A SPECIAL ONE-SIDED APPROXIMATION PROBLEM

Let P be a nonnegative perfect spline of degree n on [a, b] satisfying P(j)(a) = P(j)(b) = 0 (j = 0, ..., n - 1). We wish to approximate P from below in the L1-norm by nonnegative splines of degree n - 1 with the same fixed knots as P. We show that a unique best approximation exists (differing, in general, from the best approximation without the nonnegativity restriction), and describe the zero structure of the error function. In addition, we discuss the semi-infinite programming approach and deduce relationships with mononsplines and quadrature formulas. © 1990.