrow T(\varphi)$$
Show that if $f \in \mathcal{F}_{sr}$ then the map $T_f$ defined by $$\forall \varphi \in \mathcal{D} \quad T_f(\varphi) = \int_{-\infty}^{+\infty} f(x) \varphi(x) \mathrm{d}x$$ defines a distribution on $\mathcal{D}$.

Let $U$ be the function defined by $$\begin{cases} U(x) = 1 & \text{if } x \geqslant 0 \\ U(x) = 0 & \text{if } x < 0 \end{cases}$$ Justify that $U$ defines a distribution on $\mathcal{D}$.

Let $a$ be a real number. a) Show that the map $\delta_a$ which associates to every $\varphi \in \mathcal{D}$ the value $\varphi(a)$ is a distribution. b) Using the sequence of functions $(\varphi_n)_{n \in \mathbb{N}^*}$ of elements of $\mathcal{D}$ defined by $$\forall t \in \mathbb{R}, \varphi_n(t) = \begin{cases} \exp\left(\frac{(t-a)^2}{(t-a+1/n)(t-a-1/n)}\right) & \text{if } t \in ]a-1/n, a+1/n[ \\ 0 & \text{otherwise} \end{cases}$$ show that $\forall f \in \mathcal{F}_{sr}, T_f \neq \delta_a$.

If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by $$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$ Justify that $T'$ is a distribution on $\mathcal{D}$.

If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by $$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$
Let $f$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$. If $f$ is of class $\mathcal{C}^1$, show that $(T_f)' = T_{f'}$. Adapt this result to the case where $f$ is piecewise of class $\mathcal{C}^1$.

If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by $$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$ Let $U$ be the function defined by $$\begin{cases} U(x) = 1 & \text{if } x \geqslant 0 \\ U(x) = 0 & \text{if } x < 0 \end{cases}$$ Show that $T_U' = \delta_0$.

If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by $$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$
We consider the map $T$ which associates to every function $\varphi$ of $\mathcal{D}$ the real number $T(\varphi)$ defined by $$T(\varphi) = \int_{-1}^{0} t\varphi(t) \mathrm{d}t + \int_{0}^{+\infty} \varphi(t) \mathrm{d}t$$
a) Show that $T$ is a regular distribution. b) Calculate the derivative of this distribution.

If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by $$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$ If $f$ is an element of $\mathcal{F}_{sr}$ and if $a$ is a real number, we set $$\lim_{x \rightarrow a^-} f(x) = f(a^-) \quad \text{and} \quad \lim_{x \rightarrow a^+} f(x) = f(a^+)$$ The difference $f(a^+) - f(a^-)$, called the jump at $a$, is denoted $\sigma(a)$. a) Let $a_1, \ldots, a_p$ be real numbers such that $a_1 < \ldots < a_p$. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a piecewise $\mathcal{C}^1$ function. We further assume that $f$ is continuous on $]-\infty, a_1[ \cup ]a_1, a_2[ \cup \ldots \cup ]a_p, +\infty[$. Show that $$T_f' = T_{f'} + \sum_{i=1}^{p} \sigma(a_i) \delta_{a_i}$$ b) Recover by this method the results of questions II.B.3 and II.B.4.b.

We say that the sequence of distributions $(T_n)_{n \in \mathbb{N}}$ converges to the distribution $T$ if $$\forall \varphi \in \mathcal{D}, \lim_{n \rightarrow \infty} T_n(\varphi) = T(\varphi)$$
For $n$ a non-zero natural number, we consider the function $U_n$ zero on the negative reals, affine on the interval $[0, 1/n]$, equal to 1 for reals greater than $1/n$ and continuous on $\mathbb{R}$. a) Show that the sequence of regular distributions $(T_{U_n})_{n \in \mathbb{N}}$ converges to $T_U$. b) Show that $$\forall \varphi \in \mathcal{D} \quad T_{U_n}'(\varphi) = \int_0^{1/n} n\varphi(t) \mathrm{d}t$$ c) Deduce that the distribution $T_{U_n}'$ is regular and give a function $V_n$ such that $T_{V_n} = T_{U_n}'$. d) Sketch $V_n$ for $n = 1, 2, 4$. e) Show that if the sequence of distributions $(T_n)_{n \in \mathbb{N}}$ converges to the distribution $T$, then $(T_n')_{n \in \mathbb{N}}$ converges to $T'$. f) What is the limit of $T_{U_n}'$ as $n$ tends to infinity?

We say that the sequence of distributions $(T_n)_{n \in \mathbb{N}}$ converges to the distribution $T$ if $$\forall \varphi \in \mathcal{D}, \lim_{n \rightarrow \infty} T_n(\varphi) = T(\varphi)$$
For every non-zero natural number $n$, we consider the functions $$\begin{cases} f_n(x) = \dfrac{n}{1 + n^2 x^2} & \\ g_n(x) = nx^n & \text{if } x \in [0,1] \text{ and zero elsewhere} \\ h_n(x) = n^2 \sin nx & \text{if } x \in [-\pi/n, \pi/n] \text{ and zero elsewhere} \end{cases}$$
a) Verify that they belong to $\mathcal{F}_{sr}$. b) Study the variations of the functions $f_n, g_n$ and $h_n$ then sketch their graphs for $n = 1$ and $n = 2$. c) Study the convergence of the sequences of distributions $(T_{f_n}), (T_{g_n})$ and $(T_{h_n})$.

Let $\Omega$ be a non-empty open set of $\mathbb{R}^2$ and $P$ a polynomial of two variables, such that $P(x,y) = 0$ for all $(x,y) \in \Omega$.
a) Show that for all $(x,y) \in \Omega$, the open set $\Omega$ contains a subset of the form $I \times J$, where $I$ and $J$ are non-empty open intervals of $\mathbb{R}$ containing $x$ and $y$ respectively.
The use of a drawing will be appreciated; however, this drawing will not constitute a proof.
b) Deduce that $P$ is the zero polynomial.
One may reduce to studying polynomial functions of one variable.

Does this result hold if the set $\Omega$ has infinitely many elements but is not assumed to be open?

Let $m \in \mathbb{N}$. Justify that the vector space $\mathcal{P}_m$ is finite-dimensional and determine its dimension.

Determine a harmonic polynomial of degree 1, then of degree 2.

a) Show that the set of harmonic polynomials is a vector subspace of $\mathcal{P}$.
b) For all $m \geqslant 2$, we denote by $\Delta_m$ the restriction of $\Delta$ to $\mathcal{P}_m$. Show that $\operatorname{dim}(\operatorname{ker} \Delta_m) \geqslant 2m+1$.
c) What can be deduced about the dimension of the vector space of harmonic polynomials?

Determine a harmonic polynomial $H$ that satisfies $H(x,y) = f(x,y)$ for all $(x,y) \in C(0,1)$, where $f(x,y) = xy$.

Determine a harmonic polynomial $H$ that satisfies $H(x,y) = f(x,y)$ for all $(x,y) \in C(0,1)$, where $f(x,y) = x^4 - y^4$.

We take for $\Omega$ (only in this question) the interior of the equilateral triangle with vertices $(1,0), (-1/2, \sqrt{3}/2)$ and $(-1/2, -\sqrt{3}/2)$. We define, for all $\lambda \in \mathbb{R}^*$ and all pairs $(x_0, y_0) \in \mathbb{R}^2$: $$\Omega_{x_0, y_0, \lambda} = \left\{ \lambda(x,y) + (x_0, y_0) \mid (x,y) \in \Omega \right\}$$ Draw a figure on which both $\Omega$ and $\Omega_{2,1,1/2}$ appear.

Let $f : \Omega \rightarrow \mathbb{R}$ be a harmonic application of class $C^2$ such that $\partial_1 f$ and $\partial_2 f$ are of class $C^2$ on $\Omega$. Show that the applications $\partial_1 f$ and $\partial_2 f$ are also harmonic on $\Omega$.

Let $\lambda \in \mathbb{R}^*$ and $(x_0, y_0) \in \mathbb{R}^2$ be fixed. We define: $$\Omega_{x_0, y_0, \lambda} = \left\{ \lambda(x,y) + (x_0, y_0) \mid (x,y) \in \Omega \right\}$$ By which geometric transformation(s) is the set $\Omega_{x_0, y_0, \lambda}$ the image of $\Omega$? Justify that $\Omega_{x_0, y_0, \lambda}$ is an open set of $\mathbb{R}^2$.

Let $\lambda \in \mathbb{R}^*$ and $(x_0, y_0) \in \mathbb{R}^2$ be fixed. We define: $$\Omega_{x_0, y_0, \lambda} = \left\{ \lambda(x,y) + (x_0, y_0) \mid (x,y) \in \Omega \right\}$$ Let $g : \Omega_{x_0, y_0, \lambda} \rightarrow \mathbb{R}$ be a harmonic application.
Show that the application $(x,y) \mapsto g\left(\lambda(x,y) + (x_0, y_0)\right)$ is harmonic on $\Omega$.

Show that the applications $$h_1 : \left|\begin{array}{rll} \mathbb{R}^2 \backslash \{(0,0)\} & \rightarrow & \mathbb{R} \\ (x,y) & \mapsto & \ln(x^2 + y^2) \end{array}\right. \quad \text{and} \quad h_2 : \left|\begin{array}{rll} \mathbb{R}^2 \backslash \{(0,0)\} & \rightarrow & \mathbb{R} \\ (x,y) & \mapsto & \dfrac{1}{x^2 + y^2} \end{array}\right.$$ are harmonic.