  Inspired by shape constrained estimation under general nonnegative derivative
constraints, this paper considers the B-spline approximation of constrained
functions and studies the asymptotic performance of the constrained B-spline
estimator. By invoking a deep result in B-spline theory (known as de Boor's
conjecture) first proved by A. Shardin as well as other new analytic
techniques, we establish a critical uniform Lipschitz property of the B-spline
estimator subject to arbitrary nonnegative derivative constraints under the
$\ell_\infty$-norm with possibly non-equally spaced design points and knots.
This property leads to important asymptotic analysis results of the B-spline
estimator, e.g., the uniform convergence and consistency on the entire interval
under consideration. The results developed in this paper not only recover the
well-studied monotone and convex approximation and estimation as special cases,
but also treat general nonnegative derivative constraints in a unified
framework and open the door for the constrained B-spline approximation and
estimation subject to a broader class of shape constraints.
