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 $\alpha\in \R \geq 0$. 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.