19. Show that the polynomials $T _ { n }$ are functions of positive type in dimension 2 . Hint: you 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 $N$.
19. Show that the polynomials $T _ { n }$ are functions of positive type in dimension 2 .
Hint: you 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 $N$.\\