## Abstract

Existence and uniqueness of canonical points for best L_{1}-approximation from an Extended Tchebycheff (ET) system, by Hermite interpolating "polynomials" with free nodes of preassigned multiplicities, are proved. The canonical points are shown to coincide with the nodes of a "generalized Gaussian quadrature formula" of the form ∫^{b}_{a}u(t) σ(t)sign ∏ i=1 n (t-x_{i})^{vi}dt≉ ∑ i=1 n ∑ j=0 ν_{1}-2 a_{ij}u^{(j)}(x_{i}) + ∑ j=0 ν_{0-1} a_{0j}u^{(j)}(a)+ ∑ j=0 ν_{n+1}-1 a_{n+1j}u^{(j)}(b), (*) which is exact for the ET-system. In (*), ∑_{j = 0}^{vi - 2} ≡ 0 if v_{i} = 1, the v_{i} (> 0), i = 1,..., n, are the multiplicities of the free nodes and v_{0}≥0, v_{n + 1}≥ 0 of the boundary points in the L_{1}-approximation problem, ∑_{i = 0}^{n + 1} v_{i} is the dimension of the ET-system, and σ is the weight in the L_{1}-norm. The results generalize results on multiple node Gaussian quadrature formulas (v_{1},..., v_{n} all even in (*)) and their relation to best one-sided L_{1}-approximation. They also generalize results on the orthogonal signature of a Tchebycheff system (v_{0} = v_{n + 1} = 0, v_{i} = 1, i = 1,..., n, in (*)), and its role in best L_{1}-approximation. Recent works of the authors were the first to treat Gaussian quadrature formulas and orthogonal signatures in a unified way.

Original language | English |
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Pages (from-to) | 335-353 |

Number of pages | 19 |

Journal | Journal of Approximation Theory |

Volume | 48 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1986 |