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 $s \in [0,1[$. 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 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 $j \in \mathbf{N}$, $k \in \mathcal{T}_{j}$, $\ell \in \mathcal{T}_{j+1}$.

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 $(j, k) \in \mathcal{I}$, the function $\theta_{j,k}$ is continuous, affine on each interval of the form $[\ell 2^{-n}, (\ell+1) 2^{-n}]$ where $n > j$ and $\ell \in \mathcal{T}_{n}$.

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 $$\theta_{j,k}(x) = \left\{ \begin{array}{l} 1 - |2^{j+1} x - 2k - 1| \quad \text{if } x \in [k 2^{-j}, (k+1) 2^{-j}] \\ 0 \text{ otherwise} \end{array} \right.$$ Prove that for all $(j, k) \in \mathcal{I}$ and $(x, y) \in [0,1]^{2}$, we have $$|\theta_{j,k}(x) - \theta_{j,k}(y)| \leq 2^{j+1} |x - y|$$

In the rest of the second part, $f$ is an element of $\mathcal{C}_{0}$. For all $n \in \mathbf{N}$, let $S_{n} f$ be the function of $\mathcal{C}_{0}$ defined by $$S_{n} f = \sum_{j=0}^{n} \sum_{k \in \mathcal{T}_{j}} c_{j,k}(f) \theta_{j,k}$$ where, for all $(j, k) \in \mathcal{I}$, $$c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2} .$$ Show that $\lim_{j \rightarrow +\infty} \max_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| = 0$.

We denote $\mathcal{I} = \{(j, k) \in \mathbf{N}^{2} \mid j \in \mathbf{N} \text{ and } 0 \leq k < 2^{j}\}$. For all $(j, k) \in \mathcal{I}$, $$c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2} .$$ For all $(j, k) \in \mathcal{I}$, $(i, \ell) \in \mathcal{I}$, calculate $c_{j,k}(\theta_{i,\ell})$.

We denote $\mathcal{I} = \{(j, k) \in \mathbf{N}^{2} \mid j \in \mathbf{N} \text{ and } 0 \leq k < 2^{j}\}$; for $j \in \mathbf{N}$, $\mathcal{T}_{j} = \{k \in \mathbf{N} \mid 0 \leq k < 2^{j}\}$. Let $a_{j,k}$ be a family of real numbers indexed by $(j, k) \in \mathcal{I}$. We denote $b_{j} = \max_{k \in \mathcal{T}_{j}} |a_{j,k}|$, and we suppose that the series $\sum b_{j}$ is convergent.
For all $j \in \mathbf{N}$, let $f_{j}^{a}$ be the function defined by $$f_{j}^{a}(x) = \sum_{k \in \mathcal{T}_{j}} a_{j,k} \theta_{j,k}(x)$$ Show that the series $\sum f_{j}^{a}$ is uniformly convergent on $[0,1]$ towards a function denoted $f^{a}$, which belongs to $\mathcal{C}_{0}$ and which satisfies, for all $(j, k) \in \mathcal{I}$, $c_{j,k}(f^{a}) = a_{j,k}$.

For all $n \in \mathbf{N}$, let $S_{n} f$ be the function of $\mathcal{C}_{0}$ defined by $$S_{n} f = \sum_{j=0}^{n} \sum_{k \in \mathcal{T}_{j}} c_{j,k}(f) \theta_{j,k}$$ where, for all $(j, k) \in \mathcal{I}$, $$c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2} .$$ Suppose $f$ is of class $\mathcal{C}^{1}$. Show that there exists a constant $M \geq 0$ such that for all $(j, k) \in \mathcal{I}$, $|c_{j,k}(f)| \leq M 2^{-j}$.
Deduce that the sequence of functions $S_{n} f$ is uniformly convergent on $[0,1]$ when $n$ tends to $\infty$.

For all $(j, k) \in \mathcal{I}$, $$c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2} .$$ Suppose $f$ is of class $\mathcal{C}^{2}$. Show that there exists a constant $M' \geq 0$ such that for all $(j, k) \in \mathcal{I}$, $|c_{j,k}(f)| \leq M' 4^{-j}$.

For all $n \in \mathbf{N}$, let $S_{n} f$ be the function of $\mathcal{C}_{0}$ defined by $$S_{n} f = \sum_{j=0}^{n} \sum_{k \in \mathcal{T}_{j}} c_{j,k}(f) \theta_{j,k}$$ Show that for all $n \in \mathbf{N}$ and all $\ell \in \mathcal{T}_{n+1}$, the function $S_{n} f$ is affine on the interval $[\ell 2^{-n-1}, (\ell+1) 2^{-n-1}]$.

For all $n \in \mathbf{N}$, let $S_{n} f$ be the function of $\mathcal{C}_{0}$ defined by $$S_{n} f = \sum_{j=0}^{n} \sum_{k \in \mathcal{T}_{j}} c_{j,k}(f) \theta_{j,k}$$ where, for all $(j, k) \in \mathcal{I}$, $$c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2} .$$ Let $n \in \mathbf{N}$. Suppose that for all $\ell \in \mathcal{T}_{n}$, $(S_{n-1} f)(\ell 2^{-n}) = f(\ell 2^{-n})$. Show that we also have that for all $\ell \in \mathcal{T}_{n+1}$, $(S_{n} f)(\ell 2^{-n-1}) = f(\ell 2^{-n-1})$.
One may distinguish cases according to the parity of $\ell$.

Deduce that for all $n \in \mathbf{N}$ and all $\ell \in \mathcal{T}_{n+1}$, $(S_{n} f)(\ell 2^{-n-1}) = f(\ell 2^{-n-1})$.

Deduce from question 9 that for all $f$ in $\mathcal{C}_{0}$, $\lim_{n \rightarrow +\infty} \|f - S_{n} f\|_{\infty} = 0$.

Let $n \in \mathbf{N}$. Show that $S_{n}$ is a projector on $\mathcal{C}_{0}$, whose subordinate norm (to $\|\cdot\|_{\infty}$) equals 1.

We assume that the conditions of question 12 are satisfied and that $\lambda(0) \neq 0$. Show that $E \circ F = F \circ E + H - H^{-1}$ if and only if for all $i \in \mathbf{Z}$, $$\mu(i) = \mu(i-1) + \lambda(0) q^{-2i} - \lambda(0)^{-1} q^{2i}$$

We assume that the conditions of questions 12 and 14 are satisfied and that $\lambda(0) \neq 0$.
15a. Show that $\lambda$ and $\mu$ are periodic on $\mathbf{Z}$, with periods dividing $\ell$.
15b. Show that the period of $\lambda$ is equal to $\ell$.
15c. Show that the period of $\mu$ is also equal to $\ell$.

We assume that the conditions of questions 12 and 14 are satisfied and that $\lambda(0) \neq 0$. Let $C = (q - q^{-1}) E \circ F + q^{-1} H + q H^{-1}$ with $H^{-1}$ the inverse of $H$.
16a. Show that $C = (q - q^{-1}) F \circ E + q H + q^{-1} H^{-1}$.
16b. For $i \in \mathbf{Z}$, show that $v_i$ is an eigenvector of $C$.
16c. Deduce that $C$ is a homothety of $V$ and calculate its ratio $R(\lambda(0), \mu(0), q)$ in terms of $\lambda(0), \mu(0)$ and $q$.
16d. We fix $q$ and $\lambda(0)$. Show that the map $\mu(0) \mapsto R(\lambda(0), \mu(0), q)$ is a bijection from $\mathbf{C}$ to $\mathbf{C}$.
16e. We fix $q$ and $\mu(0)$. Show that the map $\lambda(0) \mapsto R(\lambda(0), \mu(0), q)$ is a surjection from $\mathbf{C}^*$ to $\mathbf{C}$ but not a bijection.

Let $\ell, W_{\ell}, a, P_a$ be as in Part II. We say that an element $\phi$ of $\mathcal{L}(V)$ is compatible with $P_a$ if $P_a \circ \phi \circ P_a = P_a \circ \phi$.
17a. Show that if $\phi \in \mathcal{L}(V)$ commutes with $P_a$, then $\phi$ is compatible with $P_a$.
17b. Show that $H$ and $H^{-1}$ are compatible with $P_a$.

Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$. Show that $\mathcal{U}_q$ is a subalgebra of $\mathcal{L}(V)$.

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 suppose furthermore that the function $\omega_{f}$ satisfies property $(\mathcal{P}_{2})$: for all integer $N \geq 1$, there exists a real number $c_{4}(N) > 0$, such that for all $h \in ]0,1]$, $$\omega_{f}(h) \leq c_{4}(N) (1 + |\log_{2} h|)^{-N}$$ For all integer $N \geq 1$, we set $c_{5}(N) = 3^{s} c_{1} (c_{4}(N))^{1/N}$. Show that $$n_{1} - n_{0} \leq n_{1} + 1 \leq \left(\frac{c_{4}(N)}{\omega_{f}(2^{-n_{1}})}\right)^{\frac{1}{N}}$$ and deduce $$\sum_{j=n_{0}+1}^{n_{1}} \sum_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| |\theta_{j,k}(x)| \leq c_{5}(N) |x - x_{0}|^{(1 - \frac{1}{N})s}$$

Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$. Show that $E \in \mathcal{U}_q$ and $F \in \mathcal{U}_q$.

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 suppose furthermore that $\omega_{f}$ satisfies property $(\mathcal{P}_{2})$: for all integer $N \geq 1$, there exists $c_{4}(N) > 0$ such that for all $h \in ]0,1]$, $\omega_{f}(h) \leq c_{4}(N)(1 + |\log_{2} h|)^{-N}$.
Deduce from the above that $\alpha_{f}(x_{0}) \geq s$.
One may distinguish the cases $n_{0} \geq n_{1}$ and $n_{0} < n_{1}$.