## Abstract

Let M be a normed linear space, and {M_{n}}_{1}^{∞} a sequence of increasing finite dimensional subspaces, i.e., M_{n} ⊂ M_{n + 1}, for all n. For any element f{hook} ε{lunate} M, we obviously have d(f{hook}, M_{n}) ≥ d(f{hook}, M_{n + 1}), for all n, (1) where d(f{hook}, M_{k}) is the distance, in the metric induced by the norm, from M_{k} to f{hook}. In a recent paper [2], we discussed the space M = C[a,b] with the uniform norm and with M_{n} = [u_{0},..., u_{n - 1}], the linear subspace spanned by {u_{i}}_{0}^{n - 1}, where {u_{i}}_{0}^{∞} is an infinite Tchebycheff system. We established there that the functions for which inequality (1), for a given n, is strict for all subintervals of [a, b] are precisely those that are convex with respect to (u_{0}, u_{1},..., u_{n - 1}). The proof depended crucially on the alternance properties of the best approximants in the uniform norm. Somewhat surprisingly, analogous results are valid when the norm under consideration is the L^{2}-norm. In fact, as we show in this paper, generalized convex functions play the same role in the L^{2}-norm, for all continuous weights. Weaker results for the L^{2} case were obtained in [6].

Original language | English |
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Pages (from-to) | 115-127 |

Number of pages | 13 |

Journal | Journal of Approximation Theory |

Volume | 14 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1975 |

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