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

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.