Abstract
The theory of best approximation by polynomial splines has been the subject of a substantial number of papers over the past fifteen years; see, e.g., [l, 2, 5, 9-19, 22-321. These papers discuss all of the usual approximation questions, including existence, characterization, uniqueness, strong uniqueness, the existence of selections, and computational methods. Much of the theory has been extended to the class of Tchebychehian splines. The purpose of this paper is to develop a similar theory for more general classes of spline functions. In this paper we deal only with the case of fixed knots, and with the uniform and &-norms. It turns out that in order to get results analogous to those for polynomial splines, the right class of generalized spines to look at consists of those spline spaces which have a certain interlacing property connected with the solvability of interpolation problems. A complete characterization of such spline spaces was recently given in [13].
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