J. Math. Anal. Appl. 263, No. 2, 721-737 (2001); corrigendum ibid. 408, 409-413 (2013).
K. M. Kolwankar and 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 [K. M. Kolwankar and 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.

### MSC:

 26A33 Fractional derivatives and integrals 26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives

Zbl 1055.26504
Full Text:

### References:

 [1] Ben Adda, F., Geometric interpretation of the fractional derivative, J. Frac. Calc., 11, 21-51 (1997) · Zbl 0907.26005 [2] Ben Adda, F.; Cresson, J., Divergence d’échelle et différentiabilité, C. R. Acad. Sci. Paris Sér. I Math., 330, 261-264 (2000) [3] F. Ben Adda, and, J. Cresson, Calcul fractionnaire, variétés fractales et relativité d’échelle, prépublication 2000/11, de l’équipe de Math. de Besançon.; F. Ben Adda, and, J. Cresson, Calcul fractionnaire, variétés fractales et relativité d’échelle, prépublication 2000/11, de l’équipe de Math. de Besançon. [4] Cherbit, G., Dimension locale, quantité de mouvement et trajectoire, Fractals, dimension non entière et applications (1967), Masson: Masson Paris, p. 340-352 [5] K. Kolwankar, and, A. D. Gangal, Local fractional derivatives and fractal functions of several variables, in; K. Kolwankar, and, A. D. Gangal, Local fractional derivatives and fractal functions of several variables, in [6] Kolwankar, K.; Gangal, A. D., Hölder exponents of irregular signals and local fractional derivatives, Pramana J. phys., 48, 49-68 (1997) [7] Nottale, L., Scale-relativity and quantization of the universe. I. Theoretical framework, Astronom. Astrophys, 327, 867-889 (1997) [8] Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004 [9] Podlubny, I., Fractional Differential Equations. Fractional Differential Equations, Mathematics in Science and Engineering, 198 (1999), Academic Press: Academic Press San Diego · Zbl 0924.34008 [10] Ross, B.; Samko, S.; Russel Love, E., Functions that have no first order derivative might have fractional derivatives of all order less than one, Real Anal. Exchange, 20, 140-157 (1994/5) · Zbl 0820.26002 [11] Watanabe, Y., Notes on the generalized derivative of Riemann-Liouville and its application to Leibnitz’s formula, II, Tôkoku Math. J., 34, 8-41 (1931) [12] Adams, R., Sobolev Space (1975), Academic Press: Academic Press New York [13] Brézis, H., Analyse Fonctionnelle (1983), Masson: Masson Paris · Zbl 0511.46001 [14] Ciesielski, Z., On the isomorphisms of the space $$H_α$$ and m, Bul. Acad. Pol. Sci. Sér. Sci. Math. Astronom. Phys., 8, 217-222 (1960) · Zbl 0093.12301 [15] Ginzburg, A.; Karapetyants, N., Fractional integrodifferentiation in Hölder classes of variable order, Russian Acad. Sci. Dokl. Math., 50, 441-444 (1995) · Zbl 0820.26004 [16] Samko, S.; Kilbas, A.; Marichev, O., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon & Breach: Gordon & Breach London/New York · Zbl 0818.26003
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