We introduce a general notion of fractional (noninteger) derivative for functions defined on arbitrary time scales. The basic tools for the time-scale fractional calculus (fractional differentiation and fractional integration) are then developed. As particular cases, one obtains the usual time-scale Hilger derivative when the order of differentiation is one, and a local approach to fractional calculus when the time scale is chosen to be the set of real numbers.motivations to consider such local fractional derivatives is the possibility to deal with irregular signals, so common in applications of signal processing [27].A time scale is a model of time. The calculus on time scales was initiated by Aulbach and Hilger in 1988 [7], in order to unify and generalize continuous and discrete analysis [22,23]. It has a tremendous potential for applications and has recently received much attention [3,16,17,20,21]. The idea to join the two subjects -the fractional calculus and the calculus on time scales -and to develop a Fractional Calculus on Time Scales, was born with the Ph.D. thesis of Bastos [12]. See also [5,6,13,14,15,25,37,40] and references therein. Here we introduce a general fractional calculus on time scales and develop some of its basic properties.Fractional calculus is of increasing importance in signal processing [35]. This can be explained by several factors, such as the presence of internal noises in the structural definition of the signals. Our fractional derivative depends on the graininess function of the time scale. We trust that this possibility can be very useful in applications of signal processing, providing a concept of coarsegraining in time that can be used to model white noise that occurs in signal processing or to obtain generalized entropies and new practical meanings in signal processing. Indeed, let T be a time scale (continuous time T = R, discrete time T = hZ, h > 0, or, more generally, any closed subset of the real numbers, like the Cantor set). Our results provide a mathematical framework to deal with functions/signals f (t) in signal processing that are not differentiable in the time scale, that is, signals f (t) for which the equality ∆f (t) = f ∆ (t)∆t does not hold. More precisely, we are able to model signal processes for which ∆fThe time-scale calculus can be used to unify discrete and continuous approaches to signal processing in one unique setting. Interesting in applications, is the possibility to deal with more complex time domains. One extreme case, covered by the theory of time scales and surprisingly relevant also for the process of signals, appears when one fix the time scale to be the Cantor set [11,42]. The application of the local fractional derivative in a time scale different from the classical time scales T = R and T = hZ was proposed by Kolwankar and Gangal themselves: see [27,28] where nondifferentiable signals defined on the Cantor set are considered.The article is organized as follows. In Section 2 we recall the main concepts and tools necessary in the sequel. Our...
