Show that the function $\left\lvert\, \begin{aligned} & \mathbb{R}_{+}^{*} \times \mathbb{R} \rightarrow \mathbb{R} \\ & (t, x) \mapsto g_{\sqrt{\sigma^{2}+2t}}(x) \end{aligned}\right.$ satisfies conditions i and iii, where:

• [i.] the diffusion equation: $\forall(t, x) \in \mathbb{R}_{+}^{*} \times \mathbb{R},\ \frac{\partial f}{\partial t}(t, x) = \frac{\partial^{2} f}{\partial x^{2}}(t, x)$;
• [iii.] the boundary condition: $\forall x \in \mathbb{R},\ \lim_{t \rightarrow 0^{+}} f(t, x) = g_{\sigma}(x)$.

We consider two functions of class $\mathcal{C}^2$, $u: \mathbb{R}^{*+} \to \mathbb{R}$ and $v: \mathbb{R} \to \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ The function $f$ is then a function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$, called a function with separable polar variables. Show that, if $f$ is not identically zero, then $v$ is $2\pi$-periodic.

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ The function $f$ is then a function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$, called a function with separable polar variables.
Show that, if $f$ is not identically zero, then $v$ is $2\pi$-periodic.

Let $f$ be a function satisfying the diffusion equation, the three domination conditions, and the boundary condition $\lim_{t\to 0^+} f(t,x) = g_\sigma(x)$. Justify that, for any real $t > 0$ and any real $\xi$, the function $x \mapsto f(t, x) \exp(-2\mathrm{i}\pi \xi x)$ is integrable on $\mathbb{R}$.

We assume (in this question only) that $c ( x ) = 0$ and $f ( x ) = 1$ for all $x \in [ 0,1 ]$. Let $n \in \mathbb{N}^*$, $h = \frac{1}{n+1}$, and $x_i = ih$. We denote by $u$ the exact solution of problem (1): $$\left\{ \begin{array} { l } - u ^ { \prime \prime } ( x ) + c ( x ) u ( x ) = f ( x ) , x \in [ 0,1 ] \\ u ( 0 ) = u ( 1 ) = 0 \end{array} \right.$$ and $(u_i)_{0 \leq i \leq n+1}$ the unique family satisfying $$\left\{ \begin{array} { l } - \frac { 1 } { h ^ { 2 } } \left( u _ { i + 1 } + u _ { i - 1 } - 2 u _ { i } \right) + c \left( x _ { i } \right) u _ { i } = f \left( x _ { i } \right) , \text { for } 1 \leq i \leq n \\ u _ { 0 } = u _ { n + 1 } = 0 \end{array} \right.$$ Show that for all $i \in \{ 0 , \ldots , n + 1 \}$, we have
$$u _ { i } = u \left( x _ { i } \right) = \frac { 1 } { 2 } x _ { i } \left( 1 - x _ { i } \right)$$

We consider two functions of class $\mathcal{C}^2$, $u: \mathbb{R}^{*+} \to \mathbb{R}$ and $v: \mathbb{R} \to \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ Show that, if $f$ is harmonic and not identically zero on $\mathbb{R}^2 \setminus \{(0,0)\}$, then there exists a real number $\lambda$ such that $u$ is a solution of the differential equation (II.1) $$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$ and $v$ is a solution of the differential equation (II.2) $$z''(\theta) + \lambda z(\theta) = 0$$

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ The function $f$ is then a function of class $\mathcal{C}^2$ on $\mathbb{R}^2 \setminus \{(0,0)\}$, called a function with separable polar variables.
Show that, if $f$ is harmonic and not identically zero on $\mathbb{R}^2 \setminus \{(0,0)\}$, then there exists a real number $\lambda$ such that $u$ is a solution of the differential equation (II.1) $$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$ and $v$ is a solution of the differential equation (II.2) $$z''(\theta) + \lambda z(\theta) = 0$$

We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by $$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$
Show that $\varphi$ is continuous on $\mathbb { R }$ and of class $C ^ { \infty }$ on $\mathbb { R } \backslash \{ 1 \}$.

We define $\hat{f}$ on $\mathbb{R}_{+}^{*} \times \mathbb{R}$ by: $\forall(t, \xi) \in \mathbb{R}_{+}^{*} \times \mathbb{R},\ \hat{f}(t, \xi) = \int_{-\infty}^{+\infty} f(t, x) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x$. Show that, for any real number $\xi$, $\lim_{t \rightarrow 0^{+}} \hat{f}(t, \xi) = \widehat{g_{\sigma}}(\xi)$. One may use any sequence $\left(t_{n}\right)_{n \in \mathbb{N}}$ of strictly positive reals converging to zero.

We assume here that $\lambda = 0$. What are the $2\pi$-periodic solutions of the differential equation (II.2): $$z''(\theta) + \lambda z(\theta) = 0$$

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ We assume here that $\lambda = 0$. What are the $2\pi$-periodic solutions of (II.2): $$z''(\theta) + \lambda z(\theta) = 0$$

Show that, for any real $\xi$ and any real $t > 0$, $\frac{\partial \hat{f}}{\partial t}(t, \xi) = \int_{-\infty}^{+\infty} \frac{\partial f}{\partial t}(t, x) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x$.

We assume here that $\lambda = 0$. Solve the differential equation (II.1) $$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$ on $\mathbb{R}^{+*}$.

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ We assume here that $\lambda = 0$. Solve (II.1) on $\mathbb{R}^{+*}$: $$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$

By noting that $\int_{-\infty}^{+\infty} \frac{\partial f}{\partial t}(t, x) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x = \mathcal{F}\left(\frac{\partial f}{\partial t}(t, \cdot)\right)(\xi)$ and using question 7, show that, for any real $\xi$ and any real $t > 0$, $\frac{\partial \hat{f}}{\partial t}(t, \xi) = -4\pi^{2} \xi^{2} \hat{f}(t, \xi)$.

We assume here that $\lambda = 0$. Deduce, in the case $\lambda = 0$, the harmonic functions with separable polar variables.

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ We assume here that $\lambda = 0$. Deduce from this, in the case $\lambda = 0$, the harmonic functions with separable polar variables.

Show that, for all $\xi \in \mathbb{R}$, there exists a real $K(\xi)$ such that for all $t \in \mathbb{R}_{+}^{*}$, $\hat{f}(t, \xi) = K(\xi) \exp\left(-4\pi^{2} \xi^{2} t\right)$.

Let $\alpha$ be a real number. We denote $f_{\alpha} : x \longmapsto (1-x)^{-\alpha}$.
Specify the domain of definition $D$ of $f_{\alpha}$. Justify that $f_{\alpha}$ is of class $C^{1}$ on $D$ and give a first-order linear differential equation satisfied by $f_{\alpha}$ on $D$.

Let $\alpha$ be a real number. We denote $f_{\alpha} : x \longmapsto (1-x)^{-\alpha}$.
State Cauchy's theorem for a scalar first-order linear differential equation and prove that, for all $x \in ]-1,1[$, $$f_{\alpha}(x) = \sum_{n=0}^{+\infty} L_{n}(\alpha) \frac{x^{n}}{n!}.$$

Let $g$ be the sum of the power series $\sum_{n \in \mathbb{N}} \frac{\alpha_n}{n!} x^n$ with radius of convergence $R \geqslant \pi/2$. Show $$\forall x \in I, \quad 2g^{\prime}(x) = g(x)^2 + 1.$$

We consider a general balanced urn. For all real $x, u$ and $v$, we set $$H(x, u, v) = \sum_{n=0}^{+\infty} P_{n}(u,v) \frac{x^{n}}{n!}$$ defined on $D_{\rho} = ]-\rho, \rho[ \times ]0,2[^{2}$ for $\rho$ sufficiently small.
Verify that $H(0, u, v) = u^{a_{0}} v^{b_{0}}$ and then that $H$ is a solution on $D_{\rho}$ of the partial differential equation $$\frac{\partial H}{\partial x}(x,u,v) = u^{a+1} v^{b} \frac{\partial H}{\partial u}(x,u,v) + u^{c} v^{d+1} \frac{\partial H}{\partial v}(x,u,v).$$

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 $p_k(x) = x^k$. Deduce $(T(p))''$ for all $p \in F$.

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. Deduce $(T(p))''$ for all $p \in F$.

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 $k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$ For all $f \in E$, show that $T(f)$ is of class $\mathcal{C}^2$ then that $T(f)'' = -f$.