Fourier transform of BV functions and applications

TL;DR

This paper explores the Fourier transform properties of BV functions, introduces new identities and characterizations related to sets of finite perimeter, and provides sharp bounds on quadratic discrepancy, advancing the understanding of BV function analysis.

Contribution

It introduces the $L^2$-jump product, a weighted Plancherel identity, and new characterizations of finite perimeter sets via Fourier transforms, extending classical results.

Findings

Characterization of finite perimeter sets through Fourier transform.

Sharp bounds on quadratic discrepancy of BV functions.

Generalization of Beck and Montgomery's estimates.

Abstract

This paper investigates the relation between the Fourier transform of {\rm BV} (bounded variation) functions and their jump sets. We introduce the notion of $L^2$-jump product and obtain a weighted Plancherel identity for {\rm BV} functions. As a corollary, we get a newfound characterization of sets of finite perimeter in terms of their Fourier transform. Moreover, we sharpen a result of Herz on the set-theoretic derivative of the Fourier transform of characteristic functions of sets. Last, we obtain sharp bounds on the quadratic discrepancy of {\rm BV} functions, and as a consequence, we generalize the classic estimates of Beck and Montgomery.