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Zeros of sections of exponential sums

Pavel Bleher·Robert Mallison jr
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Abstract

We derive the large $n$ asymptotics of zeros of sections of a generic exponential sum. We divide all the zeros of the $n$-th section of the exponential sum into ``genuine zeros'', which approach, as $n\to\infty$, the zeros of the exponential sum, and ``spurious zeros'', which go to infinity as $n\to\infty$. We show that the spurious zeros, after scaling down by the factor of $n$, approach a ``rosette'', a finite collection of curves on the complex plane, resembling the rosette. We derive also the large $n$ asymptotics of the ``transitional zeros'', the intermediate zeros between genuine and spurious ones. Our results give an extension to the classical results of Szeg\"o about the large $n$ asymptotics of zeros of sections of the exponential, sine, and cosine functions.

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Pavel Bleher, Robert Mallison jr. 2006-05-24. Zeros of sections of exponential sums. https://arxiv.org/abs/math-ph/0605066

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