Inversion formula for the windowed special affine Fourier transform

The Windowed Special Affine Fourier Transform (WSAFT) extends classical windowed Fourier transforms through a parameter matrix, enabling precise signal adaptation and superior time-frequency localization for non-stationary signals. Unlike the special affine Fourier transform with its global kernel, WSAFT captures local frequency components that traditional methods cannot resolve. This work establishes a novel univariate integral reconstruction framework, overcoming computational limitations of conventional double-integration methods. We present that a signal can be reconstructed from its WSAFT with a univariate integral. For continuous window functions [Formula: see text] satisfying [Formula: see text], we prove that the reconstruction integral converges both in [Formula: see text]-norm for all [Formula: see text] and almost everywhere on [Formula: see text]. Through Cesàro summability, we extend these results to [Formula: see text] signals, achieving norm convergence for all [Formula: see text] and pointwise almost everywhere convergence. These theoretical findings are validated by representative examples using Gaussian window functions and signals.