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Relationship Of Upper Box Dimension Between Continuous Fractal Functions And Their Riemann–Liouville Fractional Integral

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  • WEI XIAO

    (School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, P. R. China)

Abstract

This paper considers the relationship of box dimension between a continuous fractal function and its Riemann–Liouville fractional integral. For an arbitrary fractal function f(x) it is proved that the upper box dimension of the graph of Riemann–Liouville fractional integral D−νf(x) does not exceed the upper box dimension of f(x), i.e. dim¯BΥ(D−νf,I) ≤dim¯ BΥ(f,I). This estimate shows that ν−order Riemann–Liouville fractional integral D−νf(x) does not increase the fractal dimension of the integrand f(x), which means that Riemann–Liouville fractional integration does not decrease the smoothness at least that is obvious known result for classic integration. Our result partly answers fractal calculus conjecture in [F. B. Tatom, The relationship between fractional calculus and fractals, Fractals 2 (1995) 217–229] and [Y. S. Liang and W. Y. Su, Riemann–Liouville fractional calculus of one-dimensional continuous functions, Sci. Sin. Math. 4 (2016) 423–438].

Suggested Citation

  • Wei Xiao, 2021. "Relationship Of Upper Box Dimension Between Continuous Fractal Functions And Their Riemann–Liouville Fractional Integral," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 29(08), pages 1-6, December.
    • Handle: RePEc:wsi:fracta:v:29:y:2021:i:08:n:s0218348x21502649
    DOI: 10.1142/S0218348X21502649
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