As we know, the uncertainty principle is one of the fundamental results of the WLCT, which explains how an original function interacts with its WLCT. We have provided different proofs of the WLCT properties like the orthogonality relation, inversion theorem, and complex conjugation using the direct interaction among the windowed linear canonical transform, the windowed Fourier transform and the Fourier transform, the proofs of which are simpler than those the authors proposed in. In this work, we developed this approach within the framework of the linear canonical transform. In, the authors have investigated the fundamental properties of the continuous shearlet transforms using the direct interaction between the Fourier transform and shearlet transform. According to this idea, some properties of the fractional Fourier transform can be easily obtained using the basic connection between the fractional Fourier transform and Fourier transform. In, the author has discussed that the fractional Fourier transform is intimately related to the Fourier transformation. They also have investigated its essential properties like linearity, orthogonality relation, inversion theorem, and the inequalities. The generalized transform is built by including the Fourier kernel with the LCT kernel in the definition of the windowed Fourier transform. Some authors have introduced an extension of the WFT in the LCT domain, the so-called windowed linear canonical transform (WLCT). ![]() ![]() In recent years, a number of efforts have been made with an increasing interest in expanding various types of transformations in the context of the linear canonical transform (LCT), we refer the reader to the papers. As it is well known, the classical windowed Fourier transform (WFT) is a useful mathematical tool, which has been broadly studied in quantum physics, signal processing and many other fields of science and engineering.
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