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” 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
(under the assumption that exists) and its consequences.
Reviewer’s remarks. 1. The construction is equal to zero for any nice function, which roughly speaking, behaves locally better than a Hölder function of order and is equal to infinity at all points where it has “bad” behaviour, worse than a Hölder function of order . Therefore, this construction may serve just as a kind of indicator whether a function at the point is better or worse than the power function (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
which coincides with (1) for functions which behave at a point as the power function , since
From (2), in particular, it follows that for any function whose continuity modulus satisfies the conditions that and is integrable.