Before revision:  These results cover both undersmoothed and bias corrected inference, require seemingly minimal rate restrictions , and achieve fast approximation rates . Using the uniform approximations we construct valid confidence bands which share these same advantages. While our coupling approach exploits specific features of the estimators considered, the underlying ideas may be useful in other contexts more generally. Second, we obtain valid integrated mean square error approximations for partitioning-based estimators and develop feasible tuning parameter selection. We apply our general results to three popular partition-based estimators: splines, wavelets, and piecewise polynomials (generalizing the regressogram) .
After revision:  These results cover both undersmoothed and bias corrected inference, require seemingly minimal rate restrictions and improve on approximation rates known in the literature. Finally, we apply our general results to three popular partition-based estimators: splines, wavelets, and piecewise polynomials (generalizing the regressogram) .