Abstract
Several essential properties of the linear canonical transform (LCT) are provided. Some results related to the sampling theorem in the LCT domain are investigated. Generalized wave and heat equations on the real line are introduced, and their solutions are constructed using the sampling formulae. Some examples are presented to illustrate our results.
Highlights
The linear canonical transform (LCT) is one of nontrivial generalizations of the Fourier transform (FT)
Since the LCT is a general form of the fractional Fourier transform (FrFT) and it is closely related to the FT, it is possible to extend wave and heat equations into the LCT domain
We find a relation that the solution of the wave equation using the FT and the FrFT is a special case of the solution of these generalized wave equation
Summary
The linear canonical transform (LCT) is one of nontrivial generalizations of the Fourier transform (FT). The fundamental solution of the generalized wave equation was obtained using the fractional Fourier transform (FrFT) by [10]. Since the LCT is a general form of the FrFT and it is closely related to the FT, it is possible to extend wave and heat equations into the LCT domain. As far as we observe, up to now, the solution of generalized heat equation using the sampling formulae and the LCT has not yet been published in the literature. We emphasize that our solutions are nontrivial generalizations of the solutions of the heat equation using the sampling formulae and FT methods.
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