The general theory of superoscillations and supershifts in several variables

TL;DR

This paper develops a general method to generate superoscillatory functions and supershifts in multiple variables using infinite order differential operators, extending previous one-variable results.

Contribution

It introduces a novel approach for constructing multivariable superoscillations and supershifts based on infinite order differential operators and growth conditions.

Findings

Method for generating multivariable superoscillations

Extension of results to supershifts in several variables

Framework based on holomorphic functions and differential operators

Abstract

In this paper we describe a general method to generate superoscillatory functions of several variables starting from a superoscillating sequence of one variable. Our results are based on the study of suitable infinite order differential operators on holomorphic functions with growth conditions of exponential type, where additional constraints are required when dealing with infinite order differential operators whose symbol is a function that is holomorphic in some open set, but not necessarily entire. The results proved for the superoscillating sequence in several variables are extended to sequences of supershifts in several variables.