Lukács Theorem

Let rho(x) be an mth 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; (1+x)[C(x)]^2+(1-x)[D(x)]^2   for m odd,

where A(x), B(x), C(x), and D(x) are real polynomials whose degrees do not exceed m.

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

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Lukács Theorem

Cite this as:

Weisstein, Eric W. "Lukács Theorem." From MathWorld--A Wolfram Web Resource.

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