2014/09/18 by Michèle Caputo, M. Cristina Caputo, Delfim F. M. Torres
Engineering · Mathematics · #Derivative (finance) #Dual (grammatical number) #Duality (order theory) #Fractional Differential Equations Solutions #Fractional calculus #Function (biology) #Left and right #Mathematical analysis #Mathematics #Nonlinear Differential Equations Analysis #Numerical methods in engineering #Operator (biology) #Pure mathematics #math.CA #math.OC #msc:26A33 #msc:49J05
paper · pdf · doi:10.1016/j.sigpro.2014.09.026
published as Signal Process. 107 (2015), 265--271 · This is a preprint of a paper whose final and definite form will appear in the international journal Signal Processing, ISSN 0165-1684. Paper submitted Dec/2013; revised Apr, July and Sept 2014; accepted for publication 18/Sept/2014
arxiv created 2014/09/18 · openalex publication_date 2014/09/27 · arxiv updated 2014/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We prove duality between the left and right fractional derivatives, independently on the type of fractional operator. Main result asserts that the right derivative of a function is the dual of the left derivative of the dual function or, equivalently, the left derivative of a function is the dual of the right derivative of the dual function. Such duality between left and right fractional operators is useful to obtain results for the left operators from analogous results on the right operators and vice versa. We illustrate the usefulness of our duality theory by proving a fractional integration by parts formula for the right Caputo derivative and by proving a Tonelli-type theorem that ensures the existence of minimizer for fractional variational problems with right fractional operators.