On the zeros of cosine polynomials: solution to a problem of Littlewood

Abstract

No progress seems to have been made on this in the last half century. We show that this is false.Theorem. There exists a cosine polynomial N j=1 cos(n j θ) with the n j integral and all different so that the number of its real zeros in the period [−π, π) is O N 5/6 log N .

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“…In fact no progress appears to have been made on this in the last half century. In a recent paper [38] we showed that this is false. There exist cosine polynomials The book [13] deals with a number of related topics.…”

On the Weyl–Heisenberg frames generated by simple functions

“…Recently the problem has also been studied extensively for polynomials with restricted coefficients (e.g., p(z) = N n=0 a n z n where a n ∈ {0, 1} (Newman polynomial) or a n ∈ {−1, 1} (Littlewood polynomial) (see [1,3,21] respectively and the references therein)). There are also numerical results in connection with number theory [9] and analysis [2,19].…”

“…We say that (g, a, b) is a Weyl-Heisenberg frame (WH-frame for short) for L 2 holds for every f ∈ L 2 (R). We refer to [4,10,13] for some background materials and the recent development of this frame theory and related questions.…”

Lower bounds for the number of zeros of cosine polynomials in the period: a problem of Littlewood

“…In fact no progress appears to have been made on this in the last half-century. In a recent paper [2] we showed that this is false. There exists a cosine polynomial N m=1 cos(n m θ) with the n m integral and all different so that the number of its real zeros in the period is O(N 9/10 (log N ) 1/5 ) (here the frequencies n m = n m (N ) may vary with N ).…”