Extremal Trigonometric Polynomials

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 .


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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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