Spline monotone approximation with linear differential operators

Abstract

Let f∈C³[a,b] and L be a linear differential operator such that L(f)≥0. Then there exists a sequence Q_n, n≥1, of polynomial splines with equally spaced knots, such that Q^(r), approximates f^(r), 0≤r≤s, simultaneously in the uniform norm. This approximation is given through inequalities with rates, involving a measure of smoothness to f^(s); so that L (Q_n)≥0. The encountered cases are the continuous, periodic and discrete.