Approximation by local L-splines corresponding to a linear differential operator of the second order

For the class of functions W∞L2 = {f : f′ ∈ AC, ∥L2(D)f∥∞ ≤ 1}, where L 2(D) is a linear differential operator of the second order whose characteristic polynomial has only real roots, we construct a noninterpolating linear positive method of exponential spline approximation possessing extremal and smoothing properties and locally inheriting the monotonicity of the initial data (the values of a function f ∈ W∞L2 at the points of a uniform grid). The approximation error is calculated exactly for this class of functions in the uniform metric. © 2006 Pleiades Publishing, Inc.