Duality for the left and right fractional derivatives
Abstract
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.
Keywords
Cite
@article{arxiv.1409.5319,
title = {Duality for the left and right fractional derivatives},
author = {M. Cristina Caputo and Delfim F. M. Torres},
journal= {arXiv preprint arXiv:1409.5319},
year = {2014}
}
Comments
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