Higher order Gr\"unwald approximations of fractional derivatives and fractional powers of operators

TL;DR

This paper develops higher order Gr"unwald approximations for fractional derivatives, establishing stability and consistency, and extends the theory to operators on Banach spaces using Fourier multiplier techniques.

Contribution

It introduces a new Carlson-type inequality for periodic Fourier multipliers and generalizes the approximation theory to operators on Banach spaces.

Findings

Established stability and consistency of higher order Gr"unwald approximations

Proved regularity results using Fourier multiplier inequalities

Extended the approximation framework to generators of bounded groups on Banach spaces

Abstract

We give stability and consistency results for higher order Gr\"unwald-type formulae used in the approximation of solutions to fractional-in-space partial differential equations. We use a new Carlson-type inequality for periodic Fourier multipliers to gain regularity and stability results. We then generalise the theory to the case where the first derivative operator is replaced by the generator of a bounded group on an arbitrary Banach space.