On the zeros of cosine polynomials: solution to a problem of Littlewood
成果类型：
Article
署名作者：
Borwein, P.; Erdelyi, T.; Ferguson, R.; Lockhart, R.
刊物名称：
ANNALS OF MATHEMATICS
ISSN/ISSBN：
0003-486X
DOI：
10.4007/annals.2008.167.1109
发表日期：
2008
页码：
1109-1117
关键词：
摘要：
Littlewood in his 1968 monograph Some Problems in Real and Complex Analysis [12, Problem 22] poses the following research problem, which appears to be still open: PROBLEM. If the n(j) are integral and all different, what is the lower bound on the number of real zeros of Sigma(N)(j=1) COS(n(j)theta)? Possibly N-1, or not much less. 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 Sigma(N)(j=1) COS(n(j)theta) with the n(j) integral and all different so that the number of its real zeros in the period [-pi,pi) is O (N-5/6 log N).