Strong Stieltjes distributions and orthogonal Laurent polynomials with applications to quadratures and Padé approximation
Authors:
C. Díaz-Mendoza, P. González-Vera and M. Jiménez-Paiz
Journal:
Math. Comp. 74 (2005), 1843-1870
MSC (2000):
Primary 42C05, 41A55
DOI:
https://doi.org/10.1090/S0025-5718-05-01763-1
Published electronically:
June 7, 2005
MathSciNet review:
2164100
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Starting from a strong Stieltjes distribution $\phi$, general sequences of orthogonal Laurent polynomials are introduced and some of their most relevant algebraic properties are studied. From this perspective, the connection between certain quadrature formulas associated with the distribution $\phi$ and two-point Padé approximants to the Stieltjes transform of $\phi$ is revisited. Finally, illustrative numerical examples are discussed.
- D.H. Bailey, Y. Hida, X. S. Li, B. Thompson. ARPREC: An Arbitrary Precision Computation Package. Manuscript, 2002.
- George A. Baker Jr., G. S. Rushbrooke, and H. E. Gilbert, High-temperature series expansions for the spin-${1\over 2}$ Heisenberg model by the method of irreducible representations of the symmetric group, Phys. Rev. (2) 135 (1964), A1272–A1277. MR 171545
- A. Bultheel, C. Díaz-Mendoza, P. González-Vera, and R. Orive, Quadrature on the half-line and two-point Padé approximants to Stieltjes functions. II. Convergence, J. Comput. Appl. Math. 77 (1997), no. 1-2, 53–76. ROLLS Symposium (Leipzig, 1996). MR 1440004, DOI https://doi.org/10.1016/S0377-0427%2896%2900122-7
- A. Bultheel, C. Díaz-Mendoza, P. González-Vera, and R. Orive, Quadrature on the half line and two-point Padé approximants to Stieltjes functions. III. The unbounded case, J. Comput. Appl. Math. 87 (1997), no. 1, 95–117. MR 1488823, DOI https://doi.org/10.1016/S0377-0427%2897%2900180-5
- Adhemar Bultheel, Carlos Díaz-Mendoza, Pablo González-Vera, and Ramon Orive, Estimates of the rate of convergence for certain quadrature formulas on the half-line, Continued fractions: from analytic number theory to constructive approximation (Columbia, MO, 1998) Contemp. Math., vol. 236, Amer. Math. Soc., Providence, RI, 1999, pp. 85–100. MR 1665364, DOI https://doi.org/10.1090/conm/236/03491
- A. Bultheel, C. Díaz-Mendoza, P. González-Vera, and R. Orive, On the convergence of certain Gauss-type quadrature formulas for unbounded intervals, Math. Comp. 69 (2000), no. 230, 721–747. MR 1651743, DOI https://doi.org/10.1090/S0025-5718-99-01107-2
- A. Bultheel, C. Díaz-Mendoza, P. González-Vera, and R. Orive, Orthogonal Laurent polynomials and quadrature formulas for unbounded intervals. I. Gauss-type formulas, Rocky Mountain J. Math. 33 (2003), no. 2, 585–608. 2001: a mathematics odyssey (Grand Junction, CO). MR 2021367, DOI https://doi.org/10.1216/rmjm/1181069968
- A. Bultheel, P. González-Vera, E. Hendriksen, and Olav Njåstad, Orthogonal rational functions and quadrature on the real half line, J. Complexity 19 (2003), no. 3, 212–230. Numerical integration and its complexity (Oberwolfach, 2001). MR 1984110, DOI https://doi.org/10.1016/S0885-064X%2803%2900002-5
- A. Bultheel, P. González-Vera, and R. Orive, Quadrature on the half-line and two-point Padé approximants to Stieltjes functions. I. Algebraic aspects, Proceedings of the International Conference on Orthogonality, Moment Problems and Continued Fractions (Delft, 1994), 1995, pp. 57–72. MR 1379119, DOI https://doi.org/10.1016/0377-0427%2895%2900100-X
- André Draux, Polynômes orthogonaux formels, Lecture Notes in Mathematics, vol. 974, Springer-Verlag, Berlin, 1983 (French). Applications. MR 690769
- T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach Science Publishers, New York-London-Paris, 1978. Mathematics and its Applications, Vol. 13. MR 0481884
- Lyle Cochran and S. Clement Cooper, Orthogonal Laurent polynomials on the real line, Continued fractions and orthogonal functions (Loen, 1992) Lecture Notes in Pure and Appl. Math., vol. 154, Dekker, New York, 1994, pp. 47–100. MR 1263248
- R. Cruz-Barroso, P. González-Vera. Orthogonal Laurent polynomials and quadratures on the unit circle and the real half-line. Submitted (2003).
- G. Freud. Orthogonal polynomials. Pergamon Press, Oxford, 1971.
- Walter Gautschi, Numerical analysis, Birkhäuser Boston, Inc., Boston, MA, 1997. An introduction. MR 1454125
- Pablo González-Vera and Olav Njåstad, Convergence of two-point Padé approximants to series of Stieltjes, J. Comput. Appl. Math. 32 (1990), no. 1-2, 97–105. Extrapolation and rational approximation (Luminy, 1989). MR 1091780, DOI https://doi.org/10.1016/0377-0427%2890%2990421-U
- Philip E. Gustafson and Brian A. Hagler, Gaussian quadrature rules and numerical examples for strong extensions of mass distribution functions, J. Comput. Appl. Math. 105 (1999), no. 1-2, 317–326. Continued fractions and geometric function theory (CONFUN) (Trondheim, 1997). MR 1690598, DOI https://doi.org/10.1016/S0377-0427%2899%2900023-0
- E. Hendriksen, A characterization of classical orthogonal Laurent polynomials, Nederl. Akad. Wetensch. Indag. Math. 50 (1988), no. 2, 165–180. MR 952513
- E. Hendriksen and H. van Rossum, Orthogonal Laurent polynomials, Nederl. Akad. Wetensch. Indag. Math. 48 (1986), no. 1, 17–36. MR 834317
- Einar Hille, Analytic function theory. Vol. II, Introductions to Higher Mathematics, Ginn and Co., Boston, Mass.-New York-Toronto, Ont., 1962. MR 0201608
- Guillermo López Lagomasino and Jesús Illán González, The interpolation methods of numerical integration and their connection with rational approximation, Cienc. Mat. (Havana) 8 (1987), no. 2, 31–44 (Spanish, with English summary). MR 939222
- William B. Jones, Olav Njȧstad, and W. J. Thron, Two-point Padé expansions for a family of analytic functions, J. Comput. Appl. Math. 9 (1983), no. 2, 105–123. MR 709210, DOI https://doi.org/10.1016/0377-0427%2883%2990034-1
- William B. Jones and Olav Njåstad, Orthogonal Laurent polynomials and strong moment theory: a survey, J. Comput. Appl. Math. 105 (1999), no. 1-2, 51–91. Continued fractions and geometric function theory (CONFUN) (Trondheim, 1997). MR 1690578, DOI https://doi.org/10.1016/S0377-0427%2899%2900027-8
- William B. Jones and W. J. Thron, Two-point Padé tables and $T$-fractions, Bull. Amer. Math. Soc. 83 (1977), no. 3, 388–390. MR 447543, DOI https://doi.org/10.1090/S0002-9904-1977-14284-5
- W.B. Jones, W.J. Thron. Orthogonal Laurent polynomials and gaussian quadrature. Quantum Mechanics in Mathematics, Chemistry, and Physics (K.E. Gustafson and W.P. Reinhardt, eds.), Plenum Press, New York, 1981, 449-455.
- William B. Jones, W. J. Thron, and Haakon Waadeland, A strong Stieltjes moment problem, Trans. Amer. Math. Soc. 261 (1980), no. 2, 503–528. MR 580900, DOI https://doi.org/10.1090/S0002-9947-1980-0580900-4
- G. López Lagomasino and A. Martínez Finkelshtein, Rate of convergence of two-point Padé approximants and logarithmic asymptotics of Laurent-type orthogonal polynomials, Constr. Approx. 11 (1995), no. 2, 255–286. MR 1342387, DOI https://doi.org/10.1007/BF01203418
- Arne Magnus, On the structure of the two-point Padé table, Analytic theory of continued fractions (Loen, 1981) Lecture Notes in Math., vol. 932, Springer, Berlin-New York, 1982, pp. 176–193. MR 690461
- J. H. McCabe, A formal extension of the Padé table to include two point Padé quotionts, J. Inst. Math. Appl. 15 (1975), 363–372. MR 381246
- O. Njåstad, W.J. Thron. The theory of sequences of orthogonal L-polynomials. In H. Waadeland & H. Wallin, eds., Padé Approximants and Continued fractions, Det Kongelige Norkse Videnskabers Selskab Skrifter (No. 1), 1983, 54-91.
- A. Sri Ranga, Another quadrature rule of highest algebraic degree of precision, Numer. Math. 68 (1994), no. 2, 283–294. MR 1283343, DOI https://doi.org/10.1007/s002110050062
- A. Sri Ranga, Symmetric orthogonal polynomials and the associated orthogonal $L$-polynomials, Proc. Amer. Math. Soc. 123 (1995), no. 10, 3135–3141. MR 1291791, DOI https://doi.org/10.1090/S0002-9939-1995-1291791-7
- A. Sri Ranga and Walter Van Assche, Blumenthal’s theorem for Laurent orthogonal polynomials, J. Approx. Theory 117 (2002), no. 2, 255–278. MR 1903057, DOI https://doi.org/10.1006/jath.2002.3700
- Thomas Jan Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse Math. (6) 4 (1995), no. 3, J76–J122 (French). Reprint of Ann. Fac. Sci. Toulouse 8 (1894), J76–J122. MR 1607517, DOI https://doi.org/10.5802/afst.808
Retrieve articles in Mathematics of Computation with MSC (2000): 42C05, 41A55
Retrieve articles in all journals with MSC (2000): 42C05, 41A55
Additional Information
C. Díaz-Mendoza
Affiliation:
Department of Mathematical Analysis, La Laguna University, 38271 La Laguna, Tenerife, Canary Islands, Spain
Email:
cjdiaz@ull.es
P. González-Vera
Affiliation:
Department of Mathematical Analysis, La Laguna University, 38271 La Laguna, Tenerife, Canary Islands, Spain
M. Jiménez-Paiz
Affiliation:
Department of Mathematical Analysis, La Laguna University, 38271 La Laguna, Tenerife, Canary Islands, Spain
Keywords:
Strong Stieltjes distributions,
orthogonal Laurent polynomials,
quadrature formulas,
Stieltjes transform,
two-point Padé approximants.
Received by editor(s):
May 5, 2003
Received by editor(s) in revised form:
May 4, 2004
Published electronically:
June 7, 2005
Additional Notes:
This work was supported by the Scientific Research Projects of the Ministerio de Ciencia y Tecnología and Comunidad Autónoma de Canarias under contracts BFM2001-3411 and PI 2002/136, respectively.
Article copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.