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 $c_{3} = (1 - 2^{-s})^{-1} 2^{s} c_{1}$, $$\sum_{j=n_{0}+1}^{+\infty} \sum_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| |\theta_{j,k}(x_{0})| \leq c_{3} |x - x_{0}|^{s}$$

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 $2^{-n_{0}-1} < |x - x_{0}| \leq 2^{-n_{0}}$, and $n_{1}$ the unique natural integer such that $\omega_{f}(2^{-n_{1}-1}) < 2^{-n_{0} s} \leq \omega_{f}(2^{-n_{1}})$.
Show that for all $n \geq n_{1}$, we have $$\|f - S_{n} f\|_{\infty} \leq 2^{s+1} |x - x_{0}|^{s}$$ One may use the results of questions 9a and 9c.

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 $2^{-n_{0}-1} < |x - x_{0}| \leq 2^{-n_{0}}$, and $n_{1}$ the unique natural integer such that $\omega_{f}(2^{-n_{1}-1}) < 2^{-n_{0} s} \leq \omega_{f}(2^{-n_{1}})$.
Show that when $n_{0} < n_{1}$, we have $$\sum_{j=n_{0}+1}^{n_{1}} \sum_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| |\theta_{j,k}(x)| \leq c_{1} 3^{s} (n_{1} - n_{0}) |x - x_{0}|^{s} .$$

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 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 $$\sum_{j=n_{0}+1}^{n_{1}} \sum_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| |\theta_{j,k}(x)| \leq c_{5}(N) |x - x_{0}|^{(1 - \frac{1}{N})s}$$

We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$.
For $N \in \mathbb { N } ^ { * }$ and $x > 0$, we set $$r _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N + 1 } e ^ { - 1 / x }, \quad R _ { N } ( x ) = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t$$
Show that, if $N \in \mathbb { N } ^ { * }$ and $x > 0$, $$\left| R _ { N } ( x ) \right| \leqslant \left| r _ { N } ( x ) \right|$$ Deduce that $R _ { N + 1 } ( x ) = o \left( r _ { N } ( x ) \right)$ when $x \rightarrow 0$.

We are given $f : \mathbb { R } ^ { 2 } \rightarrow \mathbb { C }$ continuous and 1-periodic in each of its arguments, with zero average $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 } = 0$. We consider the system with $x=0$: $$F'(t) = f(\alpha(t)), \quad \alpha'(t) = \omega, \quad F(0)=0, \quad \alpha(0)=(0,0).$$ Suppose that $\omega$ is not resonant.
Show that, if $f$ is a trigonometric polynomial, then $F$ is bounded on $\mathbb { R }$.

Prove that the sequence $(y_k)_{k \geqslant 2}$ is increasing, where $y_k$ denotes the maximum of the determinant on $\mathcal{Y}_k$.

We set for every integer $n \geqslant 0$, $$B _ { n } = \sum _ { k = 0 } ^ { n } S ( n , k )$$ where $S(n,k)$ denotes the number of partitions of $\llbracket 1, n \rrbracket$ into $k$ parts.
Show that the sequence $\left( \frac { B _ { n } } { n ! } \right) _ { n \in \mathbb { N } }$ is bounded by 1.

Let $\alpha$ be a real strictly greater than 1. The Riemann series $\sum_{n \geqslant 1} \frac{1}{n^{\alpha}}$ converges to a real that we will denote $\ell$. We denote by $\left(S_{n}\right)_{n \in \mathbb{N}}$ the sequence defined by $S_{0}=0$ and $\forall n \geqslant 1, S_{n}=\sum_{k=1}^{n} \frac{1}{k^{\alpha}}$.
a) Show that $\forall n \geqslant 1, \frac{1}{\alpha-1} \frac{1}{(n+1)^{\alpha-1}} \leqslant \ell-S_{n} \leqslant \frac{1}{\alpha-1} \frac{1}{n^{\alpha-1}}$.
b) Deduce that $\left(S_{n}\right)_{n \in \mathbb{N}}$ belongs to $E^{c}$ and give its convergence rate.

Let $x \in \mathcal{D}_{\zeta}$ and let $n \in \mathbb{N}$ such that $n \geqslant 2$. Show: $$\int_{n}^{n+1} \frac{\mathrm{~d}t}{t^{x}} \leqslant \frac{1}{n^{x}} \leqslant \int_{n-1}^{n} \frac{\mathrm{~d}t}{t^{x}}$$

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Show that there exists $A \in \mathbb{R}_{+}^{*}$ such that $$\forall k \in \mathbb{N}^{*}, \forall x \in {]-1,1[}, \quad \left|f^{(k)}(x)\right| \leqslant k! \left(A + \frac{1}{(x+1)^{k+1}}\right)$$

We assume $c(x) = 0$ for all $x \in [0,1]$, $f \in \mathcal{C}([0,1],\mathbb{R})$ satisfies $|f(y)-f(z)| \leq K|y-z|^\alpha$ for some $\alpha \in ]0,1]$ and $K \geq 0$. Let $u$ be the solution of problem (1) and define $\hat{B}_{n+1}u$ as above. Let $n \in \mathbb { N } ^ { * }$ such that $n \geq 2$. We set $\chi _ { n + 1 } = \hat { B } _ { n + 1 } u - u$.
(a) Show that
$$\left\| \chi _ { n + 1 } ^ { \prime \prime } \right\| _ { \infty } \leq \left\| f - B _ { n - 1 } f \right\| _ { \infty } + \frac { 1 } { n + 1 } \| f \| _ { \infty } + K \frac { 1 } { ( n + 1 ) ^ { \alpha } }$$
(b) Show that for all $x \in [ 0,1 ]$ there exists $\xi \in [ 0,1 ]$ such that
$$\chi _ { n + 1 } ( x ) = - \frac { 1 } { 2 } x ( 1 - x ) \chi _ { n + 1 } ^ { \prime \prime } ( \xi )$$
Hint: for $x \in ]0,1[$ one may consider the function
$$h ( t ) = \chi _ { n + 1 } ( t ) - \frac { \chi _ { n + 1 } ( x ) } { x ( 1 - x ) } t ( 1 - t ) , \quad t \in [ 0,1 ]$$

Deduce that there exists a constant $M \geq 0$ such that for all $n \in \mathbb { N } ^ { * }$, we have
$$\left\| u - \hat { B } _ { n + 1 } u \right\| _ { \infty } \leq \frac { M } { n ^ { \alpha / 2 } }$$

We consider the Hilbert polynomials $$\left\{ \begin{array} { l } H _ { 0 } ( X ) = 1 \\ H _ { n } ( X ) = \frac { 1 } { n ! } \prod _ { k = 0 } ^ { n - 1 } ( X - k ) = \frac { X ( X - 1 ) \cdots ( X - n + 1 ) } { n ! } \quad \text { for all } n \in \mathbb { N } ^ { * } \end{array} \right.$$
Demonstrate that, for all $x \in ] - 1,1 \left[ , \sum _ { i = 0 } ^ { + \infty } \left( \sum _ { j = 0 } ^ { + \infty } \left| a _ { i , j } ( x ) \right| \right) = \exp \left( \frac { | x | } { \sqrt { 1 - | x | } } \right) - 1 \right.$.

We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. For $z \in \mathcal { D }$, we denote $\Phi _ { p } ( z ) = \sum _ { n = 0 } ^ { + \infty } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } z ^ { n }$. We admit that the function $\Phi _ { p }$ is bounded on $\mathcal { D }$.
Demonstrate that, for all $p \in \mathbb { N }$, there exist a real $K _ { p }$ and a natural integer $N _ { p }$ such that $$\forall n \geqslant N _ { p } , \quad \left| a _ { n } \right| \leqslant \frac { K _ { p } } { n ^ { p } }$$

We have $\pi_n(x) = \frac{\lfloor 2^n x \rfloor}{2^n}$ for all $(x,n) \in \mathbb{R} \times \mathbb{N}$.
Establish $$\forall (x,n) \in \mathbb{R} \times \mathbb{N}, \quad \pi_n(x) \leqslant x < \pi_n(x) + \frac{1}{2^n}.$$

Show $$\forall n \in \mathbb{N}^{\star}, \forall s \in ]1, +\infty[, \quad \sum_{k=n+1}^{+\infty} \frac{1}{(2k-1)^s} \leqslant \frac{1}{2(s-1)} \frac{1}{(2n-1)^{s-1}}.$$

We set $L_n = \lfloor n^{1/3} \rfloor$, where $\lfloor t \rfloor$ denotes the integer part of a real number $t$. Let $x, y \in \{0,1\}^n$ such that $x \neq y$. Set for $z \in \mathbb{C}$, $A_{x,y}(z) = \sum_{k=0}^{n-1}(x_k - y_k)z^k$.
a. Justify the existence of $\theta_0 \in \left[-\frac{\pi}{L_n}, \frac{\pi}{L_n}\right]$ such that $\left|A_{x,y}\left(e^{i\theta_0}\right)\right| \geq \frac{1}{n^{L_n - 1}}$.
b. Prove that $$\sum_{j=0}^{n-1} \left|\mathbb{E}\left[O_j(x)\right] - \mathbb{E}\left[O_j(y)\right]\right| \cdot \left|\frac{e^{i\theta_0} - (1-p)}{p}\right|^j \geq \frac{p}{n^{L_n - 1}}$$
c. Justify the existence of an integer $j_n(x,y)$ such that $0 \leq j_n(x,y) \leq n-1$ and $$\left|\mathbb{E}\left[O_{j_n(x,y)}(x)\right] - \mathbb{E}\left[O_{j_n(x,y)}(y)\right]\right| \geq \frac{p}{n^{L_n}} \exp\left(-\frac{1-p}{2p^2} \cdot \frac{\pi^2}{L_n^2} n\right).$$

In Friedman's urn model ($a_{0} = 1, b_{0} = 0, a = d = 0, b = c = 1$), we have for all integers $n$ and all $t \in ]0,1[$, $$g_{n}(t) = \frac{1}{n!} \sum_{p=1}^{+\infty} p^{n} t^{p} (1-t)^{n+1}.$$ Fix an integer $n \geqslant 2$.
Show that $\sum_{p=n+1}^{+\infty} p^{n} t^{p} (1-t)^{n+1} \underset{t \rightarrow 0^{+}}{=} O(t^{n+1})$.

For all $(x,n) \in \mathbb{R} \times \mathbb{N}$, we define $\pi_n(x) = \frac{\lfloor 2^n x \rfloor}{2^n}$.
Establish $$\forall (x,n) \in \mathbb{R} \times \mathbb{N}, \quad \pi_n(x) \leqslant x < \pi_n(x) + \frac{1}{2^n}.$$

Let $\left( a _ { n } \right) _ { n \in \mathbb{N} }$ and $\left( b _ { n } \right) _ { n \in \mathbb{N} }$ be two sequences of elements of $\mathbb{R}^{+*}$. We assume that $\left( a _ { n } \right) _ { n \in \mathbb{N} }$ is decreasing and that $$\forall n \in \mathbb{N} , \quad \sum _ { k = 0 } ^ { n } a _ { k } b _ { n - k } = 1 .$$ We set, for $n \in \mathbb{N}$: $$B _ { n } = \sum _ { k = 0 } ^ { n } b _ { k } .$$ Let $m$ and $n$ be two natural integers such that $m > n$. Show that $$a _ { n } \leq \frac { 1 } { B _ { n } } \quad \text{and} \quad 1 \leq a _ { n } B _ { m - n } + a _ { 0 } \left( B _ { m } - B _ { m - n } \right) .$$

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

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