Positive quadrature formulas III: asymptotics of weights
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Abstract:
First we discuss briefly our former characterization theorem for positive interpolation quadrature formulas (abbreviated qf), provide an equivalent characterization in terms of Jacobi matrices, and give links and applications to other qf, in particular to Gauss-Kronrod quadratures and recent rediscoveries. Then for any polynomial $t_n$ which generates a positive qf, a weight function (depending on $n$) is given with respect to which $t_n$ is orthogonal to $\mathbb {P}_{n-1}$. With the help of this result an asymptotic representation of the quadrature weights is derived. In general the asymptotic behaviour is different from that of the Gaussian weights. Only under additional conditions do the quadrature weights satisfy the so-called circle law. Corresponding results are obtained for positive qf of Radau and Lobatto type.References
- Helmut Brass, Quadraturverfahren, Studia Mathematica: Skript, vol. 3, Vandenhoeck & Ruprecht, Göttingen, 1977 (German). MR 0443305
- D. Calvetti, G. H. Golub, W. B. Gragg, and L. Reichel, Computation of Gauss-Kronrod quadrature rules, Math. Comp. 69 (2000), no. 231, 1035–1052. MR 1677474, DOI 10.1090/S0025-5718-00-01174-1
- Abdelkrim Ezzirani and Allal Guessab, A fast algorithm for Gaussian type quadrature formulae with mixed boundary conditions and some lumped mass spectral approximations, Math. Comp. 68 (1999), no. 225, 217–248. MR 1604332, DOI 10.1090/S0025-5718-99-01001-7
- Géza Freud, Orthogonale Polynome, Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften, Mathematische Reihe, Band 33, Birkhäuser Verlag, Basel-Stuttgart, 1969 (German). MR 0481888
- Walter Gautschi, Orthogonal polynomials and quadrature, Electron. Trans. Numer. Anal. 9 (1999), 65–76. Orthogonal polynomials: numerical and symbolic algorithms (Leganés, 1998). MR 1749799
- Walter Gautschi, The circle theorem and related theorems for Gauss-type quadrature rules, Electron. Trans. Numer. Anal. 25 (2006), 129–137. MR 2280368
- Gene H. Golub and John H. Welsch, Calculation of Gauss quadrature rules, Math. Comp. 23 (1969), 221-230; addendum, ibid. 23 (1969), no. 106, loose microfiche suppl, A1–A10. MR 0245201, DOI 10.1090/S0025-5718-69-99647-1
- Allal Guessab, On a new family of Gaussian quadrature formulae of Birkhoff type and some lumped mass spectral approximations. II. Generalized Laguerre case on semi-infinite domains, Int. J. Pure Appl. Math. 3 (2002), no. 2, 193–239. MR 1937650
- Dirk P. Laurie, Anti-Gaussian quadrature formulas, Math. Comp. 65 (1996), no. 214, 739–747. MR 1333318, DOI 10.1090/S0025-5718-96-00713-2
- Dirk P. Laurie, Calculation of Gauss-Kronrod quadrature rules, Math. Comp. 66 (1997), no. 219, 1133–1145. MR 1422788, DOI 10.1090/S0025-5718-97-00861-2
- Giovanni Monegato, An overview of the computational aspects of Kronrod quadrature rules, Numer. Algorithms 26 (2001), no. 2, 173–196. MR 1829797, DOI 10.1023/A:1016640617732
- I.P. Natanson, Constructive Function Theory, Vol. II, F. Ungar Publ. Co., New York, 1965.
- Paul Nevai, A new class of orthogonal polynomials, Proc. Amer. Math. Soc. 91 (1984), no. 3, 409–415. MR 744640, DOI 10.1090/S0002-9939-1984-0744640-X
- Geno Nikolov, On the weights of nearly Gaussian quadrature formulae, East J. Approx. 7 (2001), no. 1, 115–120. MR 1834159
- Franz Peherstorfer, Characterization of positive quadrature formulas, SIAM J. Math. Anal. 12 (1981), no. 6, 935–942. MR 635246, DOI 10.1137/0512079
- Franz Peherstorfer, Characterization of quadrature formula. II, SIAM J. Math. Anal. 15 (1984), no. 5, 1021–1030. MR 755862, DOI 10.1137/0515079
- Franz Peherstorfer, Linear combinations of orthogonal polynomials generating positive quadrature formulas, Math. Comp. 55 (1990), no. 191, 231–241. MR 1023052, DOI 10.1090/S0025-5718-1990-1023052-9
- Franz Peherstorfer, Finite perturbations of orthogonal polynomials, J. Comput. Appl. Math. 44 (1992), no. 3, 275–302. MR 1199259, DOI 10.1016/0377-0427(92)90002-F
- Franz Peherstorfer, On positive quadrature formulas, Numerical integration, IV (Oberwolfach, 1992) Internat. Ser. Numer. Math., vol. 112, Birkhäuser, Basel, 1993, pp. 297–313. MR 1248412
- Franz Peherstorfer, Zeros of linear combinations of orthogonal polynomials, Math. Proc. Cambridge Philos. Soc. 117 (1995), no. 3, 533–544. MR 1317495, DOI 10.1017/S0305004100073357
- F. Peherstorfer, A special class of polynomials orthogonal on the unit circle including the associated polynomials, Constr. Approx. 12 (1996), no. 2, 161–185. MR 1393285, DOI 10.1007/s003659900008
- Franz Peherstorfer and Knut Petras, Stieltjes polynomials and Gauss-Kronrod quadrature for Jacobi weight functions, Numer. Math. 95 (2003), no. 4, 689–706. MR 2013124, DOI 10.1007/s00211-002-0412-2
- H. J. Schmid, A note on positive quadrature rules, Rocky Mountain J. Math. 19 (1989), no. 1, 395–404. Constructive Function Theory—86 Conference (Edmonton, AB, 1986). MR 1016190, DOI 10.1216/RMJ-1989-19-1-395
- J. Shohat, On mechanical quadratures, in particular, with positive coefficients, Trans. Amer. Math. Soc. 42 (1937), no. 3, 461–496. MR 1501930, DOI 10.1090/S0002-9947-1937-1501930-6
- Barry Simon, Orthogonal polynomials on the unit circle. Part 1, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Classical theory. MR 2105088, DOI 10.1090/coll054.1
- Barry Simon, Orthogonal polynomials on the unit circle. Part 1, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Classical theory. MR 2105088, DOI 10.1090/coll054.1
- M.-R. Skrzipek, Generalized associated polynomials and their application in numerical differentiation and quadrature, Calcolo 40 (2003), no. 3, 131–147. MR 2025599, DOI 10.1007/s10092-003-0074-1
- G. Sottas and G. Wanner, The number of positive weights of a quadrature formula, BIT 22 (1982), no. 3, 339–352. MR 675668, DOI 10.1007/BF01934447
- Miodrag M. Spalević, On generalized averaged Gaussian formulas, Math. Comp. 76 (2007), no. 259, 1483–1492. MR 2299784, DOI 10.1090/S0025-5718-07-01975-8
- G. Szegö, Über den asymptotischen Ausdruck von Polynomen, die durch eine Orthogonalitätseigenschaft definiert sind, Math. Ann. 86 (1922), no. 1-2, 114–139 (German). MR 1512082, DOI 10.1007/BF01458575
- G. Szegő, Orthogonal Polynomials Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. Soc., Providence, RI, 3rd ed., 1967.
- Vilmos Totik, Asymptotics for Christoffel functions for general measures on the real line, J. Anal. Math. 81 (2000), 283–303. MR 1785285, DOI 10.1007/BF02788993
- Yuan Xu, Quasi-orthogonal polynomials, quadrature, and interpolation, J. Math. Anal. Appl. 182 (1994), no. 3, 779–799. MR 1272153, DOI 10.1006/jmaa.1994.1121
- Yuan Xu, A characterization of positive quadrature formulae, Math. Comp. 62 (1994), no. 206, 703–718. MR 1223234, DOI 10.1090/S0025-5718-1994-1223234-0
Additional Information
- Franz Peherstorfer
- Affiliation: Abteilung für Dynamische Systeme und Approximationstheorie, Institut für Analysis, J.K. Universität Linz, Altenberger Strasse 69, 4040 Linz, Austria
- Email: franz.peherstorfer@jku.at
- Received by editor(s): June 6, 2007
- Received by editor(s) in revised form: September 4, 2007
- Published electronically: May 1, 2008
- Additional Notes: The author was supported by the Austrian Science Fund FWF, project no. P20413-N18.
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 77 (2008), 2241-2259
- MSC (2000): Primary 65D32; Secondary 42C05
- DOI: https://doi.org/10.1090/S0025-5718-08-02119-4
- MathSciNet review: 2429883