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Consider the trigonometric polynomial

of degree

such that not all of the

and

are zero. Babenko's theorem (1984) states that the measure of the subset of

for which

is at least

. The unique extremal polynomial, positive exactly on the interval

and normalized so that

, is constructed and shown for small values of

The extremal polynomial shown is

Its zeros are located at

. All the zeros are double except those at

H. L. Montgomery and U. M. A. Vorhauer, "Biased Trigonometric Polynomials," The American Mathematical Monthly, 114(9), 2007 pp. 804–808.

Additonal information can be found at Wikipedia.

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Extremal Trigonometric Polynomials