Barycentric formulae for some optimal rational approximants involving blaschke products

Abstract

We want to approximate the value Lf of some bounded linear functional L (e.g., an integral or a function evaluation) for f∈H2 by a linear combination j=0 n ajf j where fj:=fzj for some points zj in the unit disk and the numbers aj are to be chosen independent of fj . Using ideas of Sard, Larkin has shown that, for the error Lf-j=0 n ajf j to be minimal, aj must be chosen such that j=0 n ajf j=Lf⊥ for the rational function f⊥z= j=0 n k =0 n 1-zd kzj/k =0 n 1-zd kzlj zfj, in which ljz are the Lagrange polynomials. Evaluating f⊥ as given above requires O ( n 2) operations for every z . We give here formulae, patterned after the barycentric formulae for polynomial, trigonometric and rational interpolation, which permit the evaluation of f⊥ in O ( n ) operations for every z , once some weights (that are independent of z ) 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. —Author's Abstract