Convolution and Fourier Transform: from Gaussian and Lorentzian Functions to q-Gaussian Tsallis Functions

A discussion is here proposed regarding the Voigt function, that is the convolution of Gaussian and Lorentzian functions, and the Lévy and q-Gaussian Tsallis distributions. The Voigt and q-Gaussian functions can be used as line shapes in Raman spectroscopy for fitting spectra. Using the convolution theorem, we can obtain the relaxations which are producing the Voigt line shape. To determine the relaxation governing the q-Gaussian line shape, we need to use the Lévy symmetric distribution, since the direct Fourier transform of the q-Gaussian is a very complicated function. According to the work by Deng, 2010, the q-Gaussian functions are mimicking the Lévy functions in an excellent manner. Being the Fourier transform of the Lévy function a stretched exponential relaxation, we can argue that the same mechanism is producing the q-Gaussian line shape. Moreover, using the convolution theorem for the q-Gaussians, we can further generalize the relaxation mechanism.