On uniform approximation by splines

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