  In this paper, superconvergence points are located for the approximation of
the Riesz derivative of order $\alpha$ using classical Lobatto-type polynomials
when $\alpha \in (0,1)$ and generalized Jacobi functions (GJF) for arbitrary
$\alpha > 0$, respectively. For the former, superconvergence points are zeros
of the Riesz fractional derivative of the leading term in the truncated
Legendre-Lobatto expansion. It is observed that the convergence rate for
different $\alpha$ at the superconvergence points is at least $O(N^{-2})$
better than the optimal global convergence rate. Furthermore, the interpolation
is generalized to the Riesz derivative of order $\alpha > 1$ with the help of
GJF, which deal well with the singularities. The well-posedness, convergence
and superconvergence properties are theoretically analyzed. The gain of the
convergence rate at the superconvergence points is analyzed to be
$O(N^{-(\alpha+3)/2})$ for $\alpha \in (0,1)$ and $O(N^{-2})$ for $\alpha > 1$.
Finally, we apply our findings in solving model FDEs and observe that the
convergence rates are indeed much better at the predicted superconvergence
points.
