Show that there exist $K$ polynomials $L_{1}, \ldots, L_{K}$ in $\mathbb{R}_{K-1}[X]$ such that, for any function $f \in \mathcal{C}^{K}([0,1])$, the polynomial $P = \sum_{j=1}^{K} f\left(x_{j}\right) L_{j}$ satisfies $$\forall \ell \in \llbracket 1, K \rrbracket, \quad P\left(x_{\ell}\right) = f\left(x_{\ell}\right).$$
We fix $f \in \mathcal{C}^{K}([0,1])$ and denote by $P$ the polynomial determined in question Q7. For all $k \in \llbracket 0, K-1 \rrbracket$, show that there exist at least $K - k$ distinct real numbers in $[0,1]$ at which the function $f^{(k)} - P^{(k)}$ vanishes.
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.
Let $K \in \mathbb{N}^{\star}$, consider distinct real numbers $x_1 < \cdots < x_K$ in an interval $[a,b]$ (with $a < b$), and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[a,b]$ with real values satisfying: (H1) the function series $\sum f_n^{(K)}$ converges normally on $[a,b]$; (H2) for all $\ell \in \llbracket 1, K \rrbracket$ the numerical series $\sum f_n(x_\ell)$ is absolutely convergent. In the special case $[a,b] = [0,1]$, justify that the series $\sum f_n^{(k)}$ converges normally on $[a,b]$ for all $k \in \llbracket 0, K-1 \rrbracket$.
Let $K \in \mathbb{N}^{\star}$, consider distinct real numbers $x_1 < \cdots < x_K$ in an interval $[a,b]$ (with $a < b$), and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[a,b]$ with real values satisfying: (H1) the function series $\sum f_n^{(K)}$ converges normally on $[a,b]$; (H2) for all $\ell \in \llbracket 1, K \rrbracket$ the numerical series $\sum f_n(x_\ell)$ is absolutely convergent. Treat the question of showing that $\sum f_n^{(k)}$ converges normally on $[a,b]$ for all $k \in \llbracket 0, K-1 \rrbracket$ in the general case of a segment $[a,b]$ with $a < b$. One may examine $f_n \circ \sigma$ where $\sigma : [0,1] \rightarrow [a,b]$ is defined by $\sigma(t) = (1-t)a + tb$ for all $t \in [0,1]$.
Let $K \in \mathbb{N}^{\star}$, consider distinct real numbers $x_1 < \cdots < x_K$ in an interval $[a,b]$ (with $a < b$), and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[a,b]$ satisfying hypotheses (H1) and (H2). According to the result of the previous question, we set $F_k(x) = \sum_{n=0}^{+\infty} f_n^{(k)}(x)$ for all $x \in [a,b]$. Prove that $F_0$ is of class $\mathcal{C}^K$ on $[a,b]$ and that $F_0^{(k)} = F_k$ for all $k \in \llbracket 1, K \rrbracket$.
For all $n \in \mathbb{N}^{\star}$, justify that there exists a unique function $f_n \in \mathcal{C}^{2}(]0, +\infty[)$ satisfying $f_n(1) = 0$, $f_n(2) = 0$ and $f_n^{\prime\prime}(x) = (-1)^n 2^{-nx^2}$ for all $x > 0$.
Show that the function series $\sum f_n(x)$ converges normally on any segment contained in $]0, +\infty[$ and that the function $F : x \mapsto \sum_{n=1}^{+\infty} f_n(x)$ is of class $\mathcal{C}^2$ on $]0, +\infty[$.
Let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that the series $\sum a_n^2$ converges. Justify the existence of a strictly increasing sequence of natural integers $(\phi(j))_{j \in \mathbb{N}}$ satisfying $$\forall j \in \mathbb{N}, \quad \sum_{n > \phi(j)}^{+\infty} a_n^2 \leqslant \frac{1}{8^j}.$$
Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that $\sum a_n^2$ converges. For all $N \in \mathbb{N}$, let $S_N = \sum_{n=0}^N X_n a_n$. Express the expectation and variance of $S_{\phi(j+1)} - S_{\phi(j)}$ in terms of the terms of the sequence $(a_n)_{n \in \mathbb{N}}$.
Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ for all $n \in \mathbb{N}$, and let $(a_n)_{n \in \mathbb{N}}$ be a real sequence such that $\sum a_n^2$ converges. Let $S_N = \sum_{n=0}^N X_n a_n$ and let $A_j = \{|S_{\phi(j+1)} - S_{\phi(j)}| > 2^{-j}\}$. Using the sequence $(\phi(j))_{j \in \mathbb{N}}$ satisfying $\sum_{n > \phi(j)}^{+\infty} a_n^2 \leqslant \frac{1}{8^j}$, deduce the bound $\mathbb{P}(A_j) \leqslant 2^{-j}$.
Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of mutually independent random variables satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$, and let $S_N = \sum_{n=0}^N X_n a_n$. With the events $$B_{j,m} = \left\{|S_m - S_{\phi(j)}| > 2^{-j} \text{ and } \forall n \in \llbracket \phi(j), m-1 \rrbracket, \; |S_n - S_{\phi(j)}| \leqslant 2^{-j}\right\},$$ $$B_j = \left\{\max_{\phi(j)+1 \leqslant n \leqslant \phi(j+1)} |S_n - S_{\phi(j)}| > 2^{-j}\right\},$$ for all $j \in \mathbb{N}$, prove that the events $B_{j,m}$, for $m$ ranging over $\llbracket \phi(j)+1, \phi(j+1) \rrbracket$, are pairwise disjoint and that we have the equality of events $$B_j = \bigcup_{\phi(j) < m \leqslant \phi(j+1)} B_{j,m}.$$
With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$, events $A_j$, $B_j$, $B_{j,m}$), explain how to deduce the formula $$\mathbb{P}(A_j) = \sum_{m=\phi(j)+1}^{\phi(j+1)} \mathbb{P}(A_j \cap B_{j,m}).$$
With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$, events $A_j$, $B_j$, $B_{j,m}$), prove that if the event $B_j$ occurs, then there exist $m \in \llbracket \phi(j)+1, \phi(j+1) \rrbracket$ and $\alpha \in \{-1, +1\}$ such that the event $$\left\{\left|\alpha S_{\phi(j+1)} - \alpha S_m + S_m - S_{\phi(j)}\right| > 2^{-j}\right\} \cap B_{j,m}$$ also occurs. One may express $S_m - S_{\phi(j)}$ in terms of the two numbers $\alpha S_{\phi(j+1)} - \alpha S_m + S_m - S_{\phi(j)}$ with $\alpha = \pm 1$.
With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$, events $A_j$, $B_j$, $B_{j,m}$), deduce that $$\mathbb{P}(B_j) \leqslant 2\mathbb{P}(A_j).$$
With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$, events $B_j$), denote by $B$ the event $\bigcap_{J \in \mathbb{N}} \bigcup_{j \geqslant J} B_j$. Show the equality $\mathbb{P}(B) = 0$.
With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$, events $B_j$, $B_{j,m}$), show that the event $$\left\{\exists J \in \mathbb{N}, \quad \forall j \geqslant J, \quad \forall n \in \llbracket \phi(j)+1, \phi(j+1) \rrbracket, \quad |S_n - S_{\phi(j)}| \leqslant 2^{-j}\right\}$$ occurs with probability 1.
With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$), deduce that the event $$\left\{\text{the sequence } \left(S_{\phi(j)}\right)_{j \in \mathbb{N}} \text{ is convergent}\right\}$$ also has probability 1. One may examine the series $\sum |S_{\phi(j+1)} - S_{\phi(j)}|$.
With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$), conclude that the event $$\left\{\text{the series } \sum X_n a_n \text{ is convergent}\right\}$$ has probability 1.
We fix $K \in \mathbb{N}^{\star}$ and consider a sequence of random variables $(X_n)_{n \in \mathbb{N}}$ satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$, distinct real numbers $x_1 < \cdots < x_K$ in $[0,1]$, and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[0,1]$ satisfying: (H1) the function series $\sum f_n^{(K)}$ converges normally on $[0,1]$; (H2') for all $\ell \in \llbracket 1, K \rrbracket$, the numerical series $\sum f_n(x_\ell)^2$ is convergent. Show that one of the two hypotheses (H2') or (H2) (where (H2) states that for all $\ell \in \llbracket 1, K \rrbracket$ the numerical series $\sum f_n(x_\ell)$ is absolutely convergent) implies the other.
We fix $K \in \mathbb{N}^{\star}$ and consider a sequence of random variables $(X_n)_{n \in \mathbb{N}}$ satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ (mutually independent), distinct real numbers $x_1 < \cdots < x_K$ in $[0,1]$, and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[0,1]$ satisfying hypotheses (H1) and (H2'). Show that the event $$\left\{\text{for all } \ell \in \llbracket 1, K \rrbracket, \text{ the series } \sum X_n f_n(x_\ell) \text{ is convergent}\right\}$$ has probability 1.
We fix $K \in \mathbb{N}^{\star}$ and consider a sequence of random variables $(X_n)_{n \in \mathbb{N}}$ satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ (mutually independent), distinct real numbers $x_1 < \cdots < x_K$ in $[0,1]$, and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[0,1]$ satisfying hypotheses (H1) and (H2'). Let $P_n \in \mathbb{R}_{K-1}[X]$ be a polynomial satisfying $P_n(x_\ell) = f_n(x_\ell)$ for all $\ell \in \llbracket 1, K \rrbracket$ (cf. question 7). Show that the event $$\left\{\begin{array}{l}
\text{for all } k \in \llbracket 0, K \rrbracket, \text{ the function series } \sum X_n (f_n - P_n)^{(k)} \text{ is uniformly convergent on } [0,1], \\
\text{the function } \sum_{n=0}^{+\infty} X_n (f_n - P_n) \text{ is of class } \mathcal{C}^K, \\
\text{for all } k \in \llbracket 0, K \rrbracket, \left(\sum_{n=0}^{+\infty} X_n (f_n - P_n)\right)^{(k)} = \sum_{n=0}^{+\infty} X_n (f_n - P_n)^{(k)}
\end{array}\right\}$$ has probability 1.
We fix $K \in \mathbb{N}^{\star}$ and consider a sequence of random variables $(X_n)_{n \in \mathbb{N}}$ satisfying $\mathbb{P}(X_n = -1) = \mathbb{P}(X_n = 1) = \frac{1}{2}$ (mutually independent), distinct real numbers $x_1 < \cdots < x_K$ in $[0,1]$, and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[0,1]$ satisfying hypotheses (H1) and (H2'). Show that the event $$\left\{\begin{array}{l}
\text{for all } k \in \llbracket 0, K \rrbracket, \text{ the function series } \sum X_n f_n^{(k)} \text{ is uniformly convergent on } [0,1], \\
\text{the function } \sum_{n=0}^{+\infty} X_n f_n \text{ is of class } \mathcal{C}^K, \\
\text{for all } k \in \llbracket 0, K \rrbracket, \left(\sum_{n=0}^{+\infty} X_n f_n\right)^{(k)} = \sum_{n=0}^{+\infty} X_n f_n^{(k)}
\end{array}\right\}$$ has probability 1.