Summary: One important challenge in nonparametric density and regression-function estimation is spatially inhomogeneous smoothness. This is often modelled by Besov-type smoothness constraints. With this type of constraints, Donoho and Johnstone (1998) and Delyon and Juditsky (1996) studied asymptotic-minimax optimal wavelet estimators with thresholding, while Lepski, Mammen and Spokoiny (1997) proposed a variable-bandwidth selection for kernel estimators that also achieved the asymptotic-minimax rates. However, a second challenge in many applications of nonparametric curve estimation is that the function must be nonnegative or order-constrained. Dechevsky and MacGibbon (1999) constructed wavelet- and kernel-based estimators under positivity constraints that satisfied these constraints and also achieved asymptotic-minimax rates over the appropriate smoothness classes. Here we show how to replace the integral in their definition by a quadrature formula in order to numerically construct the estimators, so that the new ‘quadrature’ estimators enjoy the positivity- and smoothness-preserving properties of the ones of Dechevsky and MacGibbon, and also are asymptotic-minimax optimal.