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Error bounds for Gauss-Turán quadrature formulae of analytic functions. (English) Zbl 1030.41018
Authors’ abstract: We study the kernels of the remainder term \(R_{n,s}(f)\) of Gauss-Turán quadrature formulas \[ \int _{-1}^1f(t)w(t)dt=\sum _{\nu =1}^n\sum _{i=0}^{2s}A_{i,\nu }f^{i}(\tau _\nu)+R_{n,s}(f)\quad (n\in N,s\in N_0) \] for classes of analytic functions on elliptical contours with foci at \(\pm 1\), when the weight \(w\) is one of the special Jacobi weights \(w^{(\alpha ,\beta)}(t)=(1-t)^\alpha (1+t)^\beta \) \((\alpha =\beta =-1/2\); \(\alpha =\beta =1/2+s\); \(\alpha =-1/2,\beta =1/2+s\); \(\alpha =1/2+s\), \(\beta =-1/2)\). We investigate the location on the contour where the modulus of the kernel attains its maximum value. Some numerical examples are included’.

MSC:
41A55 Approximate quadratures
65D30 Numerical integration
65D32 Numerical quadrature and cubature formulas
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[1] S. Bernstein, Sur les polynomes orthogonaux relatifs à un segment fini, J. Math. Pures Appl. 9 (1930), 127-177. · JFM 56.0947.04
[2] L. Chakalov, Über eine allgemeine Quadraturformel, C.R. Acad. Bulgar. Sci. 1 (1948), 9-12. · Zbl 0037.04401
[3] Lubomir Tschakaloff, General quadrature formulae of Gaussian type, East J. Approx. 1 (1995), no. 2, 261 – 276. Translated from the 1954 Bulgarian original by Borislav Bojanov [Borislav D. Boyanov]. · Zbl 0852.41023
[4] L. Chakalov, Formules générales de quadrature mécanique du type de Gauss, Colloq. Math. 5 (1957), 69-73. · Zbl 0081.05801
[5] J. D. Donaldson and David Elliott, A unified approach to quadrature rules with asymptotic estimates of their remainders, SIAM J. Numer. Anal. 9 (1972), 573 – 602. · Zbl 0264.65020
[6] 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
[7] 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
[8] Walter Gautschi and Gradimir V. Milovanović, \?-orthogonality and construction of Gauss-Turán-type quadrature formulae, J. Comput. Appl. Math. 86 (1997), no. 1, 205 – 218. Special issue dedicated to William B. Gragg (Monterey, CA, 1996). · Zbl 0890.65021
[9] W. Gautschi and S. E. Notaris, Erratum to: ”Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szegő type” [J. Comput. Appl. Math. 25 (1989), no. 2, 199 – 224; MR0988057 (90d:65045)], J. Comput. Appl. Math. 27 (1989), no. 3, 429. · Zbl 0691.41028
[10] 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
[11] 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
[12] A. Ghizzetti and A. Ossicini, Quadrature formulae, Academic Press, New York, 1970. · Zbl 0194.36901
[13] Aldo Ghizzetti and Alessandro Ossicini, Sull’esistenza e unicità delle formule di quadratura gaussiane, Rend. Mat. (6) 8 (1975), 1 – 15 (Italian, with English summary). Collection of articles dedicated to Mauro Picone on the occasion of his ninetieth birthday. · Zbl 0303.65021
[14] G. H. Golub and J. Kautský, Calculation of Gauss quadratures with multiple free and fixed knots, Numer. Math. 41 (1983), no. 2, 147 – 163. · Zbl 0525.65010
[15] L. Gori Nicolò-Amati, On the behaviour of the zeros of some s-orthogonal polynomials, in Orthogonal Polynomials and Their Applications, 2nd Int. Symp. (Segovia, 1986), Monogr. Acad. Cienc. Exactas, Fis., Quim., Nat., Zaragoza, 1988, pp. 71-85.
[16] Laura Gori and Charles A. Micchelli, On weight functions which admit explicit Gauss-Turán quadrature formulas, Math. Comp. 65 (1996), no. 216, 1567 – 1581. · Zbl 0853.65027
[17] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 6th ed., Academic Press, Inc., San Diego, CA, 2000. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. · Zbl 0981.65001
[18] David Hunter and Geno Nikolov, On the error term of symmetric Gauss-Lobatto quadrature formulae for analytic functions, Math. Comp. 69 (2000), no. 229, 269 – 282. · Zbl 0946.41019
[19] D. V. Ionescu, Restes des formules de quadrature de Gauss et de Turán, Acta Math. Acad. Sci. Hungar. 18 (1967), 283 – 295 (French). · Zbl 0159.20905
[20] Samuel Karlin and Allan Pinkus, Gaussian quadrature formulae with multiple nodes, Studies in spline functions and approximation theory, Academic Press, New York, 1976, pp. 113 – 141. · Zbl 0345.41020
[21] Gradimir V. Milovanović, Construction of \?-orthogonal polynomials and Turán quadrature formulae, Numerical methods and approximation theory, III (Niš, 1987) Univ. Niš, Niš, 1988, pp. 311 – 328. · Zbl 0643.65011
[22] Gradimir V. Milovanović, Quadratures with multiple nodes, power orthogonality, and moment-preserving spline approximation, J. Comput. Appl. Math. 127 (2001), no. 1-2, 267 – 286. Numerical analysis 2000, Vol. V, Quadrature and orthogonal polynomials. · Zbl 0970.65023
[23] Gradimir V. Milovanović and Miodrag M. Spalević, A numerical procedure for coefficients in generalized Gauss-Turán quadratures, Filomat 9 (1995), 1 – 8. · Zbl 0845.65005
[24] Gradimir V. Milovanović and Miodrag M. Spalević, Construction of Chakalov-Popoviciu’s type quadrature formulae, Proceedings of the Third International Conference on Functional Analysis and Approximation Theory, Vol. II (Acquafredda di Maratea, 1996), 1998, pp. 625 – 636. · Zbl 0906.65023
[25] G. V. Milovanovic and M. M. Spalevic, Quadrature formulae connected to \(\sigma\)-orthogonal polynomials, J. Comput. Appl. Math. 140 (2002), 619-637. · Zbl 0997.65045
[26] Alessandro Ossicini, Costruzione di formule di quadratura di tipo Gaussiano, Ann. Mat. Pura Appl. (4) 72 (1966), 213 – 237 (Italian). · Zbl 0143.38603
[27] Alessandro Ossicini, Le funzioni di influenza nel problema di Gauss sulle formule di quadratura, Matematiche (Catania) 23 (1968), 7 – 30 (Italian). · Zbl 0176.14402
[28] A. Ossicini, M. R. Martinelli, and F. Rosati, Characteristic functions and \?-orthogonal polynomials, Rend. Mat. Appl. (7) 14 (1994), no. 2, 355 – 366 (Italian, with English and Italian summaries). · Zbl 0812.33002
[29] A. Ossicini and F. Rosati, Funzioni caratteristiche nelle formule di quadratura gaussiane con nodi multipli, Boll. Un. Mat. Ital. (4) 11 (1975), no. 3, suppl., 224 – 237 (Italian, with French summary). Collection of articles dedicated to Giovanni Sansone on the occasion of his eighty-fifth birthday. · Zbl 0311.41024
[30] A. Ossicini and F. Rosati, Comparison theorems for the zeros of \?-orthogonal polynomials, Calcolo 16 (1979), no. 4, 371 – 381 (1980) (Italian, with French summary). · Zbl 0441.33017
[31] A. Ossicini and F. Rosati, \?-orthogonal Jacobi polynomials, Rend. Mat. Appl. (7) 12 (1992), no. 2, 399 – 403 (Italian, with English and Italian summaries). · Zbl 0765.33007
[32] Paraschiva Pavel, On the remainder of some Gaussian formulae, Studia Univ. Babeş-Bolyai Ser. Math.-Phys. 12 (1967), no. 2, 65 – 70 (English, with Romanian and Russian summaries). · Zbl 0206.46802
[33] Paraschiva Pavel, On some quadrature formulas of Gaussian type, Studia Univ. Babeş-Bolyai Ser. Math.-Phys. 13 (1968), no. 1, 51 – 58 (Romanian, with Russian and French summaries).
[34] Paraschiva Pavel, On the remainder of certain quadrature formulas of Gauss-Christoffel type, Studia Univ. Babeş-Bolyai Ser. Math.-Phys. 13 (1968), no. 2, 67 – 72 (Romanian, with Russian and French summaries).
[35] 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
[36] T. Popoviciu, Sur une généralisation de la formule d’integration numérique de Gauss, Acad. R. P. Romîne Fil. Iasi Stud. Cerc. Sti. 6 (1955), 29-57. · Zbl 0068.05003
[37] Thomas Schira, The remainder term for analytic functions of Gauss-Lobatto quadratures, J. Comput. Appl. Math. 76 (1996), no. 1-2, 171 – 193. · Zbl 0866.41023
[38] Thomas Schira, The remainder term for analytic functions of symmetric Gaussian quadratures, Math. Comp. 66 (1997), no. 217, 297 – 310. · Zbl 0854.41025
[39] Miodrag M. Spalević, Product of Turán quadratures for cube, simplex, surface of the sphere, \overline\?^{\?}_{\?},\?^{\?²}_{\?}, J. Comput. Appl. Math. 106 (1999), no. 1, 99 – 115. · Zbl 0937.65027
[40] M. M. Spalevic, Calculation of Chakalov-Popoviciu’s quadratures of Radau and Lobatto type, ANZIAM J. (formerly as J. Aust. Math. Soc. B) 43 (2002), 429-447. · Zbl 0998.65035
[41] Dumitru D. Stancu, Sur une classe de polynômes orthogonaux et sur des formules générales de quadrature à nombre minimum de termes, Bull. Math. Soc. Sci. Math. Phys. R. P. Roumaine (N.S.) 1 (49) (1957), 479 – 498 (French).
[42] D. D. Stancu, Sur certaines formules générales d’intégration numérique, Acad. R. P. Romîne. Stud. Cerc. Mat. 9 (1958), 209 – 216 (Romanian, with Russian and French summaries). · Zbl 0083.35402
[43] A. H. Stroud and D. D. Stancu, Quadrature formulas with multiple Gaussian nodes, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (1965), 129 – 143. · Zbl 0141.13803
[44] P. Turán, On the theory of the mechanical quadrature, Acta Sci. Math. Szeged 12(1950), 30-37. · Zbl 0041.44417
[45] Graziano Vincenti, On the computation of the coefficients of \?-orthogonal polynomials, SIAM J. Numer. Anal. 23 (1986), no. 6, 1290 – 1294. · Zbl 0632.65014
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