Chebyshev approximations in subspaces of spline functions

The main purpose of this paper is to consider strict approximations from subspaces of spline functions of degree m-1 with k fixed knots. Rice defines the strict approximation which is a particular unique best Chebyshev approximation for problems defined on a finite set. In order to determine best approximations on an interval I we define a sequence of strict approximations on finite subsets of I where the subsets fill up the interval. It is shown that the sequences always converge if k≤m. In the case k>m the sequences are convergent if we restrict ourselves to problems defined on certain subsets of I. It seems to be natural to denote these limits as strict approximations. To be able to compute these functions we also develop a Remez type algorithm.