Error bounds for polynomial spline interpolation

New upper and lower bounds for the L2 and L- norms of derivatives of the error in polynomial spline interpolation are derived. These results improve corresponding results of Ahlberg, Nilson, and Walsh, cf. (1), and Schultz and Varga, cf. (5). 1. Introduction. In this paper, we derive new bounds for the L2 and LX norms of derivatives of the error in polynomial spline interpolation. These bounds improve and generalize the known error bounds, cf. (1) and (5), in the following important ways: (1) these bounds can be explicitly calculated and are not merely asymptotic error bounds such as those given in (1) and (5); (2) explicit lower bounds are given for the error for a class of functions; (3) the degree of regularity required of the func- tion, f, being interpolated is extended, i.e., in L1) and (5) we demand that the mth or 2mth derivative of f be in L2, if we are interpolating by splines of degree 2m - 1, while here we demand only that some pth derivative of f, where m ? p ? 2m, be in L2; and (4) bounds are given for high-order derivatives of the interpolation errors. 2. Notations. Let - o < a < b < o and for each positive integer, m, let Km(a, b) denote the collection of all real-valued functions u(x) defined on (a, b) such that u E Cm'-(a, b) and such that Dm-lu is absolutely continuous, with Dmu E L2 (a, b), where Du _ du/dx denotes the derivative of u. For each nonnegative integer, M,