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About non-differentiable functions. (English) Zbl 0995.26006

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 α 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 α f(x)=Γ(1+α)lim tx f(t)-f(x) |t-x| α (1)

(under the assumption that d α f(x) exists) and its consequences.

Reviewer’s remarks. 1. The construction d α f(x) is equal to zero for any nice function, which roughly speaking, behaves locally better than a Hölder function of order λ>α and is equal to infinity at all points where it has “bad” behaviour, worse than a Hölder function of order λ<α. 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| α (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 α f(x)=1 Γ(1-α)lim tx f(t)-f(x) |t-x| α +αsign(t-x) x t f(t)-f(s) |t-s| 1+α ds(2)

which coincides with (1) for functions f(t) which behave at a point x as the power function |t-x| α , since

x t (t-x) α -(s-x) α (t-s) 1+α ds=B(1+α,-α)+1 α,t>x·

From (2), in particular, it follows that d α f(x)0 for any function whose continuity modulus ω(f,δ) satisfies the conditions that lim δ0 ω(f,δ) δ α =0, and ω(f,δ) δ 1+α is integrable.


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