caa a22 4500 465751296 CHVBK 20180323111834.0 cr unu---uuuuu 170327e19900301xx s 000 0 eng 10.1007/BF02247966 doi (NATIONALLICENCE)springer-10.1007/BF02247966 Berrut J. Université de Fribourg Mathématiques-Pérolles, CH-1700, Fribourg, Switzerland aut Barycentric formulae for some optimal rational approximants involving blaschke products [Elektronische Daten] [J. Berrut] Baryzentrische Formeln für einige optimale rationale Approximationen mit Blaschke-Produkten We want to approximate the valueLf of some bounded linear functionalL (e.g., an integral or a function evaluation) forf∈H 2 by a linear combination Σ j=0 j=0 a j f j , wheref j:=f(z j) for some pointsz j in the unit disk and the numbersa j are to be chosen independent off j. Using ideas of Sard, Larkin has shown that, for the errorLf−Σ j=0 j=0 a j f j to be minimal,a j must be chosen such that Σ j=0 j=0 a j f j =Lf ⊥ for the rational function $$f^ \bot (z) = \sum\nolimits_{j = 0}^n {\{ \prod\nolimits_{k = 0}^n {(1 - \bar z_k z_j )/\prod\nolimits_{k = 0}^n {(1 - \bar z_k z)} } \} l_j } (z)f_j $$ , in whichl j (z) are the Lagrange polynomials. Evaluatingf ⊥ as given above requriesO(n 2) operations for everyz. We give here formulae, patterned after the barycentric formulae for polynomial, trigonometric and rational interpolation, which permit the evaluation off ⊥ inO(n) operations for everyz, once some weights (that are independent ofz) have been computed. Moreover, we show that certain rational approximants introduced by F. Stenger (Math. Comp., 1986) can be interpreted as special cases of Larkin's interpolants, and are therefore optimal in the sense of Sard for the corresponding points. Springer-Verlag, 1990 Approximation nationallicence interpolation nationallicence rational function nationallicence Blaschke product nationallicence barycentric formula nationallicence Computing Springer-Verlag 44/1(1990-03-01), 69-82 0010-485X 44:1<69 1990 44 607 https://doi.org/10.1007/BF02247966 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF02247966 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Berrut J. Université de Fribourg Mathématiques-Pérolles, CH-1700, Fribourg, Switzerland aut NATIONALLICENCE 773 0- Computing Springer-Verlag 44/1(1990-03-01), 69-82 0010-485X 44:1<69 1990 44 607 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer