The Taylor Integral and a Generalization of the Discrete Fourier Transform

TL;DR

This paper introduces a novel integral based on Taylor measures that generalizes the discrete Fourier transform, explores its properties, and discusses its invertibility and applications in mathematical sciences.

Contribution

It proposes a new integral framework that encompasses many mathematical concepts and extends the discrete Fourier transform with conditions for invertibility.

Findings

The new integral generalizes the discrete Fourier transform.

Conditions for invertibility of the integral are identified.

Applications to mathematical sciences are demonstrated.

Abstract

We propose a new integral based on Taylor measures, study its properties extensively, and we illustrate that it includes many concepts from mathematics as special cases. In particular, the new integral emerges as a generalization of the discrete Fourier transform, and we identify general conditions for it to be invertible when applied to any real or complex sequence. Applications to the mathematical sciences are also presented.