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An inequality for the coefficients of a cosine polynomial. (English) Zbl 0833.26012
Summary: We prove: If \({1 \over 2} + \sum^n_{k = 1} a_k(n) \cos (kx) \geq 0\) for all \(x \in [0,2 \pi)\), then \(1 - a_k(n) \geq {1 \over 2} {k^2 \over n^2}\) for \(k = 1, \ldots, n\). The constant 1/2 is the best possible.
MSC:
26D05 Inequalities for trigonometric functions and polynomials
42A05 Trigonometric polynomials, inequalities, extremal problems
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