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

摘要

We demonstrate that fractional Laplacian (FL) is the principal characteristic operator of harmonic systems with self-similar interparticle interactions.We show that the FL represents the '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)~(α/2) holding for α ∈ R ≥ 0. We give an explicit proof that the regularized representation of the FL gives for integer powers α/2 ∈ N_0 a distributional representation of the integer powers of standard Laplacian including the trivial unity operator for α→0. We demonstrate that self-similar harmonic systems are governed in a distributional sense by this regularized representation of the FL which therefore can be conceived as characteristic footprint of self-similarity.