Integral representation of superoscillations via complex Borel measures and their convergence

TL;DR

This paper provides a unified framework for superoscillations by defining them as sequences of holomorphic functions with integral representations via complex Borel measures, converging to plane waves.

Contribution

It introduces a general definition of superoscillations that encompasses existing examples and clarifies their mathematical representation and convergence properties.

Findings

Unified integral representation of superoscillations

Convergence to plane waves in exponential function space

Clarification of the relationship between different superoscillation representations

Abstract

In the last decade there has been a growing interest in superoscillations in various fields of mathematics, physics and engineering. However, while in applications as optics the local oscillatory behaviour is the important property, some convergence to a plane wave is the standard characterizing feature of a superoscillating function in mathematics and quantum mechanics. Also there exists a certain discrepancy between the representation of superoscillations either as generalized Fourier series, as certain integrals or via special functions. The aim of this work is to close these gaps and give a general definition of superoscillations, covering the well-known examples in the existing literature. Superoscillations will be defined as sequences of holomorphic functions, which admit integral representations with respect to complex Borel measures and converge to a plane wave in the space $\mathcal{A}_1(\mathbb{C})$ of exponentially bounded entire functions.