A novel definition of real Fourier transform

TL;DR

This paper introduces a new definition of the Fourier transform that preserves the realness of functions after transformation, maintaining key properties and simplifying certain aspects of analysis.

Contribution

It proposes a novel Fourier transform definition that ensures the transform of a real function remains real, with proven inversion and retained desirable properties.

Findings

Transform of real functions remains real under the new definition

Inversion theorem is proven for the new Fourier transform

Shares properties with the classical Fourier transform

Abstract

We propose a novel definition of Fourier transform, with the property that the transform of a real function is again a real function (without doubling the number of real components). We prove the inversion theorem for the novel definition, and show that it shares the good properties of the usual definition.