Optimal nodes for interpolation in Hardy spaces

--ZusammenfassungBarycentric Formulae for Some Optimal Rational Approximants Involving Blaschke Products. We want to approximate the value Lf of some bounded linear functional L (e.g., an integral or a function evaluation) for f ~ H 2 by a linear combination ~7=o aJj, where fj := f(zi) for some points z i in the unit disk and the numbers a~ are to be chosen independent of fj. Using ideas of Sard, Larkin has shown that, for the error Lf-~7=oa~f~ to be minimal, a i must be chosen such that ~,,Lagrange polynomials. Evaluating fi 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 off" 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. AMS Subject Classifications:

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Weighted singular value decomposition basis of Szegő kernel and its applications to signal reconstruction and denoising

Wang

Tan

Lin

2023

Journal of Computational and Applied Mathematics

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Optimal nodes for interpolation in Hardy spaces

Cited by 6 publications

References 9 publications

Hardy classes and related spaces of analytic functions in the unit circle, polydisc, and ball

Hardy classes and related spaces of analytic functions in the unit circle, polydisc, and ball

Barycentric formulae for some optimal rational approximants involving blaschke products

Weighted singular value decomposition basis of Szegő kernel and its applications to signal reconstruction and denoising

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