Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$.
Show that the sequence of functions $\left(x \mapsto \prod_{k=1}^{n} p_k^{\nu_{p_k}(x)}\right)_{n \geqslant 1}$ from $\mathbb{N}^*$ to $\mathbb{N}^*$ converges pointwise to the identity function.

Let $n$ be a non-zero natural number. Deduce from questions 3 and 14 that $$\forall P \in \mathbb{C}_n[X], \quad \forall x \in [-1,1], \quad \left|P'(x)\sqrt{1-x^2}\right| \leqslant n \|P\|_{L^\infty([-1,1])}$$

Let $E$ be the set of continuous functions $f$ from $I$ to $\mathbb{R}$ such that $f^2 w$ is integrable on $I$.
Show that, for all functions $f$ and $g$ in $E$, the product $fgw$ is integrable on $I$. You may use the inequality $\forall (a,b) \in \mathbb{R}^2, |ab| \leqslant \frac{1}{2}(a^2 + b^2)$, after justifying it.

Let $n$ be a non-zero natural number. Show that $$\forall Q \in \mathbb{C}_{n-1}[X], \quad |Q(1)| \leqslant n \sup_{-1 \leqslant x \leqslant 1} \left|Q(x)\sqrt{1-x^2}\right|.$$ One may consider $f : \theta \mapsto Q(\cos\theta)\sin\theta$ and verify that $f \in \mathcal{S}_n$.

Let $E$ be the set of continuous functions $f$ from $I$ to $\mathbb{R}$ such that $f^2 w$ is integrable on $I$.
Show that $E$ is an $\mathbb{R}$-vector space.

Let $n$ be a non-zero natural number. Let $R \in \mathbb{C}_{n-1}[X]$ and $t \in [-1,1]$. Show that $$|R(t)| \leqslant n \sup_{-1 \leqslant x \leqslant 1} \left|R(x)\sqrt{1-x^2}\right|.$$ One may consider the polynomial $S_t(X) = R(tX)$.

Let $E$ be the set of continuous functions $f$ from $I$ to $\mathbb{R}$ such that $f^2 w$ is integrable on $I$. For all functions $f$ and $g$ in $E$, we set $$\langle f, g \rangle = \int_I f(x) g(x) w(x)\,\mathrm{d}x.$$
Show that we thus define an inner product on $E$.

Let $n$ be a non-zero natural number. Deduce that, for all $P$ in $\mathbb{C}_n[X]$, $$\left\|P'\right\|_{L^\infty([-1,1])} \leqslant n^2 \|P\|_{L^\infty([-1,1])}$$

We assume that, for every integer $k \in \mathbb{N}$, the function $x \mapsto x^k w(x)$ is integrable on $I$. Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of orthogonal polynomials associated with the weight $w$ (monic, $\deg(p_n) = n$, and $\langle p_i, p_j \rangle = 0$ for $i \neq j$).
Let $n \in \mathbb{N}^*$. We denote by $x_1, \ldots, x_k$ the distinct roots of $p_n$ that are in $\mathring{I}$ and $m_1, \ldots, m_k$ their respective multiplicities. We consider the polynomial $$q(X) = \prod_{i=1}^k (X - x_i)^{\varepsilon_i}, \quad \text{with } \varepsilon_i = \begin{cases} 1 & \text{if } m_i \text{ is odd} \\ 0 & \text{if } m_i \text{ is even.} \end{cases}$$
By studying $\langle p_n, q \rangle$, show that $p_n$ has $n$ distinct roots in $\mathring{I}$.

Let $n$ be a non-zero natural number. Can there be equality in the inequality $$\left\|P'\right\|_{L^\infty([-1,1])} \leqslant n^2 \|P\|_{L^\infty([-1,1])}?$$

Let $f \in L^1(\mathbb{R})$, where the Fourier transform of $f$ is defined by $$\forall \xi \in \mathbb{R}, \quad \hat{f}(\xi) = \int_{-\infty}^{+\infty} f(x) \mathrm{e}^{-\mathrm{i}x\xi} \,\mathrm{d}x$$ Show that, for every function $f \in L^1(\mathbb{R})$, $\hat{f}$ is defined and continuous on $\mathbb{R}$.

Show that the map $f \mapsto \hat{f}$ is a continuous linear map from the normed vector space $(L^1(\mathbb{R}), \|\cdot\|_1)$ to the normed vector space $(L^\infty(\mathbb{R}), \|\cdot\|_\infty)$.

In $\mathbb { R } [ X ]$ equipped with an inner product $( \cdot \mid \cdot )$, let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system (a sequence of polynomials that is orthogonal and where each $V_n$ is monic of degree $n$). Show that, for all $n \in \mathbb { N }$, the family $\left( V _ { 0 } , V _ { 1 } , \ldots , V _ { n } \right)$ is an orthogonal basis of the vector space $\mathbb { R } _ { n } [ X ]$ of polynomials with real coefficients of degree at most $n$.

We call a cycle of length $k$ with values in $\llbracket 1,n \rrbracket$, any $(k+1)$-tuple $\vec{\imath} = (i_{1}, i_{2}, \ldots, i_{k}, i_{1})$ of elements of $\llbracket 1,n \rrbracket$. We denote $|\vec{\imath}|$ the number of distinct vertices of the cycle $\vec{\imath}$.
Show that the number of cycles of length $k$ in $\llbracket 1,n \rrbracket$ passing through $\ell$ distinct vertices is at most $n^{\ell} \ell^{k}$.

Let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system in $\mathbb { R } [ X ]$ equipped with an inner product $( \cdot \mid \cdot )$. Let $n \in \mathbb { N }$ and $P \in \mathbb { R } [ X ]$ such that $\operatorname { deg } P < n$. Show that $\left( V _ { n } \mid P \right) = 0$.

We call a cycle of length $k$ with values in $\llbracket 1,n \rrbracket$, any $(k+1)$-tuple $\vec{\imath} = (i_{1}, i_{2}, \ldots, i_{k}, i_{1})$ of elements of $\llbracket 1,n \rrbracket$. We denote $|\vec{\imath}|$ the number of distinct vertices of the cycle $\vec{\imath}$.
Deduce that $$\frac{1}{n^{1+k/2}} \sum_{\substack{\vec{\imath} \in \llbracket 1,n \rrbracket^{k} \\ |\vec{\imath}| \leqslant (k+1)/2}} \left|\mathbb{E}\left(X_{i_{1}i_{2}} X_{i_{2}i_{3}} \cdots X_{i_{k-1}i_{k}} X_{i_{k}i_{1}}\right)\right| \xrightarrow{n \rightarrow +\infty} 0.$$

Let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system in $\mathbb { R } [ X ]$ equipped with an inner product $( \cdot \mid \cdot )$. Let $\left( W _ { n } \right) _ { n \in \mathbb { N } }$ be another orthogonal system. Show that $\forall n \in \mathbb { N } , W _ { n } = V _ { n }$.

We classify cycles of length $k$ into three subsets:

• the set $\mathcal{A}_{k}$, consisting of cycles where at least one edge appears only once;
• the set $\mathcal{B}_{k}$, consisting of cycles where all edges appear exactly twice;
• the set $\mathcal{C}_{k}$, consisting of cycles where all edges appear at least twice and there exists at least one that appears at least three times.

Show that, if the cycle $(i_{1}, i_{2}, \ldots, i_{k}, i_{1})$ belongs to $\mathcal{A}_{k}$, then $$\mathbb{E}\left(X_{i_{1}i_{2}} X_{i_{2}i_{3}} \cdots X_{i_{k-1}i_{k}} X_{i_{k}i_{1}}\right) = 0.$$

Let $[ a , b ]$ be a closed bounded interval of $\mathbb { R }$. If $\phi : [ a , b ] \rightarrow [ a , b ]$ is continuous, show that $\phi$ has at least one fixed point.

Let $g : [ 0,1 ] \rightarrow [ 0,1 ]$ be an increasing function (but not necessarily continuous). Show that $g$ has at least one fixed point. Hint: one may consider the set $$E = \{ x \in [ 0,1 ] ; x \leqslant g ( x ) \} .$$

Let $C$ be a non-empty, convex and closed subset of $\mathbb{R}^d$ and $x \in \mathbb{R}^d$, consider: $$\inf_{y \in C} \|x - y\|^2. \tag{1}$$ Show that (1) has a unique solution (that is, there exists a unique $y \in C$ such that $\|x - y\|^2 \leqslant \|x - z\|^2$ for all $z \in C$) which we will call the projection of $x$ onto $C$ and denote $\operatorname{proj}_C(x)$. Show that $x = \operatorname{proj}_C(x)$ if and only if $x \in C$.

Let $C$ be a non-empty, convex and closed subset of $\mathbb{R}^d$ and $x \in \mathbb{R}^d$, consider: $$\inf_{y \in C} \|x - y\|^2. \tag{1}$$ Show that (1) has a unique solution (that is, there exists a unique $y \in C$ such that $\|x - y\|^2 \leq \|x - z\|^2$ for all $z \in C$) which we will call the projection of $x$ onto $C$ and denote $\operatorname{proj}_C(x)$. Show that $x = \operatorname{proj}_C(x)$ if and only if $x \in C$.

Show that $\langle \cdot , \cdot \rangle$ is an inner product on $\mathbb { R } _ { n - 1 } [ X ]$, where $$\langle P , Q \rangle = \sum _ { k = 1 } ^ { n } P \left( a _ { k } \right) Q \left( a _ { k } \right).$$

Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that the functions $g$ and $D$ are odd.

Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that the functions $g$ and $D$ are periodic with period 1.