Publication: Gaussian quadrature formulae on the unit circle
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2002-03-01
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Elsevier
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Abstract
Let μ be a probability measure on [0,2π]. In this paper we shall be concerned with the estimation of integrals of the form $$ I_{\mu }(f):=\frac{1}{2\pi }\int_{0}^{2\pi }f(e^{i\theta })d\mu (\theta ) $$.
For this purpose we will construct quadrature formulae which are exact in a certain linear subspace of Laurent polynomials. The zeros of Szegö polynomials are chosen as nodes of the corresponding quadratures. We will study this quadrature formula in terms of error expressions and convergence, as well as, its relation with certain two-point Padé approximants for the Herglotz–Riesz transform of μ. Furthermore, a comparison with the so-called Szegö quadrature formulae is presented through some illustrative numerical examples.
For this purpose we will construct quadrature formulae which are exact in a certain linear subspace of Laurent polynomials. The zeros of Szegö polynomials are chosen as nodes of the corresponding quadratures. We will study this quadrature formula in terms of error expressions and convergence, as well as, its relation with certain two-point Padé approximants for the Herglotz–Riesz transform of μ. Furthermore, a comparison with the so-called Szegö quadrature formulae is presented through some illustrative numerical examples.
Description
25 pages, no figures.-- MSC2000 codes: 41A55; 33C45.
MR#: MR1933236 (2003k:65022)
Zbl#: Zbl 1013.41015
MR#: MR1933236 (2003k:65022)
Zbl#: Zbl 1013.41015
Keywords
Laurent polynomials, Positive measure, Quadrature formula, Two-point Padé approximants, Rate of convergence
Bibliographic citation
Journal of Computational and Applied Mathematics, 2002, vol. 140, n. 1-2, p. 159-183