Superoscillations in the hypercomplex setting

TL;DR

This paper extends the concept of superoscillations to hypercomplex functions, exploring two notions of hyperholomorphic functions and their superoscillatory properties, enriching the mathematical framework beyond the complex case.

Contribution

It introduces a hypercomplex extension of superoscillatory functions, utilizing the Fueter-Sce theorem to explore two types of hyperholomorphic functions and their superoscillatory behavior.

Findings

Two notions of hyperholomorphic functions are identified.

Superoscillatory theories are developed for hypercomplex functions.

The hypercomplex setting enriches the mathematical understanding of superoscillations.

Abstract

Superoscillatory functions represent a counterintuitive phenomenon in physics but also in mathematics, where a band-limited function oscillates faster than its highest Fourier component. They appear in various contexts, including quantum mechanics, as a result of a weak measurement introduced by Y. Aharonov and collaborators. These functions can be extended to the complex variable and are a specific instance of the more general notion of supershift. The aim of this paper is to extend the notion of superoscillatory functions to the hypercomplex setting. This extension is richer than the complex case since the Fueter-Sce extension theorem for Clifford-valued functions provides two notions of hyperholomorphic functions. We will explore these notions and address the corresponding superoscillating theories.