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arXiv · 1501.01942

A regularized representation of the fractional Laplacian in n dimensions and its relation to Weierstrass-Mandelbrot type fractal functions

Thomas Michelitsch·Gérard Maugin·Shahram Derogar·Rahman Mujibur
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Abstract

We demonstrate that the fractional Laplacian (FL) is the principal characteristic operator of harmonic systems with {\it self-similar} interparticle interactions. We show that the FL represents the "{\it fractional continuum limit}" of a discrete "self-similar Laplacian" which is obtained by Hamilton's variational principle from a discrete spring model. We deduce from generalized self-similar elastic potentials regular representations for the FL which involve convolutions of symmetric finite difference operators of even orders extending the standard representation of the FL. Further we deduce a regularized representation for the FL $-(-Δ)^{\fracα{2}}$ holding for $α\in \R \geq 0$. We give an explicit proof that the regularized representation of the FL gives for integer powers $\fracα{2} \in \N\_0$ a distributional representation of the standard Laplacian operator $Δ$ including the trivial unity operator for $α\rightarrow 0$. We demonstrate that self-similar {\it harmonic} systems are {\it all} governed in a distributional sense by this {\it regularized representation of the FL} which therefore can be conceived as characteristic footprint of self-similarity.

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Thomas Michelitsch, Gérard Maugin, Shahram Derogar, Rahman Mujibur. 2015-01-08. A regularized representation of the fractional Laplacian in n dimensions and its relation to Weierstrass-Mandelbrot type fractal functions. https://doi.org/10.1093/imamat%2Fhxu018

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