Real Analysis Exchange

On the gaps between zeros of trigonometric polynomials.

Gady Kozma and Ferencz Oravecz

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We show that for every finite set \(0\notin S\subset \mathbb{Z}^{d}\) with the property \(-S=S\), every real trigonometric polynomial \(f\) on the \(d\) dimensional torus \(\mathbb{T}^{d}=\mathbb{R}^{d}/\mathbb{Z}^{d}\) with spectrum in \(S\) has a zero in every closed ball of diameter \(D\left(S\right)\), where \(D\left(S\right)=\sum _{\lambda \in S}\frac{1}{4||\lambda ||_{2}}\), and investigate tightness in some special cases.

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Real Anal. Exchange, Volume 28, Number 2 (2002), 447-454.

First available in Project Euclid: 20 July 2007

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Primary: 42A05: Trigonometric polynomials, inequalities, extremal problems 42B99: None of the above, but in this section 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10} 26C10: Polynomials: location of zeros [See also 12D10, 30C15, 65H05]

trigonometric polynomials zeros roots


Kozma, Gady; Oravecz, Ferencz. On the gaps between zeros of trigonometric polynomials. Real Anal. Exchange 28 (2002), no. 2, 447--454.

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