Uniform Approximation by Perfect Splines

The problem of uniform approximation of a continuous function on a closed interval is considered. In the case of approximation by the class W-(n) of functions whose nth derivative is bounded by 1 almost everywhere, a criterion for a best approximation element is known. This criterion, in particular, requires that the approximating function coincide on some subinterval with a perfect spline of degree n with finitely many knots. Since perfect splines belong to the class W-(n), we study the following restriction of the problem: a continuous function is approximated by the set of perfect splines with an arbitrary finite number of knots. We establish the existence of a perfect spline that is a best approximation element both in W-(n) and in this set. Therefore, the values of the best approximation in the problems are equal. We also show that the best approximation elements in this set satisfy a criterion similar to the criterion for a best approximation element in W-(n). The set of perfect splines is shown to be everywhere dense in W-(n).