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

TL;DR

This paper introduces a regularized, convolution-based representation of the fractional Laplacian derived from self-similar elastic potentials, linking it to harmonic systems and fractal functions, and providing a distributional interpretation for integer powers.

Contribution

It presents a novel regularized integral representation of the fractional Laplacian based on self-similar elastic potentials, connecting fractional calculus with fractal and harmonic systems.

Findings

Regularized representation involves convolutions of finite difference operators.

Representation extends to all real non-negative powers of the Laplacian.

Self-similar harmonic systems are characterized by this regularized fractional Laplacian.

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 $\alpha\in \R \geq 0$. We give an explicit proof that the regularized representation of the FL gives for integer powers $\frac{\alpha}{2} \in \N\_0$ a distributional representation of the standard Laplacian operator $\Delta$ including the trivial unity operator for $\alpha\rightarrow 0$. 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.