Integral transforms Mellin transform

GENERALIZED INTEGRAL TRANSFORMATIONS, A.H. Zemanian. Graduate-level study of recent generalizations of the Laplace, Mellin, Hankel, K. Weierstrass, convolution and other simple transformations. Bibliography. 320pp. 5H x 8H. 65375-7 Pa. 7.95... 

We make now, similarly as is common with the different integral transforms, a correspondence table between the stochastic variable and the associated characteristic function. Note, there are several integral transforms. The most well-known integral transformation might be the Fourier transform. Further, we emphasize the Laplace transform, the Mellin transform, and the Hilbert transform. These transformations are useful for the solution of various differential equations, in communications technology, all ranges of the frequency analysis, also for optical problems and much other more. We designate the stochastic variable with X. The associated characteristic function should be... 

Taking Mellin transforms with respect to A and inverting, we can estimate the integral representation... 

Complex fractional moments Earthquake ground motion Fractional calculus Gaussian zero-mean excitation Mellin transform Riesz fractional integrals Stochastic analysis... 

In this entry some relevant examples on the use of fractional calculus to earthquake ground motion modeled as a stationary normal colored noise are presented. Applications of fractional calculus for the description of mono- and multivariate earthquake accelerations and exact filter equation are obtained in an integral form involving Riesz fractional operator in zero. The latter are related to complex spectral moments by Mellin transform operator. Other relevant application in probability may be found in Cottone et al. (2010) and Di Paola and Pinnola (2012). 

By using the same cmicepts, a very large niun-ber of other problems may be solved. Such an example the probability density function of a random variable may be obtained with the same technique here used for representing cross-correlations in terms of FSMs. It follows that Fokker-Planck equation, Kolmogorov-Feller equation, Einstein-Smoluchowski equation, and path integral solution (Cottone et al. 2008) may be solved in terms of FSM. Moreover, wavelet transform and classical or fractional differential equations may be easily solved by using fractional calculus and Mellin transform in complex domain. 

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