Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. We seek to solve the Dirichlet problem on the unit disk; we need to determine the function or functions $f$ defined and continuous on $\overline{D(0,1)}$, of class $\mathcal{C}^2$ on $D(0,1)$, and such that $$\begin{cases} \Delta f = 0 \text{ on } D(0,1) \\ \forall t \in \mathbb{R}, f(\cos(t), \sin(t)) = h(t) \end{cases}$$ For this, we set, for any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show that the function $z \mapsto \frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}$ is expandable in a power series for $|z| < 1$ and calculate its power series expansion. Deduce that the function $(x,y) \mapsto g(x + \mathrm{i}y)$ is a harmonic function on $D(0,1)$.

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, set $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z)\, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show that, for any complex number $z$ such that $|z| < 1$, $\frac{1}{2\pi} \int_0^{2\pi} \mathcal{P}(t,z)\, \mathrm{d}t = 1$.

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show that, for any complex number $z$ such that $|z| < 1$, $\frac{1}{2\pi} \int_0^{2\pi} \mathcal{P}(t,z) \, \mathrm{d}t = 1$.

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, set $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z)\, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show that, for all $r \in [0,1[$ and all real $t$ and $\theta$, $$\mathcal{P}\left(t, r\mathrm{e}^{\mathrm{i}\theta}\right) = \frac{1 - r^2}{1 - 2r\cos(t-\theta) + r^2}$$

Let $h$ be a function from $\mathbb{R}$ to $\mathbb{R}$, continuous and $2\pi$-periodic on $\mathbb{R}$. For any complex number $z$ such that $|z| < 1$, $$g(z) = \frac{1}{2\pi} \int_0^{2\pi} h(t) \mathcal{P}(t,z) \, \mathrm{d}t \quad \text{where} \quad \mathcal{P}(t,z) = \operatorname{Re}\left(\frac{\mathrm{e}^{\mathrm{i}t} + z}{\mathrm{e}^{\mathrm{i}t} - z}\right)$$ Show that, for all $r \in [0,1[$ and all real $t$ and $\theta$, $$\mathcal{P}\left(t, r\mathrm{e}^{\mathrm{i}\theta}\right) = \frac{1 - r^2}{1 - 2r\cos(t-\theta) + r^2}$$

We are given a polynomial $P \in \mathbb{C}[X]$ of degree $d$. Show that, for all $z \in \mathbb{C}$ and all $r \in \mathbb{R}$, we have: $$P(z) = \frac{1}{2\pi} \int_0^{2\pi} P\left(z + re^{i\theta}\right) d\theta$$

We are given a polynomial $P \in \mathbb{C}[X]$ of degree $d$. Deduce from question 3.17 that: $$\|P\|_{\mathbb{D}} = \|P\|_{\partial\mathbb{D}}$$ One may apply question 3.17 to an element $z \in \mathbb{D}$ such that $|P(z)| = \|P\|_{\mathbb{D}}$.

We are given a polynomial $P \in \mathbb{C}[X]$ of degree $d$. Show that, for all $z \in \mathbb{C}$: $$|P(z)| \leq \|P\|_{\partial\mathbb{D}} \max\{1, |z|\}^d.$$ One may apply question 3.18 to the polynomials $P(X)$ and $Q(X) = X^d P\left(X^{-1}\right)$.

We fix two non-zero natural integers $n$ and $m$ as well as two polynomials $Q \in \mathbb{C}[X]$ and $R \in \mathbb{C}[X]$ of degrees $n$ and $m$ respectively. We introduce the polynomial $P = QR$ and we denote by $\lambda$ its leading coefficient and $\alpha_1, \ldots, \alpha_{n+m}$ its roots counted with multiplicity.
Show that there exist $u$ and $v$ in $\partial\mathbb{D}$ such that: $$\|Q\|_{\mathbb{D}} \|R\|_{\mathbb{D}} \leq |\lambda| \cdot \prod_{i=1}^{n+m} \max\left\{\left|u - \alpha_i\right|, \left|v - \alpha_i\right|\right\}.$$

We fix two non-zero natural integers $n$ and $m$ as well as two polynomials $Q \in \mathbb{C}[X]$ and $R \in \mathbb{C}[X]$ of degrees $n$ and $m$ respectively. We introduce the polynomial $P = QR$ and we denote by $\lambda$ its leading coefficient and $\alpha_1, \ldots, \alpha_{n+m}$ its roots counted with multiplicity.
Deduce from question 3.20 that: $$\|Q\|_{\mathbb{D}} \|R\|_{\mathbb{D}} \leq M(S)$$ where $S$ is the polynomial defined by: $$S(X) = (X-1)^{m+n} P\left(\frac{uX - v}{X - 1}\right)$$

We fix two non-zero natural integers $n$ and $m$ as well as two polynomials $Q \in \mathbb{C}[X]$ and $R \in \mathbb{C}[X]$ of degrees $n$ and $m$ respectively. We introduce the polynomial $P = QR$. We set $w = \frac{v}{u}$. Show that: $$M(S) \leq \|P\|_{\mathbb{D}} \exp\left(\frac{n+m}{2\pi} \int_0^{2\pi} \ln\left(\max\left\{\left|e^{i\theta} - 1\right|, \left|e^{i\theta} - w\right|\right\}\right) d\theta\right)$$

We fix two non-zero natural integers $n$ and $m$ as well as two polynomials $Q \in \mathbb{C}[X]$ and $R \in \mathbb{C}[X]$ of degrees $n$ and $m$ respectively. We introduce the polynomial $P = QR$. We set $C = \exp\left(\frac{I}{2\pi}\right)$ with: $$I = \int_0^{2\pi} \ln\left(\max\left\{\left|e^{i\theta} - 1\right|, \left|e^{i\theta} + 1\right|\right\}\right) d\theta$$ Using the previous questions, show that: $$\|Q\|_{\mathbb{D}} \|R\|_{\mathbb{D}} \leq C^{n+m} \|P\|_{\mathbb{D}}$$

We set $C = \exp\left(\frac{I}{2\pi}\right)$ with $I = \int_0^{2\pi} \ln\left(\max\left\{\left|e^{i\theta}-1\right|, \left|e^{i\theta}+1\right|\right\}\right)d\theta$. For each natural integer $k \geq 2$, we set: $$\begin{aligned} & Q_k(X) = \prod_{\zeta \in U}(X - \zeta), \\ & R_k(X) = \prod_{\zeta \in V}(X - \zeta), \end{aligned}$$ where $U$ denotes the set of $k$-th roots of unity $\zeta$ such that $|\zeta - 1| \leq |\zeta + 1|$ and $V$ the set of $k$-th roots of unity that are not in $U$. By bounding below the quotient: $$\frac{\left\|Q_k\right\|_{\mathbb{D}} \left\|R_k\right\|_{\mathbb{D}}}{\left\|Q_k R_k\right\|_{\mathbb{D}}},$$ show that: $$C = \inf\left\{\lambda \in \mathbb{R} \left\lvert\, \begin{array}{c} \forall Q \in \mathbb{C}[X]\backslash\{0\},\quad \forall R \in \mathbb{C}[X]\backslash\{0\}, \\ \|Q\|_{\mathbb{D}} \|R\|_{\mathbb{D}} \leq \lambda^{\operatorname{deg}(QR)} \|QR\|_{\mathbb{D}} \end{array} \right.\right\},$$ where $\operatorname{deg}(QR)$ denotes the degree of $QR$.

Let $I = [a,b]$ be a segment of $\mathbb{R}$, and let $n$ and $m$ be two non-zero natural integers. We are given two distinct real numbers $c$ and $d$ and we set: $$J = \begin{cases} [c,d] & \text{if } c < d \\ [d,c] & \text{if } d < c. \end{cases}$$ Let $A \in \mathbb{C}_n[X]$ and $B \in \mathbb{C}_m[X]$ be two non-zero polynomials. Show that there exist polynomials $C \in \mathbb{C}_n[X]$ and $D \in \mathbb{C}_m[X]$ satisfying the following properties: $$\begin{gathered} \|A\|_I = \|C\|_J, \quad \|B\|_I = \|D\|_J, \quad \|AB\|_I = \|CD\|_J \\ A(a) = C(c), \quad B(b) = D(d). \end{gathered}$$

Let $I = [a,b]$ be a segment of $\mathbb{R}$, and let $n$ and $m$ be two non-zero natural integers. We recall: $$C_{n,m}^I = \sup\left\{\left.\frac{\|Q\|_I \|R\|_I}{\|QR\|_I}\right\rvert\, Q \in \mathbb{C}_n[X]\backslash\{0\}, R \in \mathbb{C}_m[X]\backslash\{0\}\right\} \in \mathbb{R} \cup \{+\infty\}$$ Deduce from question 4.27 that the quantity $C_{n,m}^I$ does not depend on the segment $I$.

We choose $I = [-1,1]$ and write $C_{n,m}$ instead of $C_{n,m}^I$. We fix an extremal pair $(Q_0, R_0)$, i.e., $Q_0$ and $R_0$ are monic and $\frac{\|Q_0\|_I \|R_0\|_I}{\|Q_0 R_0\|_I} = C_{n,m}$.
Let $J$ be a segment contained in $I$ such that $\left\|Q_0\right\|_J = \left\|Q_0\right\|_I$ and $\left\|R_0\right\|_J = \left\|R_0\right\|_I$. Show that: $$\left\|Q_0 R_0\right\|_J = \left\|Q_0 R_0\right\|_I$$

We choose $I = [-1,1]$ and write $C_{n,m}$ instead of $C_{n,m}^I$. We fix an extremal pair $(Q_0, R_0)$. Deduce from questions 4.27 and 4.29 that there exists an extremal pair $(Q_1, R_1)$ such that: $$\left\|Q_1\right\|_I = \left|Q_1(-1)\right| \quad \text{and} \quad \left\|R_1\right\|_I = \left|R_1(1)\right|$$

We choose $I = [-1,1]$ and write $C_{n,m}$ instead of $C_{n,m}^I$. Let $(Q_1, R_1)$ be an extremal pair such that $\left\|Q_1\right\|_I = \left|Q_1(-1)\right|$ and $\left\|R_1\right\|_I = \left|R_1(1)\right|$. Let $n_1$ and $m_1$ be the degrees of $Q_1$ and $R_1$ respectively. We set $Q_2 = X^{n-n_1} Q_1$ and $R_2 = X^{m-m_1} R_1$.
Show that $(Q_2, R_2)$ is a good extremal pair, i.e., $Q_2$ and $R_2$ are monic of degrees $n$ and $m$ respectively, $\frac{\|Q_2\|_I \|R_2\|_I}{\|Q_2 R_2\|_I} = C_{n,m}$, and $\|Q_2\|_I = |Q_2(-1)|$, $\|R_2\|_I = |R_2(1)|$.

Let $n$ be a non-zero natural number. Let $f$ be in $\mathcal{S}_n$, where $\mathcal{S}_n$ is the $\mathbb{C}$-vector space of functions $f : \mathbb{R} \rightarrow \mathbb{C}$ satisfying $$\exists (a_0, \ldots, a_n) \in \mathbb{C}^{n+1}, \quad \exists (b_1, \ldots, b_n) \in \mathbb{C}^n, \quad \forall t \in \mathbb{R}, \quad f(t) = a_0 + \sum_{k=1}^{n}\left(a_k \cos(kt) + b_k \sin(kt)\right)$$
Show that there exists $U \in \mathbb{C}_{2n}[X]$ such that, for all $\theta \in \mathbb{R}$, $f(\theta) = \mathrm{e}^{-\mathrm{i}n\theta} U(\mathrm{e}^{\mathrm{i}\theta})$.

Let $n$ be a non-zero natural number. Let $f$ be in $\mathcal{S}_n$, where $\mathcal{S}_n$ is the $\mathbb{C}$-vector space of functions $f : \mathbb{R} \rightarrow \mathbb{C}$ satisfying $$\exists (a_0, \ldots, a_n) \in \mathbb{C}^{n+1}, \quad \exists (b_1, \ldots, b_n) \in \mathbb{C}^n, \quad \forall t \in \mathbb{R}, \quad f(t) = a_0 + \sum_{k=1}^{n}\left(a_k \cos(kt) + b_k \sin(kt)\right)$$ Show that there exists $U \in \mathbb{C}_{2n}[X]$ such that, for all $\theta \in \mathbb{R}$, $f(\theta) = \mathrm{e}^{-\mathrm{i}n\theta} U(\mathrm{e}^{\mathrm{i}\theta})$.

Let $n$ be a non-zero natural number. Let $f$ be in $\mathcal{S}_n$. For $k$ in $\llbracket 1, 2n \rrbracket$, we denote $\varphi_k = \frac{\pi}{2n} + \frac{k\pi}{n}$ and $\omega_k = \mathrm{e}^{\mathrm{i}\varphi_k}$.
Verify that, for all $k \in \llbracket 1, 2n \rrbracket$, $\frac{2\omega_k}{(1 - \omega_k)^2} = \frac{-1}{2\sin(\varphi_k/2)^2}$ and deduce from questions 11 and 12 that $$\forall \theta \in \mathbb{R}, \quad f'(\theta) = \frac{1}{2n} \sum_{k=1}^{2n} f(\theta + \varphi_k) \frac{(-1)^k}{2\sin(\varphi_k/2)^2}.$$

Let $n$ be a non-zero natural number. For $k$ in $\llbracket 1, 2n \rrbracket$, we denote $\varphi_k = \frac{\pi}{2n} + \frac{k\pi}{n}$ and $\omega_k = \mathrm{e}^{\mathrm{i}\varphi_k}$. Let $f \in \mathcal{S}_n$.
Verify that, for all $k \in \llbracket 1, 2n \rrbracket$, $\frac{2\omega_k}{(1-\omega_k)^2} = \frac{-1}{2\sin(\varphi_k/2)^2}$ and deduce from questions 11 and 12 that $$\forall \theta \in \mathbb{R}, \quad f'(\theta) = \frac{1}{2n} \sum_{k=1}^{2n} f(\theta + \varphi_k) \frac{(-1)^k}{2\sin(\varphi_k/2)^2}.$$

Let $M \in \mathcal{B}_{n}$. We define $$b^{\prime}(M) = \sup\left\{\varphi_{M}(z); z \in \mathbf{C} \backslash \mathbb{D}\right\}$$ where $\varphi_{M}: z \in \mathbf{C} \backslash \mathbb{D} \longmapsto (|z|-1)\left\|R_{z}(M)\right\|_{\mathrm{op}}$, and $b^{\prime}(M) \leq b(M)$.
We fix $r \in ]1, +\infty[$ and $(X,Y) \in \Sigma_{n}^{2}$. For $\rho \in \mathbf{R}^{+*}$, we denote $\mathbb{D}_{\rho} = \{z \in \mathbf{C}; |z| \leq \rho\}$.
Show that there exists an element $F_{r}$ of $\mathcal{R}_{n}$ whose poles are all in $\mathbb{D}_{1/r}$ and such that the following two properties are satisfied: $$\begin{gathered} \forall z \in \mathbf{C} \backslash \mathbb{D}_{1/r}, \quad \left|F_{r}(z)\right| \leq \frac{b^{\prime}(M)}{r|z|-1} \\ \forall k \in \mathbf{N}, \quad X^{T}M^{k}Y = \frac{r^{k+1}}{2\pi} \int_{-\pi}^{\pi} F_{r}\left(e^{it}\right) e^{i(k+1)t} \mathrm{~d}t \end{gathered}$$

Let $M \in \mathcal{B}_{n}$. We define $$b^{\prime}(M) = \sup\left\{\varphi_{M}(z); z \in \mathbf{C} \backslash \mathbb{D}\right\}$$ where $\varphi_{M}: z \in \mathbf{C} \backslash \mathbb{D} \longmapsto (|z|-1)\left\|R_{z}(M)\right\|_{\mathrm{op}}$.
We fix $r \in ]1, +\infty[$ and $(X,Y) \in \Sigma_{n}^{2}$. For $\rho \in \mathbf{R}^{+*}$, we denote $\mathbb{D}_{\rho} = \{z \in \mathbf{C}; |z| \leq \rho\}$.
By using the previous question, integration by parts and inequality (3) from question $17$, show that $$\forall k \in \mathbf{N}, \quad \left|X^{T}M^{k}Y\right| \leq \frac{r^{k+1}}{(k+1)(r-1)} n b^{\prime}(M).$$

Show that:
$$( \cos ( t ) ) ^ { 2 p } = \frac { 1 } { 2 ^ { 2 p } } \left( \binom { 2 p } { p } + 2 \sum _ { k = 0 } ^ { p - 1 } \binom { 2 p } { k } \cos ( 2 ( p - k ) t ) \right)$$
Hint: One may develop $\left( \frac { e ^ { \mathrm{i} t } + e ^ { - \mathrm{i} t } } { 2 } \right) ^ { 2 p }$.