Uniqueness of best φ-approximation from the set of splines with infinitely many simple knots Original Research Article

Let J be an open interval and denote by SΠ the set of all the splines of degree at most n−1 with simple knots in Π, a countably infinite set of points in J, n⩾2. In this paper, we prove that there exists a unique best φ-approximation to a continuous function in Lφ(J) from SΠ, where φ:[0,∞)↦[0,∞) is a convex function that generalizes the pth-power functions, p⩾1.