Let $f(t)$ be a univariate polynomial of even degree $n$.
Markov-Lukacs theorem (reference,Eq.45) says this:
$f(t)$ is nonnegative in $t\in[-1,1]$ iff $$ f(t)=g(t)^2+(t+1)(1-t)h(t)^2 $$
where $g$ and $h$ are polynomials with $\operatorname{deg}(g) \leq n/2$ and $\operatorname{deg}(h) \leq (n-2)/2$.
SOS programming (reference, Theorem 6) says this:
$f(t)$ is nonnegative in $t\in[-1,1]$ iff $$ f(t)=s(t)+(t+1)(1-t)q(t) $$
where $s$ and $q$ are SOS polynomials with $\operatorname{deg}(s)\le n \text { and }\operatorname{deg}(q)\le n-2 $.
My question is: How could the first theorem be proven using the second one?
Note that, according to Exercise 3.30 of link, a univariate nonnegative polynomial can be expressed as a sum of two squares.
Thanks!