Simultaneous approximation, interpolation and Birkhoff systems

Abstract

By simultaneous approximation we mean the approximation of a function fE C?p) [a, b] by elements of the finite-dimensional subspace H c Ctkp’ [a, b] with respect to the semi-norm ]I . ]IF given by where F = {k, ,..., k,} and the ki are integers satisfying O<k,<k,<-a- < 5. ]I + ]I is the uniform norm on [a, b]. This is thus the simultaneous approximation of a function and its derivatives. The set au) of best simultaneous approximations to f, which is never empty, usually consists of more than one element. We want to determine its precise dimension. We would also like to know when the problem of approx- imating with I] . ]IF can be replaced by approximations with a simpler semi- norm. For example, if F = (0, I}, when do we find best approximations to $ with respect to ]] . ]JF just by finding the best Chebycheff approximation tof’ by elements h’, h E H and then integrating. Both questions are answered here. In [8], the case that H is the set of algebraic polynomials was dealt with. Keener [S] has given some first results for subspaces H satisfying dim H”) = n - i i = 0, l,..., m m}.‘The spaces Hci), (or m - 1) for the norm given by F = {0, l,..., which is the space of ith derivatives of the elements of H, are all assumed to be Haar subspaces for i Q m. He concludes that Q’“‘(f), the set of mth derivatives of best simultaneous approximations, consists only of one element (or, under the weaker assumption that dim Hem) = n - m + 1, Rcm)df) is at most two-dimensional). The dimension of Q(f) is, however, usually much smaller than this conclusion seems to indicate. Indeed, the best simultaneous approximation may be unique. By using the ideas of [5,8], we give the precise dimension of 125