Inversion Formula for the Windowed Fourier Transform

TL;DR

This paper provides a rigorous proof for the inversion formula of the windowed Fourier transform, demonstrating its convergence properties in various function spaces, thereby solidifying its theoretical foundation.

Contribution

It offers a rigorous mathematical proof for the inversion formula of the windowed Fourier transform, confirming its convergence almost everywhere and in $L^p$ spaces.

Findings

The inversion formula converges almost everywhere on $bR$.

The integral converges in $L^p$ for all $1<p<inite$.

The proof solidifies the theoretical basis of the windowed Fourier transform inversion.

Abstract

In this paper, we study the inversion formula for recovering a function from its windowed Fourier transform. We give a rigorous proof for an inversion formula which is known in engineering. We show that the integral involved in the formula is convergent almost everywhere on $\bbR$ as well as in $L^p$ for all $1<p<\infty$ if the function to be reconstructed is.