Let $x \in ] - 1,1 [$.
Determine the Fourier series of the function $\widetilde { h } : \mathbb { R } \rightarrow \mathbb { R }$ defined by $\widetilde { h } ( \theta ) = \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right)$.
One may use the result from question II.A.2.

Show that $\forall x \in \mathbb { R }$ and $\forall n \in \mathbb { N } ^ { * }$ $$\prod _ { k = 1 } ^ { n - 1 } \left( x ^ { 2 } - 2 x \cos \frac { 2 k \pi } { n } + 1 \right) = \left( \sum _ { k = 0 } ^ { n - 1 } x ^ { k } \right) ^ { 2 }$$

Show that $\prod _ { k = 1 } ^ { n - 1 } \sin \frac { k \pi } { 2 n } = \frac { \sqrt { n } } { 2 ^ { n - 1 } }$.

Show that $V$ is a vector subspace of $\mathbf{C}^{\mathbf{Z}}$. Given $f \in \mathbf{C}^{\mathbf{Z}}$, we define $E(f) \in \mathbf{C}^{\mathbf{Z}}$ by $E(f)(k) = f(k+1), k \in \mathbf{Z}$.

Show that $E \in \mathcal{L}(\mathbf{C}^{\mathbf{Z}})$ and that $V$ is stable under $E$.

Show that $\Gamma^{s}(x_{0})$ is a vector subspace of $\mathcal{C}$, then that, for all real numbers $s_{1}$ and $s_{2}$ satisfying $0 \leq s_{1} \leq s_{2} < 1$, we have $\Gamma^{s_{2}}(x_{0}) \subset \Gamma^{s_{1}}(x_{0})$. Finally, determine $\Gamma^{0}(x_{0})$.
Recall: Let $x_{0} \in [0,1]$. For all $s \in [0,1[$, $\Gamma^{s}(x_{0})$ is the subset of $\mathcal{C}$ formed by functions $f$ which satisfy: $$\sup_{x \in [0,1] \backslash \{x_{0}\}} \frac{|f(x) - f(x_{0})|}{|x - x_{0}|^{s}} < +\infty .$$

Let $f \in \mathcal{C}$. If $f$ is differentiable at $x_{0}$, show that $f \in \Gamma^{s}(x_{0})$ for all $s \in [0,1[$.

Show that for all $x_{0} \in ]0,1[$, there exists $f \in \mathcal{C}$ non-differentiable at $x_{0}$ such that for all $s \in [0,1[$, $f \in \Gamma^{s}(x_{0})$.

Show that $E \in \mathrm{GL}(V)$.

Let $p : [0,1] \rightarrow \mathbf{R}$, $x \mapsto \sqrt{|1 - 4x^{2}|}$. Determine the pointwise Hölder exponent of $p$ at $\frac{1}{2}$.
Recall: For all $f \in \mathcal{C}$ and all $x_{0} \in [0,1]$, $$\alpha_{f}(x_{0}) = \sup \{s \in [0,1[ \mid f \in \Gamma^{s}(x_{0})\} .$$

For $i \in \mathbf{Z}$, we define $v_i$ in $\mathbf{C}^{\mathbf{Z}}$ by $$v_i(k) = \left\{\begin{array}{l} 1 \text{ if } k = i \\ 0 \text{ if } k \neq i \end{array}\right.$$
3a. Show that the family $\{v_i\}_{i \in \mathbf{Z}}$ is a basis of $V$.
3b. Calculate $E(v_i)$.

For all $f \in \mathcal{C}$, the function $\omega_{f} : [0,1] \rightarrow \mathbf{R}_{+}$ is defined by $$\omega_{f}(h) = \sup \{|f(x) - f(y)| \mid x, y \in [0,1] \text{ and } |x - y| \leq h\} .$$ Show that $\omega_{f}$ is increasing, and continuous at 0.

For all $f \in \mathcal{C}$, the function $\omega_{f} : [0,1] \rightarrow \mathbf{R}_{+}$ is defined by $$\omega_{f}(h) = \sup \{|f(x) - f(y)| \mid x, y \in [0,1] \text{ and } |x - y| \leq h\} .$$ Show that for all $h, h' \in [0,1]$ such that $h \leq h'$, $\omega_{f}$ satisfies $$\omega_{f}(h') \leq \omega_{f}(h) + \omega_{f}(h' - h) .$$

For all $f \in \mathcal{C}$, the function $\omega_{f} : [0,1] \rightarrow \mathbf{R}_{+}$ is defined by $$\omega_{f}(h) = \sup \{|f(x) - f(y)| \mid x, y \in [0,1] \text{ and } |x - y| \leq h\} .$$ Using the result of question 3b, deduce that $\omega_{f}$ is continuous on $[0,1]$.

Let $\lambda, \mu \in \mathbf{C}^{\mathbf{Z}}$. We define the linear maps $F, H \in \mathcal{L}(V)$ respectively by $$H(v_i) = \lambda(i) v_i \quad \text{and} \quad F(v_i) = \mu(i) v_{i+1}, \quad i \in \mathbf{Z}.$$ Show that $H \circ E = E \circ H + 2E$ if and only if for all $i \in \mathbf{Z}, \lambda(i) = \lambda(0) - 2i$.

Let $s \in [0,1[$. Suppose that the function $h \mapsto \frac{\omega_{f}(h)}{h^{s}}$ is bounded on $]0,1]$. For all $x_{0} \in [0,1]$, show that $f \in \Gamma^{s}(x_{0})$.

Let $q : [0,1] \rightarrow \mathbf{R}$ defined by $$\left\{ \begin{array}{l} q(x) = x \cos\left(\frac{\pi}{x}\right) \text{ for } x > 0 \\ q(0) = 0 \end{array} \right.$$ Show that for all $x_{0} \in [0,1]$, $\alpha_{q}(x_{0}) = 1$, but that $\frac{\omega_{q}(h)}{\sqrt{h}}$ does not tend to 0 when $h$ tends to 0.

We assume that the conditions of question 4 are satisfied (i.e., $\lambda(i) = \lambda(0) - 2i$ for all $i \in \mathbf{Z}$). Let $\lambda, \mu \in \mathbf{C}^{\mathbf{Z}}$ and $F, H \in \mathcal{L}(V)$ defined by $H(v_i) = \lambda(i) v_i$ and $F(v_i) = \mu(i) v_{i+1}$. Show that $E \circ F = F \circ E + H$ if and only if for all $i \in \mathbf{Z}$, $$\mu(i) = \mu(0) + i(\lambda(0) - 1) - i^2.$$

We denote $\mathcal{I} = \{(j, k) \in \mathbf{N}^{2} \mid j \in \mathbf{N} \text{ and } 0 \leq k < 2^{j}\}$; for $j \in \mathbf{N}$, $\mathcal{T}_{j} = \{k \in \mathbf{N} \mid 0 \leq k < 2^{j}\}$. For all $(j, k) \in \mathcal{I}$, $\theta_{j,k} : [0,1] \rightarrow [0,1]$ is defined by $$\theta_{j,k}(x) = \left\{ \begin{array}{l} 1 - |2^{j+1} x - 2k - 1| \quad \text{if } x \in [k 2^{-j}, (k+1) 2^{-j}] \\ 0 \text{ otherwise} \end{array} \right.$$ Show that for all $j \in \mathbf{N}$ and all $k \in \mathcal{T}_{j+1}$, there exists a unique integer $k' \in \mathcal{T}_{j}$ such that $$[k 2^{-j-1}, (k+1) 2^{-j-1}] \subset [k' 2^{-j}, (k'+1) 2^{-j}]$$ Specify the relationship between $k$ and $k'$.

We denote $\mathcal{I} = \{(j, k) \in \mathbf{N}^{2} \mid j \in \mathbf{N} \text{ and } 0 \leq k < 2^{j}\}$; for $j \in \mathbf{N}$, $\mathcal{T}_{j} = \{k \in \mathbf{N} \mid 0 \leq k < 2^{j}\}$. For all $(j, k) \in \mathcal{I}$, $\theta_{j,k} : [0,1] \rightarrow [0,1]$ is defined by $$\theta_{j,k}(x) = \left\{ \begin{array}{l} 1 - |2^{j+1} x - 2k - 1| \quad \text{if } x \in [k 2^{-j}, (k+1) 2^{-j}] \\ 0 \text{ otherwise} \end{array} \right.$$ Calculate $\theta_{j,k}(\ell 2^{-j-1})$ for all $j \in \mathbf{N}$, $k \in \mathcal{T}_{j}$, $\ell \in \mathcal{T}_{j+1}$.

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$.
Show that there exists a unique $n_{0} \in \mathbf{N}$ such that $2^{-n_{0}-1} < |x - x_{0}| \leq 2^{-n_{0}}$.

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. We recall that $\widetilde{k}_{j}(x)$ is the integer part of $2^{j} x$. We set $$W_{j} = \sum_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| |\theta_{j,k}(x) - \theta_{j,k}(x_{0})|$$ Show that $$W_{j} \leq (|c_{j,\widetilde{k}_{j}(x)}(f)| + |c_{j,\widetilde{k}_{j}(x_{0})}(f)|) 2^{j+1} |x - x_{0}|$$

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. Let $n_{0}$ be the unique natural integer such that $2^{-n_{0}-1} < |x - x_{0}| \leq 2^{-n_{0}}$.
Show that for $j \leq n_{0}$, we have $$W_{j} \leq 4 c_{1} 2^{(1-s)j} 3^{s} |x - x_{0}|$$

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. Let $n_{0}$ be the unique natural integer such that $2^{-n_{0}-1} < |x - x_{0}| \leq 2^{-n_{0}}$.
Deduce that, by setting $c_{2} = 8(2^{1-s} - 1)^{-1} (3/2)^{s} c_{1}$, $$\sum_{j=0}^{n_{0}} \sum_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| |\theta_{j,k}(x) - \theta_{j,k}(x_{0})| \leq c_{2} |x - x_{0}|^{s}$$

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. We recall that $\widetilde{k}_{j}(x_{0})$ is the integer part of $2^{j} x_{0}$.
Show that for all $j \in \mathbf{N}$, $|c_{j,\widetilde{k}_{j}(x_{0})}(f)| \leq 2^{s(1-j)} c_{1}$. Deduce, by setting $c_{3} = (1 - 2^{-s})^{-1} 2^{s} c_{1}$, $$\sum_{j=n_{0}+1}^{+\infty} \sum_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| |\theta_{j,k}(x_{0})| \leq c_{3} |x - x_{0}|^{s}$$