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 $0 \leqslant j \leqslant n - 1$.
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 $[ a , b ]$.
\section*{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 $0 \leqslant j \leqslant n - 1$.\\