广义傅里叶变换：傅里叶、拉普拉斯、梅林和$Z$变换的统一框架
The Generalized Fourier Transform: A Unified Framework for the Fourier, Laplace, Mellin and $Z$ Transforms
摘要：
本文介绍了广义傅里叶变换（GFT），它是傅里叶变换（FT）的推广或推广。单边拉普拉斯变换（LT）是广义傅里叶变换的特例。GFT，在这项工作中提出的，有助于显着的学术文献。这项工作有许多突出贡献。首先，GFT适用于更大的一类信号，其中一些不能用FT和LT分析。例如，我们已经证明了GFT对多项式衰减函数和超指数的适用性。其次，我们证明了GFT在解决初值问题（IVP）的有效性。第三，将傅立叶变换的推广推广到其他积分变换，并以小波变换和余弦变换为例进行了说明。同样，本文也给出了广义Gamma函数。GFT的一个有趣的应用是计算任何随机变量（如柯西随机变量）的广义矩（对于其他非有限矩）。第四，我们介绍了傅立叶尺度变换（FST），利用GFT与指数映射的拓扑同构。最后，我们提出了广义离散时间傅里叶变换（GDTFT）。的DTFT和单边$z$变换被证明是建议GDTFT的特殊情况。并对GFT和GDTFT的特性进行了讨论。
英文摘要：
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.
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