Holomorphic fractional Fourier transforms

TL;DR

The paper introduces the Holomorphic Fractional Fourier Transform (HFrFT), a novel transform that smoothly interpolates between signals and their Fourier transforms while maintaining unitarity, with potential broad applications.

Contribution

The paper proposes the HFrFT, a new holomorphic transform that extends the classical FrFT with improved properties and continuous interpolation between signals and their Fourier transforms.

Findings

HFrFT spans a one-parameter family of holomorphic functions.

HFrFT is unitary for all parameter values in (0, π/2).

HFrFT provides heat kernel smoothing while preserving unitarity.

Abstract

The Fractional Fourier Transform (FrFT) has widespread applications in areas like signal analysis, Fourier optics, diffraction theory, etc. The Holomorphic Fractional Fourier Transform (HFrFT) proposed in the present paper may be used in the same wide range of applications with improved properties. The HFrFT of signals spans a one-parameter family of (essentially) holomorphic functions, where the parameter takes values in the bounded interval $t\in (0,\pi/2)$. At the boundary values of the parameter, one obtains the original signal at $t=0$ and its Fourier transform at the other end of the interval $t=\pi/2$. If the initial signal is $L^2 $, then, for an appropriate choice of inner product that will be detailed below, the transform is unitary for all values of the parameter in the interval. This transform provides a heat kernel smoothening of the signals while preserving unitarity for $L^2$-signals and continuously interpolating between the original signal and its Fourier transform.