Lp simulation for measures

TL;DR

This paper develops analogs of Lp spaces for measures motivated by Fourier analysis, extending many properties of classical Lp spaces and applying them to uncertainty principles and Fourier transform integrability.

Contribution

It introduces a new measure-based Lp space framework, extending classical properties and applying it to Fourier analysis problems.

Findings

Extended classical Lp space properties to measure-based spaces

Derived a version of the uncertainty principle in this setting

Proved integrability results for Fourier transforms of functions of bounded variation

Abstract

Being motivated by general interest as well as by certain concrete problems of Fourier Analysis, we construct analogs of the Lp spaces for measures. It turns out that most of standard properties of the usual Lp spaces for functions are extended to the measure setting. We illustrate the obtained results by examples and apply them to obtain a version of the uncertainty principle and an integrability result for the Fourier transform of a function of bounded variation.