On the convergence of interpolating periodic spline functions of high degree

Let the spline functionS _m of degree 2m?1 and period 1 be the unique solution of the interpolation problem in § 1. An interesting question was posed by Schoenberg 1], p. 125: What happens toS _m if we letm→∞? In this paper, we prove that the spline functionsS _m and their derivatives converge form→∞ to a well determined trigonometric polynomial and its derivatives. Estimates for the rate of convergence are given.