  As one myth of polynomial interpolation and quadrature, Trefethen [30]
revealed that the Chebyshev interpolation of $|x-a|$ (with $|a|<1 $) at the
Clenshaw-Curtis points exhibited a much smaller error than the best polynomial
approximation (in the maximum norm) in about $95\%$ range of $[-1,1]$ except
for a small neighbourhood near the singular point $x=a.$ In this paper, we
rigorously show that the Jacobi expansion for a more general class of
$\Phi$-functions also enjoys such a local convergence behaviour. Our assertion
draws on the pointwise error estimate using the reproducing kernel of Jacobi
polynomials and the Hilb-type formula on the asymptotic of the Bessel
transforms. We also study the local superconvergence and show the gain in order
and the subregions it occurs. As a by-product of this new argument, the
undesired $\log n$-factor in the pointwise error estimate for the Legendre
expansion recently stated in Babu\u{s}ka and Hakula [5] can be removed.
Finally, all these estimates are extended to the functions with boundary
singularities. We provide ample numerical evidences to demonstrate the
optimality and sharpness of the estimates.
