Chebyshev Approximation by Spline Functions with Free Knots

Abstract

This paper is concerned with the Chebyshev approximation of real continuous functions from the class S n,k of polynomial spline functions of degree n with k free knots. Using the notion of the tangent cone and a new sign rule for spline functions, a necessary alternant condition for local best approximations from S n,k is derived. It shows that the corresponding error function must have an alternant of a certain length with a prescribed sign on a subinterval. This condition improves results by Braess (1971) and Cromme (1982). A characterization of best approximations by fixed knots splines with coefficient constraints is also obtained.