On uniform approximation by splines

for 0 ≤ r ≤ k − 1. In particular, dist (f, S π) = O(|π| ) for f ∈ C(I), or, more generally, for f ∈ C(I), such, that f (k−1) satisfies a Lipschitz condition, a result proved earlier by different means [2]. These results are shown to be true even if I is permitted to become infinite and some of the knots are permitted to coalesce. The argument is based on a “local” interpolation scheme Pπ by splines, which is, in a way, a generalization of interpolation by broken lines, and which achieves the convergence rate (1.1). The linear projector (i.e., linear idempotent map) Pπ can be shown to be bounded independently of π. Hence, the argument supplies the fact that any sequence S πn with lim |πn| = 0 admits a corresponding uniformly bounded sequence Pπn of linear projectors on C(I) with S k πn the range of Pπn , which converges strongly to the identity. Such sequences are important for the application of Galerkin’s method and its generalizations to the approximate solution of functional equations (cf., e.g., [1]). The following standard notation will be adhered to throughout. For T some set, m(T ) denotes the Banach space of all bounded real–valued functions on T , with norm