A sampling theorem for the Riemann--Liouville fractional derivative

This paper continues the work of Matychyn (2016, Fract. Calc. Appl. Anal. 19(6), 1460–1465), where an identity was obtained expressing an analytic function as an integral over the order of its Riemann–Liouville fractional derivative. We show that, for an analytic function f, the normalized fractional derivative x^α/Γ(α+1) (D^α f)(x), regarded as a function of α, belongs to the Paley–Wiener space PW_π of functions band-limited to [−π,π]. Hence it is band-limited in the order variable and is given by the Whittaker–Shannon–Kotelnikov interpolation of the samples x^k f^(k)(x)/k!, k=0,1,2,.... Thus the normalized Riemann–Liouville derivative is the unique band-limited interpolation of the normalized classical integer-order derivatives. The main identity of the earlier paper is recovered as the integral form of this sampling expansion.