Best interpolation in a strip II: Reduction to unconstrained convex optimization

Abstract

In this paper, we study the problem of finding a real-valued function f on the interval [0, 1] with minimal L 2 norm of the second derivative that interpolates the points ( t i , y i ) and satisfies e(t) ≤ f(t) ≤ d(t) for t ∈ [0, 1]. The functions e and d are continuous in each interval ( t i , t i +1) and at t 1 and t n but may be discontinuous at t i . Based on an earlier paper by the first author [7] we characterize the solution in the case when e and d are linear in each interval ( t i , t i +1). We present a method for the reduction of the problem to a convex finite-dimensional unconstrained minimization problem. When e and d are arbitrary continuous functions we approximate the problem by a sequence of finite-dimensional minimization problems and prove that the sequence of solutions to the approximating problems converges in the norm of W 2,2 to the solution of the original problem. Numerical examples are reported.