General Parameterized Fourier Transform: A Unified Framework for the Fourier, Laplace, Mellin and $Z$ Transforms

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