In this chapter we consider polynomial spline approximation. Approximation by splines entered the theory relatively late and immediately acquired a large following; in particular because of their advantages over classical polynomials in the problem of the interpolation of functions. It turns out that, in addition to computational advantage due to the existence of bases with local supports, interpolating splines realize the minimal (for a fixed dimension) deviation for many classes of function. Interpolating polynomials do not even give the best order.

The results in this chapter will in general be connected with two aspects of spline approximation of classes of functions with the rth derivative bounded in Lp: (1) the best approximation by splines with minimal defects; (2) spline interpolation. In the first place we consider those situations when the best approximation is realized by interpolating splines and here it turns out that elementary analysis tools are sufficient if the specific properties of the polynomial splines are used. In Section 5.1 we give some basic broad-based facts to which many problems in spline interpolation can be reduced.

The methods of obtaining exact results in the estimation of the error in best approximation by splines with minimal defects (Section 5.4) are based on essentially different ideas – application of duality relations. These methods allow us to obtain solutions in situations in which the exact estimates for spline interpolation are not known.