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About non-differentiable functions. (English) Zbl 0995.26006
{\it K. M. Kolwankar} and {\it A. D. Gangal} [“Fractional differentiability of nowhere differentiable functions and dimensions”, Chaos, 6, No. 4, 505-513 (1996; Zbl 1055.26504)] introduced a certain “localization” $d^\alpha f(x)$ of the Riemann-Liouville fractional derivative to study the local behaviour of nowhere differentiable functions. The authors present a development of this notion, one of the main statements being the formula $$d^\alpha f(x)= \Gamma(1+\alpha)\lim_{ t\to x}\frac{f(t)-f(x)}{|t-x|^\alpha} \tag 1$$ (under the assumption that $d^\alpha f(x)$ exists) and its consequences. Reviewer’s remarks. 1. The construction $d^\alpha f(x)$ is equal to zero for any nice function, which roughly speaking, behaves locally better than a Hölder function of order $\lambda> \alpha$ and is equal to infinity at all points where it has “bad” behaviour, worse than a Hölder function of order $\lambda< \alpha$. Therefore, this construction may serve just as a kind of indicator whether a function $f(t)$ at the point $x$ is better or worse than the power function $|t-x|^\alpha$ (as it is in fact interpreted in the recent paper [{\it K. M. Kolwankar} and {\it J. Lévy Véhel}, “Measuring functions smoothness with local fractional derivatives”, Fract. Calc. Appl. Anal. 4, No. 3, 285-301 (2001)]. It cannot be named as a fractional derivative, even if a local one. 2. From the well-known Marchaud representation for the fractional derivative it follows immediately that $$d^\alpha f(x)= \frac{1}{\Gamma(1-\alpha)}\lim_{ t\to x}\left[ \frac{f(t)-f(x)}{|t-x|^\alpha}+ \alpha \text{sign}(t-x)\int_x^t\frac{f(t)-f(s)}{|t-s|^{1+\alpha}}ds\right]\tag 2$$ which coincides with (1) for functions $f(t)$ which behave at a point $x$ as the power function $|t-x|^\alpha$, since $$\int_x^t\frac{(t-x)^\alpha - (s-x)^\alpha}{(t-s)^{1+\alpha} } ds = B(1+\alpha,-\alpha)+\frac{1}{\alpha} , \quad t>x.$$ From (2), in particular, it follows that $d^\alpha f(x)\equiv 0$ for any function whose continuity modulus $\omega(f,\delta)$ satisfies the conditions that $\lim_{\delta\to 0}\frac{\omega(f,\delta)}{\delta^\alpha}=0,$ and $\frac{\omega(f,\delta)}{\delta^{1+\alpha}}$ is integrable.

26A33Fractional derivatives and integrals (real functions)
26A27Nondifferentiability of functions of one real variable; discontinuous derivatives
Full Text: DOI
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