Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A regularized representation of the fractional Laplacian in n dimensions
  and its relation to Weierstrass-Mandelbrot type fractal functions   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.

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The categories of this paper are: cmp-lg
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The categories of this paper are: funct-an
9
The categories of this paper are: nlin
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The categories of this paper are: alg-geom
3
The categories of this paper are: physics
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The categories of this paper are: plasm-ph
7
The categories of this paper are: hep-th
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The categories of this paper are: ao-sci
1
The categories of this paper are: cs
6
The categories of this paper are: astro-ph
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The categories of this paper are: eess
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The categories of this paper are: atom-ph
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The categories of this paper are: supr-con
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The categories of this paper are: hep-ex
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The categories of this paper are: q-fin
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The categories of this paper are: adap-org