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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

[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 $m$.


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

© 1996-9 Eric W. Weisstein