Lukács Theorem

Let $\rho(x)$ be an $m$th degree Polynomial which is Nonnegative in $[-1,1]$. Then $\rho(x)$ can be represented in the form

$$\cases{ [A(x)]^2+(1-x^2)[B(x)]^2 & for $m$\ even\cr (1+x)[C(x)]^2+(1-x)[D(x)]^2 & for $m$\ odd,\cr}$$

where $A(x)$, $B(x)$, $C(x)$, and $D(x)$ are Real Polynomials whose degrees do not exceed $m$.

References

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 4, 1975.