Generalized Gaussian quadrature formulas

Existence and uniqueness of canonical points for best L1-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 ∫bau(t) σ(t)sign ∏i=1n (t−xi)vidt≉∑i=1n ∑j=0ν1−2 aiju(j)(xi) + ∑j=0ν0−1 a0ju(j)(a)+ ∑j=0νn+1−1 an+1ju(j)(b), (∗) which is exact for the ET-system. In (∗), ∑j = 0vi − 2 ≡ 0 if vi = 1, the vi (> 0), i = 1,…, n, are the multiplicities of the free nodes and v0⩾0, vn + 1⩾ 0 of the boundary points in the L1-approximation problem, ∑i = 0n + 1 vi is the dimension of the ET-system, and σ is the weight in the L1-norm. The results generalize results on multiple node Gaussian quadrature formulas (v1,…, vn all even in (∗)) and their relation to best one-sided L1-approximation. They also generalize results on the orthogonal signature of a Tchebycheff system (v0 = vn + 1 = 0, vi = 1, i = 1,…, n, in (∗)), and its role in best L1-approximation. Recent works of the authors were the first to treat Gaussian quadrature formulas and orthogonal signatures in a unified way.