Non-local fractional derivatives. Discrete and continuous

TL;DR

This paper establishes fundamental properties, approximation methods, and convergence results for fractional discrete derivatives on integers, connecting discrete and continuous fractional calculus with harmonic analysis tools.

Contribution

It introduces maximum and comparison principles, approximation theorems, and convergence analysis for fractional discrete derivatives, linking discrete and continuous fractional calculus.

Findings

Proved maximum and comparison principles for fractional discrete derivatives.

Established approximation theorems and convergence rates to continuous derivatives.

Analyzed harmonic analysis operators related to the discrete derivatives in Lebesgue spaces.

Abstract

We prove maximum and comparison principles for fractional discrete derivatives in the integers. Regularity results when the space is a mesh of length $h$, and approximation theorems to the continuous fractional derivatives are shown. When the functions are good enough, these approximation procedures give a measure of the order of approximation. These results also allows us to prove the coincidence, for good enough functions, of the Marchaud and Gr\"unwald-Letnikov derivatives in every point and the speed of convergence to the Gr\"unwald-Letnikov derivative. The fractional discrete derivative will be also described as a Neumann-Dirichlet operator defined by a semi-discrete extension problem. Some operators related to the Harmonic Analysis associated to the discrete derivative will be also considered, in particular their behavior in the Lebesgue spaces $\ell^p(\mathbb{Z}).$