Fractional Integrals and Derivatives with Respect to a Measure

@inproceedings{AhleFractionalIA,
  title={Fractional Integrals and Derivatives with Respect to a Measure},
  author={M Z Ahle}
}
  • M Z Ahle
Fractional integrals and derivatives of functions deened on fractal subsets of the real line are introduced. Lebesgue measure is replaced by arbitrary reference measures supported on these sets. The related fractional calculus is similar to the classical case. An extension to R n via Riesz{type potentials is indicated. 0. Background In order to establish relationships between classical fractional calculus and fractal geometry it seems to be natural to follow the measure{theoretic line. Borel… CONTINUE READING
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Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach

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  • Fractional Integrals and Derivatives. Theory and…
  • 1993

Fractional Calculus with respect to a Measure. (Preprint). ) Mathematical Institute Received: University of Jena 07740 Jena { GERMANY e-mail: zaehle@minet.uni-jena

  • Fractional Calculus with respect to a Measure…

It turns out that a natural fractal variant of (?) ?1=2 is the following: Let be a nite atomless Borel measure in R n with compact support and Hausdorr dimension d > 1

  • M Zz
  • Then we deene for f 2 L 2 (), (? ) ?1=2 f

The spectral behaviour of the resulting Laplace operator for d{measures is in correspondence with the so{called Berry conjecture in related physics. Proofs and more details may be found in 2

  • The spectral behaviour of the resulting Laplace…

n (n ? d + 1) Z f(y) jy ? xj d?1 (dy)

  • n (n ? d + 1) Z f(y) jy ? xj d?1 (dy)

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