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The remainder term for analytic functions of symmetric Gaussian quadratures. (English) Zbl 0854.41025
Summary: For analytic functions the remainder term of Gaussian quadrature rules can be expressed as a contour integral with kernel \(K_n\). In this paper the kernel is studied on elliptic contours for a great variety of symmetric weight functions including especially Gegenbauer weight functions. First a new series representation of the kernel is developed and analyzed. Then the location of the maximum modulus of the kernel on suitable ellipses is determined. Depending on the weight function the maximum modulus is attained at the intersection point of the ellipse with either the real or imaginary axis. Finally, a detailed discussion for some special weight functions is given.

MSC:
41A55 Approximate quadratures
65D30 Numerical integration
65D32 Numerical quadrature and cubature formulas
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[1] Helmut Brass, Quadraturverfahren, Vandenhoeck & Ruprecht, Göttingen, 1977 (German). Studia Mathematica, Skript 3. · Zbl 0368.65014
[2] Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. · Zbl 0537.65020
[3] Géza Freud, Error estimates for Gauss-Jacobi quadrature formulae, Topics in numerical analysis (Proc. Roy. Irish Acad. Conf., University Coll., Dublin, 1972) Academic Press, London, 1973, pp. 113 – 121.
[4] Walter Gautschi, On the remainder term for analytic functions of Gauss-Lobatto and Gauss-Radau quadratures, Proceedings of the U.S.-Western Europe Regional Conference on Padé Approximants and Related Topics (Boulder, CO, 1988), 1991, pp. 209 – 226. , https://doi.org/10.1216/rmjm/1181073004 Walter Gautschi, Corrections to: ”On the remainder term for analytic functions of Gauss-Lobatto and Gauss-Radau quadratures”, Rocky Mountain J. Math. 21 (1991), no. 3, 1143. Walter Gautschi, On the remainder term for analytic functions of Gauss-Lobatto and Gauss-Radau quadratures, Proceedings of the U.S.-Western Europe Regional Conference on Padé Approximants and Related Topics (Boulder, CO, 1988), 1991, pp. 209 – 226. , https://doi.org/10.1216/rmjm/1181073004 Walter Gautschi, Corrections to: ”On the remainder term for analytic functions of Gauss-Lobatto and Gauss-Radau quadratures”, Rocky Mountain J. Math. 21 (1991), no. 3, 1143. · Zbl 0749.41026
[5] Walter Gautschi, Remainder estimates for analytic functions, Numerical integration (Bergen, 1991) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 357, Kluwer Acad. Publ., Dordrecht, 1992, pp. 133 – 145. · Zbl 0754.41026
[6] Walter Gautschi and Shikang Li, The remainder term for analytic functions of Gauss-Radau and Gauss-Lobatto quadrature rules and with multiple end points, J. Comput. Appl. Math. 33 (1990), no. 3, 315 – 329. · Zbl 0724.41024
[7] Walter Gautschi and Richard S. Varga, Error bounds for Gaussian quadrature of analytic functions, SIAM J. Numer. Anal. 20 (1983), no. 6, 1170 – 1186. · Zbl 0545.41040
[8] Walter Gautschi, E. Tychopoulos, and R. S. Varga, A note on the contour integral representation of the remainder term for a Gauss-Chebyshev quadrature rule, SIAM J. Numer. Anal. 27 (1990), no. 1, 219 – 224. · Zbl 0685.41019
[9] D.B. Hunter, Some error expansions for Gaussian quadrature, BIT 35 (1995), 64-82. · Zbl 0824.41032
[10] F. Peherstorfer, On the remainder of Gaussian quadrature formulas for Bernstein-Szegő weight functions, Math. Comp. 60 (1993), no. 201, 317 – 325. · Zbl 0796.41025
[11] T. Schira, Ableitungsfreie Fehlerabschätzungen bei numerischer Integration holomorpher Funktionen, Ph.D. Dissertation, Universität Karlsruhe, 1994. · Zbl 0809.41029
[12] Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society, Providence, R.I., 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII. · JFM 65.0278.03
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