  It is well known that the minimax rates of convergence of nonparametric
density and regression function estimation of a random variable measured with
error is much slower than the rate in the error free case. Surprisingly, we
show that if one is willing to impose a relatively mild assumption in requiring
that the error-prone variable has a compact support, then the results can be
greatly improved. We describe new and constructive methods to take full
advantage of the compact support assumption via spline-assisted semiparametric
methods. We further prove that the new estimator achieves the usual
nonparametric rate in estimating both the density and regression functions as
if there were no measurement error. The proof involves linear and bilinear
operator theories, semiparametric theory, asymptotic analysis regarding
Bsplines, as well as integral equation treatments. The performance of the new
methods is demonstrated through several simulations and a data example.
