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- W2063620147 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 v i − 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." @default.
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- W2063620147 date "1986-12-01" @default.
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- W2063620147 title "Generalized Gaussian quadrature formulas" @default.
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- W2063620147 doi "https://doi.org/10.1016/0021-9045(86)90008-0" @default.
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