On the Zeros of Cosine Polynomials: Solution to a Problem of Littlewood
نویسندگان
چکیده
Littlewood in his 1968 monograph “Some Problems in Real and Complex Analysis” [12, problem 22] poses the following research problem, which appears to still be open: Problem. “If the nj are integral and all different, what is the lower bound on the number of real zeros of PN j=1 cos(njθ)? Possibly N −1, or not much less.” No progress appears to have been made on this in the last half century. We show that this is false. Theorem. There exists a cosine polynomial PN j=1 cos(njθ) with the nj integral and all different so that the number of its real zeros in the period is O “
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Lower Bounds for the Number of Zeros of Cosine Polynomials in the Period: a Problem of Littlewood
Abstract. Littlewood in his 1968 monograph “Some Problems in Real and Complex Analysis” [9, problem 22] poses the following research problem, which appears to still be open: “If the nm are integral and all different, what is the lower bound on the number of real zeros of PN m=1 cos(nmθ)? Possibly N − 1, or not much less.” Here real zeros are counted in a period. In fact no progress appears to h...
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