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A class of positive trigonometric sums. II. (English) Zbl 0663.42002

We prove the following facts. Theorem A. Suppose that \((a_ k)^{\infty}_{k=1}\) is a nonincreasing sequence of nonnegative real numbers, such that \(a_ 1>0\) and \(2ka_{2k-1}\geq (2k+1)a_{2k}=(3k+2)a_{2k+1}\) for \(k=1,2,3,... \). Then for all positive integers N we have \[ (i)\quad a_ 1\sin \theta +a_ 2\sin 2\theta +...+a_{2N+1}\sin (2N+1)\theta >0\quad for\quad 0<\theta <\pi, \]
\[ (ii)\quad a_ 1\sin \theta +a_ 2\sin 2\theta +...+a_{2N}\sin 2N\theta >0\quad for\quad 0<\theta \leq \pi -\frac{1}{2N}\pi, \] and \[ (iii)\quad a_ 1\sin \theta +a_ 2\sin 2\theta +...+a_{N-1}\sin (N- 1)\theta +a_ N\log 2\sin N\theta >0\quad for\quad 0<\theta <\pi. \] Theorem B. Suppose that \((a_ k)^{\infty}_{k=1}\) is a nonincreasing sequence of nonnegative real numbers, such that \(a_ 1>0\) and \(2ka_{2k-1}\geq (2k+1)a_{2k}=(2k+1)a_{2k+1}\) for \(k=1,2,3,... \). Then for all positive integers N we have \[ (i)\quad a_ 1\sin \theta +a_ 2\sin 2\theta +...+a_{2N+1}\sin (2N+1)\theta >0\quad for\quad 0<\theta <\pi, \]
\[ (ii)\quad a_ 1\sin \theta +a_ 2\sin 2\theta +...+a_{2N}\sin 2N\theta >0\quad for\quad 0<\theta \leq \pi - \frac{1}{2N}\pi, \] and \[ (iii)\quad a_ 1\sin \theta +a_ 2\sin 2\theta +...+a_{N-1}\sin (N-1)\theta +\frac{3}{4}a_ N\sin N\theta >0\quad for\quad 0<\theta <\pi. \] Theorems A and B further develop (and correct) work of G. Brown and E. Hewitt [Math. Ann. (to appear) (1987; Zbl 0522.42001)], which was itself a generalization of L. Vietoris’ deep results [Österreich Akad. Wiss., Math. Naturw. Kl., S.-Ber., Abt. II 167, 125-135 (1958; Zbl 0088.274), Österreich. Akad. Wiss. Math. Naturw. Kl. Anzeiger 1959, 192-193 (1959; Zbl 0090.043)]. Theorem B provides an analogue for sines of the well-known positivity of the Rogosinski-Szegö cosine series. The proofs are in part modelled on those of Vietoris and Brown-Hewitt, and in part apply new techniques.
Reviewer: G.Brown

MSC:

42A05 Trigonometric polynomials, inequalities, extremal problems
42A32 Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)
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References:

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