Let $(A, \vec{b})$ and $(A^{\prime}, \vec{b}^{\prime})$ be in $\mathrm{SO}(2) \times \mathbb{R}^2$. Show that $M(A, \vec{b}) M\left(A^{\prime}, \vec{b}^{\prime}\right) = M\left(A A^{\prime}, A \vec{b}^{\prime} + \vec{b}\right)$.
Is the application $\Phi : \left\{ \begin{array}{cll} G & \rightarrow & \mathbb{R}^2 \\ M(A, \vec{b}) & \mapsto & \vec{b} \end{array} \right.$ surjective? Is it injective?
Show that a parametrization of $\Delta\left(q, \vec{u}_\theta\right)$ is given by $\left\{ \begin{array}{l} x(t) = q\cos\theta - t\sin\theta \\ y(t) = q\sin\theta + t\cos\theta \end{array} \right.$ when $t$ ranges over $\mathbb{R}$.
Let $\varepsilon$ and $r$ be fixed such that $0 < \varepsilon < r$. With the change of variables $q = r\cos\theta$, establish that $$\int_\varepsilon^r \frac{\mathrm{d}q}{q^2\sqrt{r^2-q^2}} = \frac{\sqrt{r^2-\varepsilon^2}}{r^2\varepsilon}$$
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: $$\forall q \in \mathbb{R}^+,\quad F(q) = \frac{1}{2\pi}\int_0^{2\pi} \hat{f}(q,\theta)\,\mathrm{d}\theta = 2\int_q^{+\infty} \frac{r\bar{f}(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r$$ 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 $(0,0)$.