Zeros of sections of exponential sums

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.