A generalized transform, grouping, Fourier, Laplace and Z transforms

The following is a summary description of some research results that were taught to students over many years and communicated by the author in IEEE Proceedings submission, in the book proposals, and poster sessions. A generalized transform, grouping Fourier and Laplace transform in the continuous-time domain and Fourier and z transform in the discrete-time domain is proposed. A generalized formalism is obtained for the continuous time domain and a similar one for the discrete time domain. The key to the existence of a generalized formalism and transform is based on a generalization of the formalism of generalized functions, and the Dirac-delta impulse in particular. A generalization of the Dirac-delta impulse is proposed, both in the continuous time domain and in the discrete time domain. The generalization of the impulse leads to a considerable extension of the domain of existence of Laplace and z transforms. Functions and sequences that have impulsive Fourier transforms and hitherto no Laplace or z transform are rendered within the region of convergence of these transforms. More importantly, a large class of functions and sequences that have no Fourier transform, such as two-sided infinite duration growing exponentials and exponentially diverging sinusoids, now have Laplace and z transforms. Two generalized impulses, namely, the Xi and Zeta impulses, defined on the complex transform plane are proposed. These generalized impulses may provide the missing link that bridges the gap between the theory of generalized functions as it applies to the Fourier transform and the more general Laplace and z transforms.