Non-local fractional derivatives. Discrete and continuous

2016/05/23 by Luciano Abadías, Abadías, Luciano, Marta de León-Contreras +3
Mathematics · #26A33 #34A08 #35R09 #35R11 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:26A33 #msc:34A08 #msc:35R09 #msc:35R11

paper · pdf · doi:10.48550/arxiv.1605.07088

arxiv created 2016/05/23 · arxiv updated 2016/05/24

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 Grünwald-Letnikov derivatives in every point and the speed of convergence to the Grünwald-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 ℓ^p(ℤ).

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