We assume that there exists a function $\varphi$ from $\mathbb{R}^+$ to $\mathbb{R}$, continuous and integrable on $\mathbb{R}^+$, such that: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \varphi\left(\sqrt{x^2+y^2}\right)$.
Justify the convergence, for any real $q \geqslant 0$, of $\int_q^{+\infty} \frac{r\varphi(r)}{\sqrt{r^2 - q^2}}\,\mathrm{d}r$.

We assume that there exists a function $\varphi$ from $\mathbb{R}^+$ to $\mathbb{R}$, continuous and integrable on $\mathbb{R}^+$, such that: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \varphi\left(\sqrt{x^2+y^2}\right)$.
Prove that the Radon transform of $f$ is defined on $\mathbb{R}^2$ and that $$\forall q \in \mathbb{R}^+,\quad \forall \theta \in \mathbb{R} \quad \hat{f}(q,\theta) = 2\int_q^{+\infty} \frac{r\varphi(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r$$

We assume that there exists a function $\varphi$ from $\mathbb{R}^+$ to $\mathbb{R}$, continuous and integrable on $\mathbb{R}^+$, such that: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \varphi\left(\sqrt{x^2+y^2}\right)$.
Deduce that $\forall q \in \mathbb{R}^+,\ \frac{1}{2\pi} \int_0^{2\pi} \hat{f}(q,\theta)\,\mathrm{d}\theta = 2\int_q^{+\infty} \frac{r\bar{f}(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r$.

We consider a function $f$ belonging to $\mathcal{B}_1$ and we recall that $$\hat{f}(q,\theta) = \int_{-\infty}^{+\infty} f(q\cos\theta - t\sin\theta,\, q\sin\theta + t\cos\theta)\,\mathrm{d}t$$
Verify that $\hat{f}$ is defined on $\mathbb{R}^2$.

We consider a function $f$ belonging to $\mathcal{B}_1$ and we recall that $$\hat{f}(q,\theta) = \int_{-\infty}^{+\infty} f(q\cos\theta - t\sin\theta,\, q\sin\theta + t\cos\theta)\,\mathrm{d}t$$
Justify that for all $q$ and all $\theta$ we have $\hat{f}(-q, \theta+\pi) = \hat{f}(q,\theta)$.

We consider a function $f$ belonging to $\mathcal{B}_1$. We set $\bar{f}(r) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos t, r\sin t)\,\mathrm{d}t$.
Prove that $\bar{f}$ is of class $C^1$ on $\mathbb{R}$.

We consider a function $f$ belonging to $\mathcal{B}_1$. We set $\bar{f}(r) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos t, r\sin t)\,\mathrm{d}t$.
Prove that the function $r \mapsto r^2 \bar{f}(r)$ is bounded on $\mathbb{R}$.

We consider a function $f$ belonging to $\mathcal{B}_1$. We set $\bar{f}(r) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos t, r\sin t)\,\mathrm{d}t$.
Show that if we further assume that $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$, then $r \mapsto r^4 \bar{f}^{\prime}(r)$ is bounded on $\mathbb{R}$.

Let $h$ be a function of class $C^1$ on $\mathbb{R}^+$. We assume that $r \mapsto r^2 h(r)$ is bounded and we set $H(q) = \int_1^{+\infty} \frac{t\, h(qt)}{\sqrt{t^2-1}}\,\mathrm{d}t$.
Show that $H$ is continuous on $]0, +\infty[$.

Let $h$ be a function of class $C^1$ on $\mathbb{R}^+$. We assume that $r \mapsto r^2 h(r)$ is bounded and we set $H(q) = \int_1^{+\infty} \frac{t\, h(qt)}{\sqrt{t^2-1}}\,\mathrm{d}t$.
Show that near $+\infty$ we have $H(q) = O\left(\frac{1}{q^2}\right)$.

Let $h$ be a function of class $C^1$ on $\mathbb{R}^+$. We assume that $r \mapsto r^2 h(r)$ is bounded and we set $H(q) = \int_1^{+\infty} \frac{t\, h(qt)}{\sqrt{t^2-1}}\,\mathrm{d}t$.
Prove that if we further assume that $r \mapsto r^4 h^{\prime}(r)$ is bounded, then the function $H$ is of class $C^1$ on $]0, +\infty[$.

We consider a function $f$ in $\mathcal{B}_1$ whose partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$. We set, with the notations of part III: $$\forall q \in \mathbb{R}^+,\quad F(q) = \frac{1}{2\pi}\int_0^{2\pi} \hat{f}(q,\theta)\,\mathrm{d}\theta = 2\int_q^{+\infty} \frac{r\bar{f}(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r$$
Justify that $F$ is of class $C^1$ on $]0, +\infty[$ and that near $+\infty$ we have $F(q) = O\left(\frac{1}{q}\right)$.

We consider a function $f$ in $\mathcal{B}_1$ whose partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$. We set: $$\forall q \in \mathbb{R}^+,\quad F(q) = \frac{1}{2\pi}\int_0^{2\pi} \hat{f}(q,\theta)\,\mathrm{d}\theta = 2\int_q^{+\infty} \frac{r\bar{f}(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r$$
Prove: $\forall \varepsilon > 0,\ \int_\varepsilon^{+\infty} \frac{F^{\prime}(q)}{q}\,\mathrm{d}q = -\frac{F(\varepsilon)}{\varepsilon} + 2\int_\varepsilon^{+\infty} \frac{1}{q^2}\left(\int_q^{+\infty} \frac{r\bar{f}(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r\right)\mathrm{d}q$.

We consider a function $f$ in $\mathcal{B}_1$ whose partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$. We set: $$\forall q \in \mathbb{R}^+,\quad F(q) = \frac{1}{2\pi}\int_0^{2\pi} \hat{f}(q,\theta)\,\mathrm{d}\theta = 2\int_q^{+\infty} \frac{r\bar{f}(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r$$
We admit that we can interchange the two integrals and therefore that $$\forall \varepsilon > 0 \quad \int_\varepsilon^{+\infty} \left(\frac{1}{q^2}\int_q^{+\infty} \frac{r\bar{f}(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r\right)\mathrm{d}q = \int_\varepsilon^{+\infty} \left(\int_\varepsilon^r \frac{r\bar{f}(r)}{q^2\sqrt{r^2-q^2}}\,\mathrm{d}q\right)\mathrm{d}r$$
Deduce that $\forall \varepsilon > 0,\ \int_\varepsilon^{+\infty} \frac{F^{\prime}(q)}{q}\,\mathrm{d}q = -2\varepsilon \int_\varepsilon^{+\infty} \frac{\bar{f}(r)}{r\sqrt{r^2-\varepsilon^2}}\,\mathrm{d}r$.

We consider a function $f$ in $\mathcal{B}_1$ such that $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$. The Radon inversion formula states: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{-1}{\pi} \int_0^{+\infty} \frac{R_{x,y}^{\prime}(q)}{q}\,\mathrm{d}q$, where $R_{x,y}(q) = \frac{1}{2\pi}\int_0^{2\pi} \hat{f}(x\cos\theta + y\sin\theta + q, \theta)\,\mathrm{d}\theta$.
Establish the Radon inversion formula for this function $f$ at the point $(x,y) = (0,0)$.

We consider a function $f$ in $\mathcal{B}_1$ such that $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$. The Radon inversion formula states: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{-1}{\pi} \int_0^{+\infty} \frac{R_{x,y}^{\prime}(q)}{q}\,\mathrm{d}q$.
Are the hypotheses made on $f$ necessary for the Radon inversion formula to be verified at the point $(x,y) = (0,0)$?

We consider a function $f$ in $\mathcal{B}_1$ such that $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$. The Radon inversion formula states: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{-1}{\pi} \int_0^{+\infty} \frac{R_{x,y}^{\prime}(q)}{q}\,\mathrm{d}q$.
Propose a method to obtain the Radon inversion formula at any pair $(x,y)$ from the formula at $(0,0)$.

For $(x,y) \in D(0,1)$ fixed, we define the complex number $z = x + iy$ and we set for $t$ real: $$\mathrm{N}(x,y,t) = \frac{1 - |z|^2}{|z - e^{it}|^2} = \frac{1 - (x^2 + y^2)}{(x - \cos t)^2 + (y - \sin t)^2}$$
In the rest of this part, the pair $(x,y)$ is fixed in $D(0,1)$.
Show that $t \mapsto \mathrm{N}(x,y,t)$ is defined and continuous on $[0, 2\pi]$.

Let $f : C(0,1) \rightarrow \mathbb{R}$ be a continuous application. We define: $$\mathrm{N}_f(x,y) = \frac{1}{2\pi} \int_0^{2\pi} \mathrm{N}(x,y,t) f(\cos t, \sin t)\, \mathrm{d}t$$ on $D(0,1)$.
In this question, we fix $t_0 \in [0,2\pi]$, $(x,y) \in D(0,1)$ and $\varepsilon > 0$. Moreover, we denote, for all real $\delta > 0$: $$I_0^\delta = \left\{ t \in [0,2\pi] \mid \|(\cos t, \sin t) - (\cos t_0, \sin t_0)\|_2 \leqslant \delta \right\}$$
a) Show that $I_0^\delta$ is an interval or the union of two disjoint intervals.
The use of a drawing will be appreciated; however, this drawing will not constitute a proof.
b) Show, using the application $f$, the existence of a real $\delta > 0$ such that $$\left| \int_{t \in I_0^\delta} \mathrm{N}(x,y,t) \left( f(\cos t, \sin t) - f(\cos t_0, \sin t_0) \right) \mathrm{d}t \right| \leqslant \frac{\varepsilon}{2}$$
c) Let $\delta > 0$ be arbitrary. Show that, if $t \in [0,2\pi] \backslash I_0^\delta$ and $\|(x,y) - (\cos t_0, \sin t_0)\|_2 \leqslant \delta/2$, then $$|\mathrm{N}(x,y,t)| \leqslant 4 \frac{1 - (x^2 + y^2)}{\delta^2}$$
d) Deduce from the previous question that, for $\delta > 0$ fixed, there exists $\eta > 0$ such that, if $\|(x,y) - (\cos t_0, \sin t_0)\|_2 \leqslant \eta$, then $$\left| \int_{t \in [0,2\pi] \backslash I_0^\delta} \mathrm{N}(x,y,t) \left( f(\cos t, \sin t) - f(\cos t_0, \sin t_0) \right) \mathrm{d}t \right| \leqslant \frac{\varepsilon}{2}$$

We consider the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = \mathrm{e}^{-\frac{1}{2}x^{2}}$ for all $x \in \mathbb{R}$.
Show that there exists a real number $M > 0$ such that $f$ is an $M$-Lipschitz function.

Let $U$ and $V$ be two random variables on $(\Omega, \mathcal{A}, P)$ possessing a second moment and such that $V$ is not almost surely zero. Show that $E\left(U^{2}\right) E\left(V^{2}\right) - E(UV)^{2} \geqslant 0$ and that $E\left(U^{2}\right) E\left(V^{2}\right) - E(UV)^{2} = 0$ if and only if there exists $\lambda \in \mathbb{R}$ such that $\lambda V + U$ is almost surely zero.

Show that for all $f \in \mathscr { C } _ { b } ^ { 2 }$, $f$ admits an entropy relative to $\mu$ and that $$\operatorname { Ent } _ { \mu } ( f ) \leqslant \int \left| f ^ { \prime } ( x ) \right| ^ { 2 } \mu ( x ) d x$$ You may consider the family of functions defined by $f _ { \delta } = \delta + f ^ { 2 }$ for $\delta > 0$.

Let $m$ be a measure. We assume that there exists a constant $C > 0$ such that, if $f : \mathbb { R } \rightarrow \mathbb { R }$ is of class $\mathscr { C } ^ { 1 }$ and of bounded derivative $f ^ { \prime }$, then $f$ admits an entropy relative to $m$ and $$\operatorname { Ent } _ { m } ( f ) \leqslant C \int \left| f ^ { \prime } ( x ) \right| ^ { 2 } m ( x ) d x \tag{1}$$ Show that $\int \left( 1 + | x | + x ^ { 2 } \right) m ( x ) d x < + \infty$.

Let $m$ be a measure. We assume that there exists a constant $C > 0$ such that, if $f : \mathbb { R } \rightarrow \mathbb { R }$ is of class $\mathscr { C } ^ { 1 }$ and of bounded derivative $f ^ { \prime }$, then $f$ admits an entropy relative to $m$ and $$\operatorname { Ent } _ { m } ( f ) \leqslant C \int \left| f ^ { \prime } ( x ) \right| ^ { 2 } m ( x ) d x \tag{1}$$ Let $f \in \mathscr { C } _ { b } ^ { 1 }$. We wish to show that $f$ admits a variance relative to $m$ and that $$\operatorname { Var } _ { m } ( f ) \leqslant \frac { C } { 2 } \int \left| f ^ { \prime } ( x ) \right| ^ { 2 } m ( x ) d x \tag{2}$$
10a. Show that $fm$ and $f ^ { 2 } m$ are integrable, and that it suffices to show (2) in the case where we additionally have $\int f ( x ) m ( x ) d x = 0$ and $\int f ( x ) ^ { 2 } m ( x ) d x = 1$.
10b. Under the hypotheses of the previous question, show (2). You may apply (1) to the family of functions $f _ { \varepsilon } = 1 + \varepsilon f$ for $\varepsilon > 0$.

Let $m$ be a measure. We assume that there exists a constant $C > 0$ such that, if $f : \mathbb { R } \rightarrow \mathbb { R }$ is of class $\mathscr { C } ^ { 1 }$ and of bounded derivative $f ^ { \prime }$, then $f$ admits an entropy relative to $m$ and $$\operatorname { Ent } _ { m } ( f ) \leqslant C \int \left| f ^ { \prime } ( x ) \right| ^ { 2 } m ( x ) d x \tag{1}$$ Let $f$ be a function in $\mathscr { C } _ { b } ^ { 1 }$, such that for all $x \in \mathbb { R }$, we have $\left| f ^ { \prime } ( x ) \right| \leqslant 1$. We denote, for $\lambda \in \mathbb { R }$, $$H ( \lambda ) = \int e ^ { \lambda f ( x ) } m ( x ) d x$$ We admit that $H$ is of class $\mathscr { C } ^ { 1 }$ and that we obtain an expression of $H ^ { \prime } ( \lambda )$ by differentiating under the integral sign in the usual manner.
11a. Show that for all $\lambda \in \mathbb { R }$, $$\lambda H ^ { \prime } ( \lambda ) - H ( \lambda ) \ln H ( \lambda ) \leqslant \frac { C \lambda ^ { 2 } } { 4 } H ( \lambda )$$
11b. Deduce that for $\lambda \geqslant 0$, $$\int e ^ { \lambda f ( x ) } m ( x ) d x \leqslant \exp \left( \lambda \int f ( x ) m ( x ) d x + \frac { C \lambda ^ { 2 } } { 4 } \right) \tag{3}$$ You may study the function $\lambda \mapsto \frac { 1 } { \lambda } \ln H ( \lambda )$.