Non-local fractional derivatives. Discrete and continuous

作者：Abadias Luciano; De Leon Contreras Marta; Torrea Jose L^*

来源：Journal of Mathematical Analysis and Applications, 2017, 449(1): 734-755.

DOI：10.1016/j.jmaa.2016.12.006

• Nonlocal operators
• Discrete fractional derivatives
• Holder estimates
• Semigroups

We prove maximum and comparison principles for the discrete fractional 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 (Holder continuous), these approximation procedures give a measure of the order of approximation. These results also allow us to prove the coincidence, for Holder continuous functions, of the Marchaud and Grunwald-Letnikov derivatives in every point and the speed of convergence to the Grunwald-Letnikov derivative. The discrete fractional derivative will be also described as a Neumaaan-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).

• 出版日期2017-5-1