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).