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

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  • Amelia Carolina Sparavigna

Abstract

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.

Suggested Citation

  • Amelia Carolina Sparavigna, 2023. "Convolution and Fourier Transform: from Gaussian and Lorentzian Functions to q-Gaussian Tsallis Functions," International Journal of Sciences, Office ijSciences, vol. 12(11), pages 7-11, November.
    • Handle: RePEc:adm:journl:v:12:y:2023:i:11:p:7-11
    DOI: 10.18483/ijSci.2732
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    References listed on IDEAS

    as
    1. Amelia Carolina Sparavigna, 2023. "q-Gaussian Tsallis Line Shapes and Raman Spectral Bands," International Journal of Sciences, Office ijSciences, vol. 12(03), pages 27-40, March.
    2. Amelia Carolina Sparavigna, 2023. "SERS Spectral Bands of L-Cysteine, Cysteamine and Homocysteine Fitted by Tsallis q-Gaussian Functions," International Journal of Sciences, Office ijSciences, vol. 12(09), pages 14-24, September.
    3. R. Hanel & S. Thurner & C. Tsallis, 2009. "Limit distributions of scale-invariant probabilistic models of correlated random variables with the q-Gaussian as an explicit example," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 72(2), pages 263-268, November.
    4. Amelia Carolina Sparavigna, 2023. "q-Gaussian Tsallis Functions and Egelstaff-Schofield Spectral Line Shapes," International Journal of Sciences, Office ijSciences, vol. 12(03), pages 47-50, March.
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    Cited by:

    1. Amelia Carolina Sparavigna, 2024. "The Fitted q-Gaussian Function, from Voigt Profile to Kubo Lineshape," International Journal of Sciences, Office ijSciences, vol. 13(03), pages 1-16, March.
    2. Amelia Carolina Sparavigna, 2024. "Kubo Lineshape and its Fitted q-Gaussian Tsallis Function," International Journal of Sciences, Office ijSciences, vol. 13(01), pages 1-9, January.

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