In this question only, we set $f(x) := \frac{1}{2}Lx^2$ for all $x \in \mathbb{R}$, where $L > 0$ is fixed. a) Show that $x_{n+1} = (1 - \tau L)x_n$, then express directly $x_n$ as a function of $x_0$ and $n$. b) We suppose $x_0 \neq 0$. Justify that $x_n \rightarrow 0$ if and only if $0 < \tau < 2/L$.

Let $F$ be an element of $\mathbb{C}[X^{\pm 1}]$. Prove that $\widehat{\xi}(\Pi(F)) = \widehat{\xi}(F)$.

Let $P$ be a polynomial and let $F$ be an element of $\mathcal{D}$. Prove that $P(\xi)(F) = \Pi(PF)$.

Let $\left( X _ { i } \right) _ { i \in [ 1 , n ] }$ be a sequence of independent random variables all following a Rademacher distribution. Show that $$\forall t \in \mathbf { R } , \quad \operatorname { ch } ( t ) \leq \mathrm { e } ^ { t ^ { 2 } / 2 }$$

Show that, for all $q \geq 2$, $$S _ { 1 , q } = ( - 1 ) ^ { q } \left( \phi _ { 1,1 } ( q - 2 ) - \ln 2 \right)$$

Let $n$ be a natural integer. Prove that $\xi^n$ is surjective and give a basis of the kernel of $\xi^n$.

Let $r$ be a nonzero natural integer. Prove that the smallest vector subspace $\mathcal{D}_r$ of $\mathcal{D}$ containing $X^{-r}$ and stable by $\xi$ has as basis $(X^{k-r})_{0 \leqslant k \leqslant r-1}$. Write the matrix of the endomorphism $\xi_{\mathcal{D}_r}$ induced by $\xi$ on $\mathcal{D}_r$ in this basis.

6. We assume that $f$ is absolutely monotone on $[ a , b ]$. Show that, for every polynomial $P \in \mathbb { R } [ X ]$ split in $] a , b [$, the function $Q ( f , P )$ is absolutely monotone on $[ a , b ]$.

Second Part

Let $I = [ - 1,1 ]$. We fix an integer $n \geqslant 2$ for this entire part. Let $f : I \rightarrow ] 0 , + \infty [$ be a continuous function. We recall that we define an inner product on $\mathbb { R } _ { n } [ X ]$ by setting, for all $P , Q \in \mathbb { R } [ X ]$,
$$\langle P , Q \rangle = \int _ { - 1 } ^ { 1 } P ( x ) Q ( x ) f ( x ) d x$$
Let $D \in \mathbb { R } _ { n } [ X ]$ be a polynomial having $n$ distinct real roots $r _ { 1 } > \cdots > r _ { n }$ in $I$. We further assume that $D \in \mathbb { R } _ { n - 1 } [ X ] ^ { \perp }$.
7a. Show that there exist real numbers $\lambda _ { 1 } , \ldots , \lambda _ { n }$ such that, for all $P \in \mathbb { R } _ { n - 1 } [ X ]$,
$$\int _ { - 1 } ^ { 1 } P ( x ) f ( x ) d x = \sum _ { i = 1 } ^ { n } \lambda _ { i } P \left( r _ { i } \right)$$
7b. Show that if $P \in \mathbb { R } _ { 2 n - 1 } [ X ]$, we have
$$\int _ { - 1 } ^ { 1 } P ( x ) f ( x ) d x = \sum _ { i = 1 } ^ { n } \lambda _ { i } P \left( r _ { i } \right)$$
Hint: one may consider the Euclidean division of $P$ by $D$. 7c. By evaluating equality 1 on the polynomial $\prod _ { \substack { 1 \leqslant j \leqslant n \\ j \neq i } } \left( X - r _ { j } \right) ^ { 2 }$, show that $\lambda _ { i } > 0$ for all $1 \leqslant i \leqslant n$. For $1 \leqslant j \leqslant n - 1$ and $t \in \mathbb { R }$, we set $f _ { j } ( t ) = \prod _ { i = 1 } ^ { j } \left( r _ { i } - t \right)$ as well as $f _ { 0 } ( t ) = 1$. If $0 \leqslant j \leqslant n - 1$ and $P , Q \in \mathbb { R } _ { n } [ X ]$, we set
$$\langle P , Q \rangle _ { j } = \left\langle P , Q f _ { j } \right\rangle$$
7d. Show that, for all $0 \leqslant j \leqslant n - 1 , \langle \cdot , \cdot \rangle _ { j }$ defines an inner product on $\mathbb { R } _ { n - j - 1 } [ X ]$. In questions 8. to 12. below, we fix a natural integer $0 \leqslant j \leqslant n - 1$.
8a. Show that there exists a unique family $q _ { 0 } , \ldots , q _ { n - j - 1 }$ of monic polynomials of $\mathbb { R } [ X ]$ such that $\operatorname { deg } \left( q _ { i } \right) = i$ for $0 \leqslant i \leqslant n - j - 1$ and such that for all $0 \leqslant i \neq i ^ { \prime } \leqslant n - j - 1$,
$$\left\langle q _ { i } , q _ { i ^ { \prime } } \right\rangle _ { j } = 0$$
8b. We set $q _ { n - j } = \prod _ { i = j + 1 } ^ { n } \left( X - r _ { i } \right)$. Show that $q _ { n - j }$ is the unique monic polynomial of degree $n - j$ satisfying, for all $0 \leqslant i \leqslant n - j - 1$,
$$\left\langle q _ { i } , q _ { n - j } \right\rangle _ { j } = 0$$
9a. Let $2 \leqslant i \leqslant n - j$. Show that there exist real numbers $a _ { i }$ and $b _ { i }$ such that
$$q _ { i } - X q _ { i - 1 } = a _ { i } q _ { i - 1 } + b _ { i } q _ { i - 2 }$$
9b. Show that
$$b _ { i } \left\langle q _ { i - 2 } , q _ { i - 2 } \right\rangle _ { j } = - \left\langle X q _ { i - 1 } , q _ { i - 2 } \right\rangle _ { j }$$
9c. Show that $b _ { i } < 0$.
10a. For $i \in \{ 0,1 \}$, show that the polynomial $q _ { i }$ has exactly $i$ roots in $\mathbb { R }$ (note that we do not require the roots to belong to the interval $I$ ).
10b. Show that, for all $1 \leqslant i \leqslant n - j$, the polynomial $q _ { i }$ has exactly $i$ distinct real roots, these roots are simple and if $x _ { 1 } < x _ { 2 }$ are two consecutive roots of $q _ { i }$, there exists a unique root of $q _ { i - 1 }$ in the interval $] x _ { 1 } , x _ { 2 } [$.
10c. Deduce that, for all $0 \leqslant i \leqslant n - j - 1$, we have $q _ { i } \left( r _ { j + 1 } \right) > 0$. For $0 \leqslant i \leqslant n - j - 1$, there therefore exists a unique real number $\alpha _ { i }$ such that
$$q _ { i + 1 } \left( r _ { j + 1 } \right) + \alpha _ { i } q _ { i } \left( r _ { j + 1 } \right) = 0$$
We fix $0 \leqslant i \leqslant n - j - 1$ and we set
$$p _ { i } = \frac { q _ { i + 1 } + \alpha _ { i } q _ { i } } { X - r _ { j + 1 } }$$
We denote $c _ { 0 } , \ldots , c _ { i } \in \mathbb { R }$ the coordinates of $p _ { i }$ in the basis $\left( q _ { 0 } , \ldots , q _ { i } \right)$ of $\mathbb { R } _ { i } [ X ]$. 11a. Show that, for $0 \leqslant \ell \leqslant i$,
$$\left\langle q _ { i + 1 } + \alpha _ { i } q _ { i } , \frac { q _ { \ell } - q _ { \ell } \left( r _ { j + 1 } \right) } { X - r _ { j + 1 } } \right\rangle _ { j } = 0$$
11b. Show that, for every integer $0 \leqslant \ell \leqslant i$, there exists a real $\gamma _ { \ell } > 0$ such that $c _ { \ell } = \gamma _ { \ell } c _ { 0 }$ and deduce that $c _ { \ell } > 0$.

6. We assume that $f$ is absolutely monotone on $[ a , b ]$. Show that, for every polynomial $P \in \mathbb { R } [ X ]$ split in $] a , b [$, the function $Q ( f , P )$ is absolutely monotone on $[ a , b ]$.

Second Part

Let $I = [ - 1,1 ]$. We fix an integer $n \geqslant 2$ for this entire part. Let $f : I \rightarrow ] 0 , + \infty [$ be a continuous function. We recall that we define an inner product on $\mathbb { R } _ { n } [ X ]$ by setting, for all $P , Q \in \mathbb { R } [ X ]$,
$$\langle P , Q \rangle = \int _ { - 1 } ^ { 1 } P ( x ) Q ( x ) f ( x ) d x$$
Let $D \in \mathbb { R } _ { n } [ X ]$ be a polynomial having $n$ distinct real roots $r _ { 1 } > \cdots > r _ { n }$ in $I$. We further assume that $D \in \mathbb { R } _ { n - 1 } [ X ] ^ { \perp }$.
Ya. Show that there exist real numbers $\lambda _ { 1 } , \ldots , \lambda _ { n }$ such that, for all $P \in \mathbb { R } _ { n - 1 } [ X ]$,
$$\int _ { - 1 } ^ { 1 } P ( x ) f ( x ) d x = \sum _ { i = 1 } ^ { n } \lambda _ { i } P \left( r _ { i } \right)$$
7b. Show that if $P \in \mathbb { R } _ { 2 n - 1 } [ X ]$, we have
$$\int _ { - 1 } ^ { 1 } P ( x ) f ( x ) d x = \sum _ { i = 1 } ^ { n } \lambda _ { i } P \left( r _ { i } \right)$$
Hint: one may consider the Euclidean division of $P$ by $D$. łc. By evaluating equality (1) on the polynomial $\prod _ { \substack { 1 \leqslant j \leqslant n \\ j \neq i } } \left( X - r _ { j } \right) ^ { 2 }$, show that $\lambda _ { i } > 0$ for all $1 \leqslant i \leqslant n$.
For $1 \leqslant j \leqslant n - 1$ and $t \in \mathbb { R }$, we set $f _ { j } ( t ) = \prod _ { i = 1 } ^ { j } \left( r _ { i } - t \right)$ as well as $f _ { 0 } ( t ) = 1$. If $0 \leqslant j \leqslant n - 1$ and $P , Q \in \mathbb { R } _ { n } [ X ]$, we set
$$\langle P , Q \rangle _ { j } = \left\langle P , Q f _ { j } \right\rangle .$$
7èd. Show that, for all $0 \leqslant j \leqslant n - 1 , \langle \cdot , \cdot \rangle _ { j }$ defines an inner product on $\mathbb { R } _ { n - j - 1 } [ X ]$./ In questions 8. to 12. below, we fix a natural integer $0 \leqslant j \leqslant n - 1$.

Verify that the set $$\mathcal{J} = \{P \in \mathbb{C}[X],\, P(u)(v) \in W\}$$ is an ideal of $\mathbb{C}[X]$.

Prove that there exists a natural integer $n$ such that $X^n \in \mathcal{J}$. Deduce that $\mathcal{J}$ is generated by the monomial $X^r$ for a suitable natural integer $r$ that we do not ask you to specify.

Problem 2, Part 2: Linear recurrence sequences with constant coefficients
We consider a sequence $\left( u _ { n } \right) _ { n \geqslant 0 }$ of complex numbers defined by the data of $u _ { 0 } , \ldots , u _ { d }$ and the linear recurrence relation $$u _ { n + d } = \sum _ { i = 0 } ^ { d - 1 } a _ { i } u _ { n + i } + b ,$$ where the $a _ { i }$ and $b$ are complex numbers. We define $P \in \mathbb { C } [ X ]$ by $P ( X ) = X ^ { d } - \sum _ { i = 0 } ^ { d - 1 } a _ { i } X ^ { i }$ and we assume that all complex roots of $P$ have modulus strictly less than 1.
For $n \geqslant 0$ we define the vector $U _ { n } \in \mathbb { C } ^ { d }$ by $U _ { n } = \left( u _ { n } , \ldots , u _ { n + d - 1 } \right)$ (recall that $U _ { n }$ is identified with a column vector). Show that the sequence $(U _ { n })$ satisfies a recurrence relation of the form $U _ { n + 1 } = A U _ { n } + B$, with $A \in \mathrm { M } _ { d } ( \mathbb { C } )$ and $B \in \mathbb { C } ^ { d }$ are elements that we shall specify.

Problem 2, Part 2: Linear recurrence sequences with constant coefficients
We consider a sequence $\left( u _ { n } \right) _ { n \geqslant 0 }$ of complex numbers defined by the data of $u _ { 0 } , \ldots , u _ { d }$ and the linear recurrence relation $$u _ { n + d } = \sum _ { i = 0 } ^ { d - 1 } a _ { i } u _ { n + i } + b ,$$ where the $a _ { i }$ and $b$ are complex numbers. We define $P \in \mathbb { C } [ X ]$ by $P ( X ) = X ^ { d } - \sum _ { i = 0 } ^ { d - 1 } a _ { i } X ^ { i }$ and we assume that all complex roots of $P$ have modulus strictly less than 1. The matrix $A \in \mathrm{M}_d(\mathbb{C})$ is as defined in question 7.
Calculate the characteristic polynomial of the matrix $A$ (one may reason by induction on $d$).

8. Show that there exists a unique family $q _ { 0 } , \ldots , q _ { n - j - 1 }$ of monic polynomials of $\mathbb { R } [ \boldsymbol { X } ]$ such that $\operatorname { deg } \left( q _ { i } \right) = i$ for $0 \leqslant i \leqslant n - j - 1$ and such that for all $0 \leqslant i \neq i ^ { \prime } \leqslant n - j - 1$,
$$\left\langle q _ { i } , q _ { i ^ { \prime } } \right\rangle _ { j } = 0 .$$
8t. We set $q _ { n - j } = \prod _ { i = j + 1 } ^ { n } \left( X - r _ { i } \right)$. Show that $q _ { n - j }$ is the unique monic polynomial of degree $n - j$ satisfying, for all $0 \leqslant i \leqslant n - j - 1$,
$$\left\langle q _ { i } , q _ { n - j } \right\rangle _ { j } = 0$$
Let $2 \leqslant i \leqslant n - j$. Show that there exist real numbers $a _ { i }$ and $b _ { i }$ such that
$$q _ { i } - X q _ { i - 1 } = a _ { i } q _ { i - 1 } + b _ { i } q _ { i - 2 }$$
9b. Show that
9c. Show that $b _ { i } < 0$.
$$b _ { i } \left\langle q _ { i - 2 } , q _ { i - 2 } \right\rangle _ { j } = - \langle \underbrace { X q _ { i - 1 } , q _ { i - 2 } } _ { \geqslant 0 } \rangle _ { j } j .$$

1. For $i \in \{ 0,1 \}$, show that the polynomial $q _ { i }$ has exactly $i$ roots in $\mathbb { R }$ (note that we do not require the roots to belong to the interval $I$ ).

10b. Show that, for all $1 \leqslant i \leqslant n - j$, the polynomial $q _ { i }$ has exactly $i$ distinct real roots, that these roots are simple and that if $x _ { 1 } < x _ { 2 }$ are two consecutive roots of $q _ { i }$, there exists a unique root of $q _ { i - 1 }$ in the interval $] x _ { 1 } , x _ { 2 } [$. 10c. Deduce that, for all $0 \leqslant i \leqslant n - j - 1$, we have $q _ { i } \left( r _ { j + 1 } \right) > 0$. For $0 \leqslant i \leqslant n - j - 1$, there therefore exists a unique real number $\alpha _ { i }$ such that
$$q _ { i + 1 } \left( r _ { j + 1 } \right) + \alpha _ { i } q _ { i } \left( r _ { j + 1 } \right) = 0$$
We fix $0 \leqslant i \leqslant n - j - 1$ and we set
$$p _ { i } = \frac { q _ { i + 1 } + \alpha _ { i } q _ { i } } { X - r _ { j + 1 } }$$
We denote $c _ { 0 } , \ldots , c _ { i } \in \mathbb { R }$ the coordinates of $p _ { i }$ in the basis $\left( q _ { 0 } , \ldots , q _ { i } \right)$ of $\mathbb { R } _ { i } [ X ]$. 11a. Show that, for $0 \leqslant \ell \leqslant i$,
$$\left\langle q _ { i + 1 } + \alpha _ { i } q _ { i } , \frac { q _ { \ell } - q _ { \ell } \left( r _ { j + 1 } \right) } { X - r _ { j + 1 } } \right\rangle _ { j } = 0 .$$
11b. Show that, for every integer $0 \leqslant \ell \leqslant i$, there exists a real $\gamma _ { \ell } > 0$ such that $c _ { \ell } = \gamma _ { \ell } c _ { 0 }$ and deduce that $c _ { \ell } > 0$.

Write, in Python language, a function \texttt{mat\_adj(graph)} which takes as argument a graph of $m \in \mathbb{N}^*$ vertices, directed or undirected, represented by a dictionary having as keys the integers from 0 to $m-1$, and for value associated with such a key the adjacency list of the corresponding vertex, and which returns, respecting the enumeration of the vertices, the adjacency matrix of this graph.
Thus \texttt{mat\_adj(\{0: [1, 2], 1: [0], 2: [0, 1, 2]\})} must return $$[[0,1,1],[1,0,0],[1,1,1]].$$

12. Show that, if $0 \leqslant j \leqslant n - 2$, for all $0 \leqslant i \leqslant n - j - 1$, the polynomial $p _ { i }$ is orthogonal to $\mathbb { R } _ { i - 1 } [ X ]$ for the inner product $\langle \cdot , \cdot \rangle _ { j + 1 }$.

13. Let $\mathscr { B } = \left( a _ { 0 } , \ldots , a _ { n } \right)$ be the unique orthogonal basis of $\left( \mathbb { R } _ { n } [ X ] , \langle \cdot , \cdot \rangle \right)$ such that $a _ { i }$ is a monic polynomial of degree $i$ for all $0 \leqslant i \leqslant n$. Show that, for all $0 \leqslant j \leqslant n - 1$, the coefficients of the polynomial $\prod _ { \ell = j + 1 } ^ { n } \left( X - r _ { \ell } \right)$ in the basis $\mathscr { B }$ are strictly positive real numbers. Hint: one may denote $\left( q _ { j , 0 } , \ldots , q _ { j , n - j } \right)$ the basis of $\left( \mathbb { R } _ { n - j } [ X ] , \langle \cdot , \cdot \rangle _ { j } \right)$ obtained in questions $8 a$ and $8 b$ and reason by descending induction on $j$.

Third Part

Let $\lambda$ be a strictly positive real number. For all real $x$ and $r$ such that $| x | < 1$ and $| r | < 1$, we set
$$F _ { \lambda } ( x , r ) = \left( 1 - 2 r x + r ^ { 2 } \right) ^ { - \lambda }$$

1. Show that the function $F _ { \lambda }$ is of class $\mathscr { C } ^ { \infty }$ on $] - 1,1 \left[ ^ { 2 } \right.$.
2. Show that for $x \in ] - 1,1 \left[ \right.$, the function $r \mapsto F _ { \lambda } ( x , r )$ is expandable as a power series in a neighborhood of 0 . For $x \in ] - 1,1 \left[ \right.$, we denote $a _ { n } ^ { ( \lambda ) } ( x )$ the $n$-th coefficient of the expansion of the function $r \mapsto F _ { \lambda } ( x , r )$ so that, for $r$ in a neighborhood of 0 ,

$$F _ { \lambda } ( x , r ) = \sum _ { n \geqslant 0 } a _ { n } ^ { ( \lambda ) } ( x ) r ^ { n }$$
16a. For $x \in ] - 1,1 \left[ \right.$, show that $a _ { 1 } ^ { ( \lambda ) } ( x ) = 2 \lambda x a _ { 0 } ^ { ( \lambda ) } ( x )$ and that, for every integer $n \geqslant 1$,
$$( n + 1 ) a _ { n + 1 } ^ { ( \lambda ) } ( x ) = 2 ( n + \lambda ) x a _ { n } ^ { ( \lambda ) } ( x ) - ( n + 2 \lambda - 1 ) a _ { n - 1 } ^ { ( \lambda ) } ( x )$$
Hint: one may begin by computing $\left( 1 - 2 x r + r ^ { 2 } \right) \frac { \partial F _ { \lambda } } { \partial r } ( x , r )$. 16b. Deduce that, for all $n \geqslant 0$, the function $a _ { n } ^ { ( \lambda ) }$ is a polynomial of degree $n$ whose leading coefficient and parity will be determined. We now assume that $\lambda > \frac { 1 } { 2 }$. For $P , Q \in \mathbb { R } [ X ]$, we set
$$\langle P , Q \rangle = \int _ { - 1 } ^ { 1 } P ( x ) Q ( x ) \left( 1 - x ^ { 2 } \right) ^ { \lambda - \frac { 1 } { 2 } } d x$$

Let $(u,v,r,s) \in (\mathbb{N}^*)^4$. Let $A = (a_{ij})_{\substack{1 \leqslant i \leqslant u \\ 1 \leqslant j \leqslant v}} \in \mathcal{M}_{u,v}(\mathbb{R})$ and $B \in \mathcal{M}_{r,s}(\mathbb{R})$. We define the Kronecker product of $A$ by $B$, and we denote $A \otimes B$, the matrix of $\mathcal{M}_{ur,vs}(\mathbb{R})$ which is defined by $uv$ blocks of size $r \times s$ in such a way that, for all $(i,j) \in \llbracket 1,u \rrbracket \times \llbracket 1,v \rrbracket$, the block with index $(i,j)$ is $a_{i,j}B$.
Besides $u,v,r$ and $s$, we are also given two non-zero natural integers, $w$ and $t$.
Show that, for all matrices $A \in \mathcal{M}_{u,v}(\mathbb{R})$, $A' \in \mathcal{M}_{v,w}(\mathbb{R})$, $B \in \mathcal{M}_{r,s}(\mathbb{R})$ and $B' \in \mathcal{M}_{s,t}(\mathbb{R})$, $$(A \otimes B)(A' \otimes B') = (AA') \otimes (BB').$$

Let $\left( X _ { i } \right) _ { i \in \mathbf { N } }$ be a sequence of independent random variables all following a Rademacher distribution. Let the map $\psi : u \in \mathbf { R } ^ { ( \mathbf { N } ) } \mapsto \sum _ { i = 0 } ^ { + \infty } u _ { i } X _ { i }$. Show that $\psi$ takes its values in $L ^ { 0 } ( \Omega )$, then that $\psi$ preserves the inner product.

Justify that the function $\psi_n$ is differentiable on $\mathbb{R}_+$ and that $\psi_n' = m_n$.
We admit that, when $\psi$ is differentiable on $\mathbb{R}_+^*$, then $(\lim \psi_n)' = \lim \psi_n'$, that is $\psi' = m$, on $\mathbb{R}_+^*$.

Let $\left( X _ { i } \right) _ { i \in \mathbf { N } }$ be a sequence of independent random variables all following a Rademacher distribution. Let the map $\psi : u \in \mathbf { R } ^ { ( \mathbf { N } ) } \mapsto \sum _ { i = 0 } ^ { + \infty } u _ { i } X _ { i }$. We denote $R = \psi \left( \mathbf { R } ^ { ( \mathbf { N } ) } \right)$. Show that for all $p , q \in \left[ 1 , + \infty \right[$, the norms $\| \cdot \| _ { p }$ and $\| \cdot \| _ { q }$ are equivalent on $R$.

19. Show that the polynomials $T _ { n }$ are functions of positive type in dimension 2.
Hint: one may use the exponential form of the cosine. We shall admit, in the rest of the problem, that for every integer $n \geqslant 0$ and every integer $N \geqslant 4$, the polynomial $a _ { n } ^ { \left( \frac { N } { 2 } - 1 \right) }$ is of positive type in dimension $N$. For an integer $N \geqslant 2$, we say that a polynomial $P \in \mathbb { R } [ X ]$ is $N$-conductive if, for every absolutely monotone function $f$ from $[ - 1,1 ]$ to $\mathbb { R }$, the polynomial $H ( f , P )$ is a function of positive type in dimension $N$.

In this part, we assume that $n$ is a power of 2: we write $n = 2 ^ { k }$ with $k \in \mathbf { N } ^ { \star }$. Let $\left( a _ { 1 } , \ldots , a _ { k } \right) \in \mathbf { R } ^ { k }$. Show that $$\alpha _ { 1 } n \left\| \left( a _ { 1 } , \ldots , a _ { k } \right) \right\| _ { 2 } ^ { \mathbf { R } ^ { k } } \leq \sum _ { \left( \varepsilon _ { 1 } , \ldots , \varepsilon _ { k } \right) \in \{ - 1,1 \} ^ { k } } \left| \sum _ { i = 1 } ^ { k } \varepsilon _ { i } a _ { i } \right| \leq \beta _ { 1 } n \left\| \left( a _ { 1 } , \ldots , a _ { k } \right) \right\| _ { 2 } ^ { \mathbf { R } ^ { k } } .$$ You may use questions 11 and 16.

20. Let $P _ { 1 }$ and $P _ { 2 }$ be two $N$-conductive polynomials. Show that if $P _ { 1 }$ is of positive type in dimension $N$, then $P _ { 1 } P _ { 2 }$ is $N$-conductive. We fix an integer $N \geqslant 4$ and an integer $n \geqslant 2$. We admit that the polynomial $a _ { n } ^ { \left( \frac { N } { 2 } - 1 \right) }$ has $n$ simple real roots $r _ { 1 } > r _ { 2 } > \cdots > r _ { n }$ in $] - 1,1 [$. Let $f : [ - 1,1 ] \rightarrow \mathbb { R }$ be an absolutely monotone function.

21. Show that the polynomial $H \left( f , \prod _ { i = 1 } ^ { n } \left( X - r _ { i } \right) \right)$ is a function of positive type in dimension $N$.