Integrability of the Fourier transform: functions of bounded variation

TL;DR

This paper explores the relationship between Fourier transforms of functions of bounded variation and their derivatives' Hilbert transforms, identifying the largest subspaces where Fourier transforms are integrable.

Contribution

It reveals new relations between Fourier and Hilbert transforms for bounded variation functions and characterizes maximal subspaces with integrable Fourier transforms.

Findings

Relations between Fourier and Hilbert transforms are established.

Largest subspaces with integrable Fourier transforms are identified.

Conditions for Fourier transform integrability are clarified.

Abstract

Certain relations between the Fourier transform of a function of bounded variation and the Hilbert transform of its derivative are revealed. The widest subspaces of the space of functions of bounded variation are indicated in which the cosine and sine Fourier transforms are integrable.