Trigonometric polynomials with many real zeros and a Littlewood-type problem

Borwein, Peter; Erdélyi, Tamás

doi:10.1090/S0002-9939-00-06021-4

Trigonometric polynomials with many real zeros and a Littlewood-type problem
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by Peter Borwein and Tamás Erdélyi PDF

Proc. Amer. Math. Soc. 129 (2001), 725-730

Abstract:

We examine the size of a real trigonometric polynomial of degree at most $n$ having at least $k$ zeros in $K := {\mathbb {R}} \ (\text {mod}\ 2\pi )$ (counting multiplicities). This result is then used to give a new proof of a theorem of Littlewood concerning flatness of unimodular trigonometric polynomials. Our proof is shorter and simpler than Littlewood’s. Moreover our constant is explicit in contrast to Littlewood’s approach, which is indirect.

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