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 $n$ 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 $n$ 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 $f \in L^1(\mathbb{R})$ and $g \in L^\infty(\mathbb{R})$. Show that $f * g$ is bounded and that $\|f * g\|_\infty \leqslant \|f\|_1 \|g\|_\infty$.

Let $\lambda > 0$ and let $f \in L^1(\mathbb{R}) \cap \mathcal{C}^1(\mathbb{R})$ such that $\hat{f} \in L^1(\mathbb{R})$ and such that $\hat{f}$ is zero outside the segment $[-\lambda, \lambda]$. We denote by $r_\lambda$ the function from $\mathbb{R}$ to $\mathbb{C}$ such that $r_\lambda(x) = r(\lambda x)$ for all real $x$.
Deduce that, if $f \in L^\infty(\mathbb{R})$, there exists a constant $C \in \mathbb{R}_+^\star$, independent of $\lambda$ and of $f$, such that $$\left\|f'\right\|_\infty \leqslant C\lambda \|f\|_\infty$$

Let $n$ be a non-zero natural number. Let $f \in \mathcal{S}_n$. Using the result of question 13, deduce that $$\forall \theta \in \mathbb{R}, \quad |f'(\theta)| \leqslant n \|f\|_{L^\infty(\mathbb{R})} \tag{I.4}$$

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

Assume that $f \in L^1(\mathbb{R})$ and $g \in L^\infty(\mathbb{R})$. Show that $f * g$ is bounded and that $\|f*g\|_\infty \leqslant \|f\|_1 \|g\|_\infty$.

Let $\lambda > 0$ and let $f \in L^1(\mathbb{R}) \cap \mathcal{C}^1(\mathbb{R})$ such that $\hat{f} \in L^1(\mathbb{R})$ and such that $\hat{f}$ is zero outside the segment $[-\lambda, \lambda]$. Deduce that, if $f \in L^\infty(\mathbb{R})$, there exists a constant $C \in \mathbb{R}_+^*$, independent of $\lambda$ and of $f$, such that $$\left\|f'\right\|_\infty \leqslant C\lambda \|f\|_\infty$$

Let $A > 2$. Show that, for every $(p,q) \in \mathbb{N}^{2}$, $$\mathbb{E}\left(\sum_{\substack{1 \leqslant i \leqslant n \\ |\Lambda_{i,n}| \geqslant A}} |\Lambda_{i,n}|^{p}\right) \leqslant \frac{1}{A^{p+2q}} \mathbb{E}\left(\sum_{i=1}^{n} |\Lambda_{i,n}|^{2(p+q)}\right)$$

Let $A > 2$. Show that, for all $(p,q) \in \mathbb{N}^{2}$, $$\mathbb{E}\left(\sum_{\substack{1 \leqslant i \leqslant n \\ |\Lambda_{i,n}| \geqslant A}} |\Lambda_{i,n}|^{p}\right) \leqslant \frac{1}{A^{p+2q}} \mathbb{E}\left(\sum_{i=1}^{n} |\Lambda_{i,n}|^{2(p+q)}\right)$$

Let $x \in \mathbb{R} \backslash \mathbb{Z}$. Let $m \in \mathbb{N}$ such that $m > |x|$. We set, for $n \in \mathbb{N}$ such that $n > m$: $$v_{m,n}(x) = \prod_{k=m+1}^{n}\left(1 - \frac{\sin^2\left(\frac{\pi x}{2n+1}\right)}{\sin^2\left(\frac{k\pi}{2n+1}\right)}\right).$$ We denote $v_m(x)$ the limit of $\left(v_{m,n}(x)\right)_{n > m}$.
Show that, for $n \in \mathbb{N}$ such that $n > m$, we have $$1 \geqslant v_{m,n}(x) \geqslant \prod_{k=m+1}^{n}\left(1 - \frac{\pi^2 x^2}{4k^2}\right)$$ and deduce that $\lim_{m \rightarrow +\infty} v_m(x) = 1$.

We consider a power series $g \in O_1$ with non-negative real coefficients. We assume that there exist $a > 0, b > 0$ such that $$g \prec a\left(I + \frac{g^2}{b - g}\right)$$ and $h$ is the function from question (11). Show, by induction on $k$, that $(g)_k \leqslant (h)_k$ for all $k \in \mathbb{N}$, conclude.

We consider the special case of a series of the form $f = I + F, F \in O_2$. We denote $G := f^\dagger - I, G \in O_2$. Assume that there exist $s > 0$ and $\alpha \in ]0,1[$ such that $\hat{F}(s) \leqslant \alpha s$. Show then that for all $d \geqslant 2, \widehat{[G]_d}((1-\alpha)s) \leqslant \alpha s$. Conclude that $$\hat{G}((1-\alpha)s) \leqslant \alpha s.$$

We assume that $|\lambda| \notin \{0,1\}$ and that $\rho(f) > 0$. We consider the series $H \in O_2$ from part E. Show that the series $H$ satisfies $\hat{H} \prec \frac{1}{\omega} \hat{F} \circ (I + \hat{H})$.

Let $\ell$ be a strictly positive integer. We are given a sequence $\left( v _ { n } \right) _ { n \geqslant 0 }$ of vectors in $\mathbb { R } ^ { \ell }$ such that the series $\sum _ { n } \left\| v _ { n + 1 } - v _ { n } \right\|$ converges.
(a) Show that the sequence $\left( v _ { n } \right) _ { n \geqslant 0 }$ is convergent.
(b) Let $v ^ { * }$ denote the limit of this sequence. Bound $\left\| v _ { n } - v ^ { * } \right\|$ by means of a remainder of the sum of the series $\sum _ { n } \left\| v _ { n + 1 } - v _ { n } \right\|$.

We fix $f \in \mathcal{C}^{K}([0,1])$ and denote by $P$ the polynomial determined in question Q7. Deduce the inequality $$\left\|f^{(k)} - P^{(k)}\right\|_{\infty} \leqslant \left\|f^{(k+1)} - P^{(k+1)}\right\|_{\infty}$$ for all $k \in \llbracket 0, K-1 \rrbracket$.

Show that there exists a constant $C > 0$ for which the interpolation inequality $$\forall f \in \mathcal{C}^{K}([0,1]), \quad \max_{0 \leqslant k \leqslant K-1} \left\|f^{(k)}\right\|_{\infty} \leqslant \left\|f^{(K)}\right\|_{\infty} + C \sum_{\ell=1}^{K} \left|f\left(x_{\ell}\right)\right|$$ is satisfied.

Show that $|F(x)| \leqslant \frac{1}{3}$ for all $x \in [1,2]$, where $F(x) = \sum_{n=1}^{+\infty} f_n(x)$.

We fix $f \in \mathcal{C}^K([0,1])$ and denote by $P$ the polynomial determined in question Q7. Deduce the inequality $\left\|f^{(k)} - P^{(k)}\right\|_\infty \leqslant \left\|f^{(k+1)} - P^{(k+1)}\right\|_\infty$ for all $k \in \llbracket 0, K-1 \rrbracket$.

Show that there exists a constant $C > 0$ for which the interpolation inequality $$\forall f \in \mathcal{C}^K([0,1]), \quad \max_{0 \leqslant k \leqslant K-1} \left\|f^{(k)}\right\|_\infty \leqslant \left\|f^{(K)}\right\|_\infty + C \sum_{\ell=1}^K \left|f\left(x_\ell\right)\right|$$ is satisfied.

For all $n \in \mathbb{N}^\star$, let $f_n \in \mathcal{C}^2(]0,+\infty[)$ satisfy $f_n(1) = 0$, $f_n(2) = 0$ and $f_n^{\prime\prime}(x) = (-1)^n 2^{-nx^2}$ for all $x > 0$. Let $F : x \mapsto \sum_{n=1}^{+\infty} f_n(x)$. Show that $|F(x)| \leqslant \frac{1}{3}$ for all $x \in [1,2]$.