Inversion formula for the windowed Fourier transform

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 $\mathbb {R}$ as well as in L^p for all 1 < p < ∞ if the function to be reconstructed is.