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

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 $-(-\Delta {)}^{\frac{\alpha }{2}}$ holding for $\alpha \in \mathbb{R}\ge 0$. We give an explicit proof that the regularized representation of the FL gives for integer powers $\frac{\alpha }{2}\in \mathbb{N}\_0$ a distributional representation of the standard Laplacian operator $\Delta$ including the trivial unity operator for $\alpha \to 0$. 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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Cited by in corpus (5)

• Fractional random walk lattice dynamics
• On recurrence of random walks with long-range steps generated by fractional Laplacian matrices on regular networks and simple cubic lattices
• A fractional generalization of the classical lattice dynamics approach
• Lattice fractional Laplacian and its continuum limit kernel on the finite cyclic chain
• Fractional Lattice Dynamics: Nonlocal constitutive behavior generated by power law matrix functions and their fractional continuum limit kernels