Strong Stieltjes distributions and orthogonal Laurent polynomials with applications to quadratures and Padé approximation

Díaz-Mendoza, C.; González-Vera, P.; Jiménez-Paiz, M.

doi:10.1090/S0025-5718-05-01763-1

Strong Stieltjes distributions and orthogonal Laurent polynomials with applications to quadratures and Padé approximation
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by C. Díaz-Mendoza, P. González-Vera and M. Jiménez-Paiz;

Math. Comp. 74 (2005), 1843-1870

DOI: https://doi.org/10.1090/S0025-5718-05-01763-1

Published electronically: June 7, 2005

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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.

References

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 10.1016/S0377-0427(96)00122-7
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 10.1016/S0377-0427(97)00180-5
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 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 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 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 10.1016/S0885-064X(03)00002-5
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 10.1016/0377-0427(95)00100-X
André Draux, Polynômes orthogonaux formels, Lecture Notes in Mathematics, vol. 974, Springer-Verlag, Berlin, 1983 (French). Applications. MR 690769, DOI 10.1007/BFb0066470
T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 481884
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 10.1016/0377-0427(90)90421-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 10.1016/S0377-0427(99)00023-0
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 Company, Boston, Mass.-New York-Toronto, 1962. MR 201608
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 10.1016/0377-0427(83)90034-1
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 10.1016/S0377-0427(99)00027-8
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 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 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 10.1007/BF01203418
Arne Magnus, On the structure of the two-point Padé table, Analytic theory of continued fractions (Loen, 1981) Notas Mat., 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 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 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 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 10.5802/afst.808