Characterization in the Fractional Case

Abstract

In Chapter 10 the class W[L^P; |υ|^r], 1 ≤ p ≤ 2, was characterized in terms of differentiability properties upon f (r even) or f~ (r odd). The question arises whether for fractional α > 0 the class W[L^P;|υ|^α is connected with a derivative of f of fractional order α. This will be shown to be the case. It is opportune to define fractional differentiation through integration of fractional order. There are at least two such definitions. If [L₁f](x) is the integral of f over (a, x), and [L _α f](x) the integral of [L_α-1;f](x) over (a, x) α = 2, 3,…, then

$$[{L_\alpha }f](x) = \frac{1}{{\Gamma (\alpha )}}\smallint _a^x{(x - u)^{\alpha - 1}}f(u)du$$

((11.0.1))