The Generalized Fourier Transform: A Unified Framework for the Fourier,
Laplace, Mellin and Z Transforms
Abstract
This paper introduces Generalized Fourier transform (GFT) that is an
extension or the generalization of the Fourier transform (FT). The
Unilateral Laplace transform (LT) is observed to be the special case of
GFT. GFT, as proposed in this work, contributes significantly to the
scholarly literature. There are many salient contribution of this work.
Firstly, GFT is applicable to a much larger class of signals, some of
which cannot be analyzed with FT and LT. For example, we have shown the
applicability of GFT on the polynomially decaying functions and super
exponentials. Secondly, we demonstrate the efficacy of GFT in solving
the initial value problems (IVPs). Thirdly, the generalization presented
for FT is extended for other integral transforms with examples shown for
wavelet transform and cosine transform. Likewise, generalized Gamma
function is also presented. One interesting application of GFT is the
computation of generalized moments, for the otherwise non-finite
moments, of any random variable such as the Cauchy random variable.
Fourthly, we introduce Fourier scale transform (FST) that utilizes GFT
with the topological isomorphism of an exponential map. Lastly, we
propose Generalized Discrete-Time Fourier transform (GDTFT). The DTFT
and unilateral $z$-transform are shown to be the special cases of the
proposed GDTFT. The properties of GFT and GDTFT have also been
discussed.