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 ∫^b_au(t) σ(t)sign ∏ i=1 n (t-x_i)^vidt≉ ∑ i=1 n ∑ j=0 ν₁-2 a_iju^(j)(x_i) + ∑ j=0 ν_0-1 a_0ju^(j)(a)+ ∑ j=0 ν_n+1-1 a_n+1ju^(j)(b), (*) which is exact for the ET-system. In (*), ∑_{j = 0}^{v_i - 2} ≡ 0 if v_i = 1, the v_i (> 0), i = 1,..., n, are the multiplicities of the free nodes and v₀≥0, v_{n + 1}≥ 0 of the boundary points in the L₁-approximation problem, ∑_{i = 0}^{n + 1} v_i 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,..., v_n 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 = v_{n + 1} = 0, v_i = 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.