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A Laplace transform of irregular growth

Jan Wiegerinck

TL;DR

This paper constructs an explicit example of a Laplace transform $\int_\gamma e^{zs} d\mu(s)$ with irregular growth, answering Korevaar’s question in Hayman’s List. It does so by building a canonical product whose zeros lie on circles of radius $2^k$, giving an exponential-type function with indicator $h(\theta)=2$, and then expressing the function via its Borel transform and a contour integral. By decomposing the transform into a bounded part and a remainder and demonstrating that the remainder cannot have regular growth, the work establishes the existence of a Laplace transform of irregular growth despite controlled exponential-type behavior. The result clarifies the distinction between regular and irregular growth in Laplace-type representations and highlights the intricate role of zero distribution and Borel transforms in growth properties of entire functions.

Abstract

We give an example of a Laplace transform $\int_γe^{ζz} dμ(ζ)$ that does not have regular growth. This answers a question in Hayman's List

A Laplace transform of irregular growth

TL;DR

This paper constructs an explicit example of a Laplace transform with irregular growth, answering Korevaar’s question in Hayman’s List. It does so by building a canonical product whose zeros lie on circles of radius , giving an exponential-type function with indicator , and then expressing the function via its Borel transform and a contour integral. By decomposing the transform into a bounded part and a remainder and demonstrating that the remainder cannot have regular growth, the work establishes the existence of a Laplace transform of irregular growth despite controlled exponential-type behavior. The result clarifies the distinction between regular and irregular growth in Laplace-type representations and highlights the intricate role of zero distribution and Borel transforms in growth properties of entire functions.

Abstract

We give an example of a Laplace transform that does not have regular growth. This answers a question in Hayman's List

Paper Structure

This paper contains 5 sections, 1 theorem, 13 equations.

Table of Contents

  1. Introduction
  2. Functions of exponential type
  3. Zeros
  4. Regular growth
  5. A Laplace transform of irregular growth

Key Result

Theorem 2.1

An entire function of exponential type has regular growth if and only if its zeros are regularly distributed.

Theorems & Definitions (1)

  • Theorem 2.1: cf. L, Chapter 3, Theorem 4