General Parameterized Fourier Transform: A Unified Framework for the Fourier, Laplace, Mellin and $Z$ Transforms
Pushpendra Singh, Anubha Gupta, Shiv Dutt Joshi
Abstract
This paper introduces the general parameterized Fourier transform (GP-FT) that is an extension of the Fourier transform (FT). This study makes a significant scholarly contribution. Firstly, GP-FT is applicable to a much larger class of signals, some of which cannot be analyzed with FT and Laplace transform (LT). For example, we have shown the applicability of GP-FT on the polynomially decaying functions and super-exponential functions. Interestingly, GP-FT provides a valid representation of signals that do not satisfy Dirichlet conditions such as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\tan (t)$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\cos (1/t)$</tex-math></inline-formula> . Unilateral Laplace transform is the special case of the proposed GP-FT. Secondly, we demonstrate the efficacy of GP-FT 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, the general Gamma function is also presented. One interesting application of GP-FT is the computation of general parameterized moments, for the otherwise non-finite moments, of any random variable such as the Cauchy random variable. Fourthly, the exponential isomorphic mapping in GP-FT leads to a general parameterized version of Mellin transform, designated as Fourier scale transform (FST). Lastly, we propose General Parameterized Discrete-Time Fourier transform (GP-DTFT). DTFT and unilateral <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula> -transform are shown to be the special cases of the proposed GP-DTFT. We have also discussed the properties of GP-FT and GP-DTFT.