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Zeros of sections of exponential sums

Pavel Bleher, Robert Mallison

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 Szegö about the large $n$ asymptotics of zeros of sections of the exponential, sine, and cosine functions.

Zeros of sections of exponential sums

Abstract

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

Paper Structure

This paper contains 14 sections, 316 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Zeros of exponential sums
  3. Zeros of sections of exponential sums
  4. Finite zeros
  5. Zeros of the main series
  6. Spurious zeros
  7. Transitional zeros
  8. Completeness of Zeros
  9. Limiting Distribution of Zeros on the Rosette
  10. Beyond Conditions P and Q
  11. Beyond Condition P
  12. Beyond Condition Q
  13. Uniform asymptotics of the zeros of $s_{n-1}(n\zeta)$ in the disk $D(1,\delta)$
  14. Uniform asymptotics of the function $s_n(n\zeta)$ in the disk $D(1,\delta)$

Figures (5)

  • Figure 1: The zeros of the $n=250$ section of exponential sum (\ref{['in6']}).
  • Figure 2: The convex hull for exponential sum (\ref{['in6']}), with $\lambda_1=8+2i,\; \lambda_2=4+7i,\;\lambda_3=-7+4i,\;\lambda_4=-6-6i,\;\lambda_5=1-8i,\;\lambda_6=6-4i,\;\lambda_7=4+4i,\;\lambda_8=-2-4i$.
  • Figure 3: The complex conjugate convex hull for exponential sum (\ref{['in6']}), and the corresponding rays $\mathcal{S}_{j,j+1}$, $j=1,\ldots,6$.
  • Figure 4: The generalized Szegö curve for $c=0.9$.
  • Figure 5: The zeros of the $n=200$ section of exponential sum (\ref{['bd4']}) for $m=4$.

Theorems & Definitions (14)

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  • ...and 4 more