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Exact solution of averaging procedure over the Cantor set

Author

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  • Stanislavsky, A.A.
  • Weron, K.

Abstract

Using functional equations with self-similar properties, we have derived the exact analytical result for convolution of a smooth function with the normalized density of the Cantor set in the limit N→∞. We have proved that the self-similar kernel of this convolution cannot be reduced explicitly to any product of a power and a log-periodic function as suggested in literature. Only its asymptotic behaviour can be expressed in terms of such a product. This clarifies the relationship between fractals and fractional calculus.

Suggested Citation

  • Stanislavsky, A.A. & Weron, K., 2002. "Exact solution of averaging procedure over the Cantor set," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 303(1), pages 57-66.
    • Handle: RePEc:eee:phsmap:v:303:y:2002:i:1:p:57-66
    DOI: 10.1016/S0378-4371(01)00487-3
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