Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Application of methods of quasicrystals theory to entire functions of exponential growth

Sergii Yu. Favorov

Abstract

Let $f$ be an entire almost periodic function with zeros in a horizontal strip of finite width; for example, any exponential polynomial with purely imaginary exponents is such a function. Let $μ$ be the measure on the set of zeros of $f$ whose masses coincide with multiplicities of zeros. We define the Fourier transform in the sense of distributions for $μ$ and prove that it is a pure point measure on $\R$ whose complex masses correspond to coefficients of Dirichlet series of the logarithmic derivative of $f$. Bases on this description and Meyer's theorem on quasicrystals, we give a simple necessary and sufficient condition for $f$ to be a finite product of sines.

Application of methods of quasicrystals theory to entire functions of exponential growth

Abstract

Let be an entire almost periodic function with zeros in a horizontal strip of finite width; for example, any exponential polynomial with purely imaginary exponents is such a function. Let be the measure on the set of zeros of whose masses coincide with multiplicities of zeros. We define the Fourier transform in the sense of distributions for and prove that it is a pure point measure on whose complex masses correspond to coefficients of Dirichlet series of the logarithmic derivative of . Bases on this description and Meyer's theorem on quasicrystals, we give a simple necessary and sufficient condition for to be a finite product of sines.

Paper Structure

This paper contains 4 sections, 1 theorem, 57 equations.

Table of Contents

  1. Introduction
  2. Almost periodic functions and measures
  3. Some properties of the Fourier transform
  4. The proof of the Theorem

Key Result

Theorem 1

Let $f(z)$ be an entire almost periodic function of exponential growth with zeros in a strip and Then the condition where $K$ is a constant, is necessary and sufficient for the identity with some $C\in{\mathbb C},\ J\in{\mathbb N},\ a,\,\alpha_j,\,\beta_j\in{\mathbb R}$.

Theorems & Definitions (3)

  • Theorem
  • Definition 1
  • Definition 2