In what follows we restrict to the case $n = 2$ from Part I.
Let $x \in C ^ { 2 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { 2 } \right) \right. \right.$ be a solution of equation (1). Show that there exist four real numbers $c _ { 1 } , c _ { 2 } , \varphi _ { 1 } , \varphi _ { 2 }$ such that: $$\forall t \in \left[ 0 , + \infty \left[ , x ( t ) = \sum _ { i = 1 } ^ { 2 } c _ { i } \cos \left( t \sqrt { \lambda _ { i } } + \varphi _ { i } \right) e _ { i } \right. \right.$$ (We recall that the two vectors $e _ { 1 } , e _ { 2 }$ are introduced in Question 4).

We denote by $C _ { 2 \pi , 2 \pi } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$ the set of continuous functions $f : \mathbb { R } ^ { 2 } \rightarrow \mathbb { C }$ such that: $$\forall \left( \theta _ { 1 } , \theta _ { 2 } \right) \in \mathbb { R } ^ { 2 } , f \left( \theta _ { 1 } + 2 \pi , \theta _ { 2 } \right) = f \left( \theta _ { 1 } , \theta _ { 2 } \right) = f \left( \theta _ { 1 } , \theta _ { 2 } + 2 \pi \right)$$
Let $f \in C _ { 2 \pi , 2 \pi } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$. Prove that $$\sup _ { \left( \theta _ { 1 } , \theta _ { 2 } \right) \in \mathbb { R } ^ { 2 } } \left| f \left( \theta _ { 1 } , \theta _ { 2 } \right) \right| = \sup _ { \left( \theta _ { 1 } , \theta _ { 2 } \right) \in [ 0,2 \pi ] ^ { 2 } } \left| f \left( \theta _ { 1 } , \theta _ { 2 } \right) \right|$$ Deduce that $\left( \theta _ { 1 } , \theta _ { 2 } \right) \mapsto \left| f \left( \theta _ { 1 } , \theta _ { 2 } \right) \right|$ is bounded on $\mathbb { R } ^ { 2 }$ and attains its supremum.

We assume that $\frac { \sqrt { \lambda _ { 1 } } } { \sqrt { \lambda _ { 2 } } }$ is not a rational number. We denote by $C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$ the set of functions $f \in C _ { 2 \pi , 2 \pi } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$ such that the two partial derivatives $\frac { \partial f } { \partial \theta _ { 1 } } , \frac { \partial f } { \partial \theta _ { 2 } }$ exist at every point of $\mathbb { R } ^ { 2 }$ and both define continuous functions on $\mathbb { R } ^ { 2 }$.
We set $\forall t \in \left[ 0 , + \infty \left[ , \theta ( t ) = \left( t \sqrt { \lambda _ { 1 } } + \varphi _ { 1 } , t \sqrt { \lambda _ { 2 } } + \varphi _ { 2 } \right) \right. \right.$.
The Ergodic Theorem states: Let $f \in C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$. Then, $$\lim _ { T \rightarrow + \infty } \frac { 1 } { T } \int _ { 0 } ^ { T } f \circ \theta ( t ) d t = ( 2 \pi ) ^ { - 2 } \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 } \tag{4}$$
Let $j , l \in \mathbb { Z }$. Prove the Ergodic Theorem in the special case of the function $\left( \theta _ { 1 } , \theta _ { 2 } \right) \mapsto f \left( \theta _ { 1 } , \theta _ { 2 } \right) = e ^ { i \theta _ { 1 } j } e ^ { i \theta _ { 2 } l }$. (In the case where $( j , l ) \neq ( 0,0 )$ one may verify that $j \sqrt { \lambda _ { 1 } } + l \sqrt { \lambda _ { 2 } }$ is non-zero and then one may calculate each side of (4) separately in this special case).

Let $\mathcal{A}$ be an open bounded non-empty subset of $\mathbb{R}^{2}$. We denote by $C(\mathcal{A})$ the set of continuous functions $f$ from $\mathbb{R}^{2}$ to $[0,1]$ such that $\forall (x,y) \in \mathbb{R}^{2} \setminus \mathcal{A},\, f(x,y) = 0$ (in other words $f$ is zero outside $\mathcal{A}$). Show that the supremum $$\sup_{f \in C(\mathcal{A})} \iint_{\mathbb{R}^{2}} f(x,y)\,dx\,dy$$ exists and defines a real number denoted $V(\mathcal{A})$.

We assume that $\frac { \sqrt { \lambda _ { 1 } } } { \sqrt { \lambda _ { 2 } } }$ is not a rational number. We fix an arbitrary element $f \in C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$. For each $k \in \mathbb { N } ^ { * }$ we set: $$\forall ( u , v ) \in \mathbb { R } ^ { 2 } , f _ { k } ( u , v ) = \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } R _ { k } \left( u - \theta _ { 1 } \right) R _ { k } \left( v - \theta _ { 2 } \right) f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 }$$
Let $k \in \mathbb { N } ^ { * }$. Prove that there exist $( 2 k + 1 ) ^ { 2 }$ complex numbers $\left( a _ { j , l } \right) _ { - k \leq j , l \leq k }$ such that for every $( u , v ) \in \mathbb { R } ^ { 2 } : f _ { k } ( u , v ) = \sum _ { - k \leq j , l \leq k } a _ { j , l } e ^ { i u j } e ^ { i v l }$. Justify that the function $f _ { k }$ satisfies the Ergodic Theorem.

We assume that $\frac { \sqrt { \lambda _ { 1 } } } { \sqrt { \lambda _ { 2 } } }$ is not a rational number. We fix an arbitrary element $f \in C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$. For each $k \in \mathbb { N } ^ { * }$ we set: $$\forall ( u , v ) \in \mathbb { R } ^ { 2 } , f _ { k } ( u , v ) = \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } R _ { k } \left( u - \theta _ { 1 } \right) R _ { k } \left( v - \theta _ { 2 } \right) f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 }$$
Let $\epsilon \in ] 0 , \pi [$ and $k \in \mathbb { N } ^ { * }$. By writing $f _ { k } ( u , v ) - f ( u , v )$ as a sum of two terms and applying Question 10, prove that for every $( u , v ) \in \mathbb { R } ^ { 2 }$: $$\left| f _ { k } ( u , v ) - f ( u , v ) \right| \leq 2 \epsilon \left( \left\| \frac { \partial f } { \partial \theta _ { 1 } } \right\| + \left\| \frac { \partial f } { \partial \theta _ { 2 } } \right\| \right) + 8 \pi \| f \| d _ { k } ( \epsilon )$$

We assume that $\frac { \sqrt { \lambda _ { 1 } } } { \sqrt { \lambda _ { 2 } } }$ is not a rational number. We fix an arbitrary element $f \in C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$. For each $k \in \mathbb { N } ^ { * }$ we set: $$\forall ( u , v ) \in \mathbb { R } ^ { 2 } , f _ { k } ( u , v ) = \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } R _ { k } \left( u - \theta _ { 1 } \right) R _ { k } \left( v - \theta _ { 2 } \right) f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 }$$
Prove the Ergodic Theorem for the function $f$. (One may set $M = 2 \left( \left\| \frac { \partial f } { \partial \theta _ { 1 } } \right\| + \left\| \frac { \partial f } { \partial \theta _ { 2 } } \right\| \right) + 8 \pi \| f \|$. For given $\epsilon > 0$, one may choose $k \in \mathbb { N } ^ { * }$ such that $d _ { k } ( \epsilon ) < \epsilon$. Next, one may apply Question 14 to $f _ { k }$ and consider $T _ { 0 } > 0$ such that for every $T \geq T _ { 0 }$: $$\left| \frac { 1 } { T } \int _ { 0 } ^ { T } f _ { k } \circ \theta ( t ) d t - ( 2 \pi ) ^ { - 2 } \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } f _ { k } \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 } \right| < \epsilon .)$$

Let $u : \mathbb{R}^{2} \rightarrow ]0,+\infty[$ be a continuous and log-concave function in the sense of Part II. Prove that the preceding inequality remains true if we replace the application $V$ by the application $\gamma$ defined for all open bounded (non-empty) subsets $\mathcal{A}$ of $\mathbb{R}^{2}$ by $$\gamma(\mathcal{A}) = \sup_{f \in C(\mathcal{A})} \iint_{\mathbb{R}^{2}} f(x,y)\,u(x,y)\,dx\,dy$$

Let $a , b \in ] 0,2 \pi [$ such that $a < b$. We denote by $\phi _ { a , b } : \mathbb { R } \rightarrow \mathbb { R }$ the continuous $2 \pi$-periodic function defined as follows. The function $\phi _ { a , b }$ is zero on $[ 0 , a ]$ and $[ b , 2 \pi ]$. For every $t \in [ a , b ] , \phi _ { a , b } ( t ) = \sin ^ { 2 } \left( \frac { \pi } { b - a } ( t - a ) \right)$.
Recall that every non-empty open set of $] - 1,1 [ ^ { 2 }$ contains a rectangle of the form $] \cos b , \cos a [ \times ] \cos d , \cos c [$ where $0 < a < b < \pi$ and $0 < c < d < \pi$.
Consider the solution $x ( t ) = \sum _ { i = 1 } ^ { 2 } \cos \left( t \sqrt { \lambda _ { i } } + \varphi _ { i } \right) e _ { i }$ of (1) obtained by taking $c _ { 1 } = c _ { 2 } = 1$ in (2). Let $\Omega$ be a non-empty open set of $\left\{ u e _ { 1 } + v e _ { 2 } \mid u , v \in ] - 1,1 [ \right\}$. Prove that there exists $t \in [ 0 , + \infty [$ such that $x ( t ) \in \Omega$. (One may reason by contradiction and justify the existence of a function of the type $\left( \theta _ { 1 } , \theta _ { 2 } \right) \mapsto \phi _ { a , b } \left( \theta _ { 1 } \right) \phi _ { c , d } \left( \theta _ { 2 } \right) = \Phi \left( \theta _ { 1 } , \theta _ { 2 } \right)$ such that $\Phi ( \theta ( t ) )$ is zero for all $t \in [ 0 , + \infty [)$.

In each of the two cases below, show that $f * g$ is defined and bounded on $\mathbb{R}$ and give an upper bound for $\|f * g\|_{\infty}$ which may involve $\|\cdot\|_{1}$, $\|\cdot\|_{2}$ or $\|\cdot\|_{\infty}$. a) $f \in L^{1}(\mathbb{R}),\ g \in C_{b}(\mathbb{R})$; b) $f, g \in L^{2}(\mathbb{R})$.

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 $L^{2}(\mathbb{R})$. Show that a function $h$ is uniformly continuous on $\mathbb{R}$ if and only if $\lim_{\alpha \rightarrow 0} \|T_{\alpha}(h) - h\|_{\infty} = 0$.

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 $T_{\alpha}(f * g) = \left(T_{\alpha}(f)\right) * g$.

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 $\left\|T_{\alpha}(f * g) - f * g\right\|_{\infty} \leqslant \left\|T_{\alpha}(f) - f\right\|_{2} \times \|g\|_{2}$.

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 $L^{2}(\mathbb{R})$. Deduce that $f * g$ is uniformly continuous on $\mathbb{R}$ in the case where $f$ 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 $L^{2}(\mathbb{R})$. Show that $f * g$ is uniformly continuous on $\mathbb{R}$ in the general case.

Assume that $f \in L^{1}(\mathbb{R})$ and $g \in C_{b}(\mathbb{R})$. a) Show that $f * g$ is continuous. b) Show that if $g$ is uniformly continuous on $\mathbb{R}$, then $f * g$ is uniformly continuous on $\mathbb{R}$.

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 $k$.

In this question I.C.3, assume that $g$ is continuous, $2\pi$-periodic and of class $C^{1}$ piecewise. a) State without proof the theorem on Fourier series applicable to continuous, $2\pi$-periodic functions of class $C^{1}$ piecewise. b) Show that $f * g$ is $2\pi$-periodic and is the sum of its Fourier series. Specify the Fourier coefficients of $f * g$ using the Fourier coefficients of $g$ and integrals involving $f$.

Let $f \in C_{b}(\mathbb{R})$ and let $(\delta_{n})$ be a sequence of functions forming an approximate identity. Show that the sequence $\left(f * \delta_{n}\right)_{n \in \mathbb{N}}$ converges pointwise to $f$ on $\mathbb{R}$.

Let $f \in C_{b}(\mathbb{R})$ and let $(\delta_{n})$ be a sequence of functions forming an approximate identity. Show that if $f$ has compact support, then the sequence $\left(f * \delta_{n}\right)_{n \in \mathbb{N}}$ converges uniformly to $f$ on $\mathbb{R}$.

For every natural number $n$, we denote by $h_{n}$ the function defined on $[-1,1]$ by $$h_{n}(t) = \frac{\left(1 - t^{2}\right)^{n}}{\lambda_{n}}$$ and zero outside $[-1,1]$, where the real number $\lambda_{n}$ is given by the formula $$\lambda_{n} = \int_{-1}^{1} \left(1 - t^{2}\right)^{n} \mathrm{~d}t$$ a) Show that the sequence of functions $\left(h_{n}\right)_{n \in \mathbb{N}}$ is an approximate identity. b) Show that if $f$ is a continuous function with support included in $\left[-\frac{1}{2}, \frac{1}{2}\right]$, then $f * h_{n}$ is a polynomial function on $\left[-\frac{1}{2}, \frac{1}{2}\right]$ and zero outside the interval $\left[-\frac{3}{2}, \frac{3}{2}\right]$. c) Deduce a proof of Weierstrass's theorem: every complex continuous function on a closed interval of $\mathbb{R}$ is the uniform limit on this interval of a sequence of polynomial functions.

Does there exist a function $g \in C_{b}(\mathbb{R})$ such that for every function $f$ in $L^{1}(\mathbb{R})$, we have $f * g = f$?

For any function $f \in L^{1}(\mathbb{R})$, the Fourier transform of $f$ is the function $\hat{f}$ defined by $$\forall x \in \mathbb{R} \quad \hat{f}(x) = \int_{\mathbb{R}} f(t) \mathrm{e}^{-\mathrm{i}xt} \mathrm{~d}t$$ For any function $f \in L^{1}(\mathbb{R})$, show that $\hat{f}$ belongs to $C_{b}(\mathbb{R})$.