proceedings of the american mathematical society Volume 118, Number 4, August 1993 ON THE ZEROS OF CERTAIN COSINE POLYNOMIALS A. B. MINGARELLI AND S. WANG (Communicated by J. Marshall Ash) Abstract. In a recent paper [The first sign change of a cosine polynomial, Proc. Amer. Math. Soc. Ill (1991), 709-716], Zeng studies the location of the first sign change of a cosine polynomial and thereby improves on results by Nulton and Stolarsky [The first sign change of a cosine polynomial, Proc. Amer. Math. Soc. 84 (1982), 55-59]. In this article we present a proof for each one of the four conjectures announced in Zeng's aforementioned paper and also discuss some extensions. 1. Introduction In this note a real cosine polynomial is an expression of the form n (1.1) f(x) = ^2 at cosXjX 1=1 where the a,■■, i = 1, ... , n , are real and 0 < Xx < ■■ ■< Xn, n>\. The study of the location of the first sign change of / was the object of a paper by Zeng [3] in which the following conjectures were presented. Conjecture 1.1 [3, p. 714]., The first sign change of the cosine polynomial (1.2) COS/li* + COSA2 H-r-cosA„x belongs to (0, n/Xx). Conjecture 1.2. The first sign change of (1.2) is a decreasing function of kx. Note. Zeng [3] proved the validity of these conjectures for n < 4 and pointed out that Conjecture 1.1 implies Conjecture 1.2. Conjecture 1.3. Let ax = 1 and let a2, ... , a„ all have the same sign. Then the first sign change of (1.1) belongs to (0, n/Xx). Conjecture 1.4. Under the assumption of Conjecture 1.3, // f(0) > 0 (resp. f(0) < 0), the first sign change of f(x) is a decreasing (resp. increasing) function of h- Received by the editors July 29, 1991 and, in revised form, November 27, 1991. 1991 Mathematics Subject Classification. Primary 33B10, 42A05. Key words and phrases. Cosine polynomial, zeros, almost periodic functions, first sign change. This research was partially funded by a grant from NSERC Canada. ©1993 American Mathematical Society 0002-9939/93 $1.00+ $.25 per page 1103 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use