  We investigate superconvergence properties of the spectral interpolation
involving fractional derivatives. Our interest in this superconvergence problem
is, in fact, twofold: when interpolating function values, we identify the
points at which fractional derivatives of the interpolant superconverge; when
interpolating fractional derivatives, we locate those points where function
values of the interpolant superconverge. For the former case, we apply various
Legendre polynomials as basis functions and obtain the superconvergence points,
which naturally unify the superconvergence points for the first order
derivative presented in [Z. Zhang, SIAM J. Numer. Anal., 50 (2012), 2966-2985],
depending on orders of fractional derivatives. While for the latter case, we
utilize Petrov-Galerkin method based on generalized Jacobi functions (GJF) [S.
Chen et al., arXiv: 1407. 8303v1] and locate the superconvergence points both
for function values and fractional derivatives. Numerical examples are provided
to verify the analysis of superconvergence points for each case.
