Abstract
Padé approximation is the rational generalization of Hermite interpolating polynomial. On its own merits, it has earned a relevant place in the theory of constructive approximation. In this article, we will develop an exhaustive analysis of two-point Padé approximations to Herglotz-Riesz transforms. We study the convergence problem when the poles are partially preassigned. In this analysis, the Stieltjes polynomials on the unit circle naturally arise. Finally, some illustrative numerical examples are discussed.
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The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
References
Brezinski, C.: Padé-type approximation and general orthogonal polynomials. ISNM, vol. 50 Birkhäuser, Basel (1980)
Walsh, J.L.: Interpolation and approximation, 5th edn. Amer. Math. Soc. Colloq. Publ., vol. 20. Amer. Math. Soc., Providence, Rhode Island, First edn. 1935 (1969)
Brezinski, C.: Partial Padé approximantion. J. Approx. Th. 54, 210–233 (1988). https://doi.org/10.1016/0021-9045(88)90020-2
Brezinski, C.: New representations of Padé, Padé-type and partial Padé approximants. J. Comput. Appl. Math. 284, 69–77 (2015). https://doi.org/10.1016/j.cam.2014.07.007
Díaz Mendoza, C., González-Vera, P., Orive, R.: On convergence of two-point Padé-type approximants. Numer. Math. 72, 295–312 (1996). https://doi.org/10.1007/s002110050171
Ambroladze, A., Wallin, H.: Convergence rates of Padé-type approximants. J. Approx. Th. 86(3), 310–319 (1996)
Simon, B.: Orthogonal polynomials on the unit circle. Colloquium publications, vol. 54. Amer. Math Soc., Providence, RI (2005)
Carathéodory, C.: ÜBer den variabilitätsbereich der koeffizienten von potenzreihen die gegebene werte nicht annehmen. Math. Anal. 64, 95–115 (1907). https://doi.org/10.1007/BF01449883
Bultheel, A.: Laurent series and their Padé approximations. Oper. Th. Adv. Appl., vol. OT-27 Birkhäuser, Basel-Boston. https://doi.org/10.1007/978-3-0348-9306-0 (1987)
Bultheel, A., González-Vera, P., Hendriksen, E., Njåstad, O.: Quadrature formulas on the unit circle and two-point Padé approximation. In: Cuyt, A.M. (ed.) Nonlinear numerical methods and rational approximation II, pp. 303–318. Kluwer (1994)
González-Vera, P., Jiménez-Paíz, M., Orive, R.: On the convergence of quadrature formulas connected to multipoint Padé-type approximantion. J. Math. Anal. Appl. 202, 747–775 (1996). https://doi.org/10.1006/jmaa.1996.0345
Díaz Mendoza, C., González-Vera, P., Orive, R.: Two-point Padé-type approximation to the cauchy transform of certain strong distributions. J. Comput. Appl. Math. 105, 229–243 (1999). https://doi.org/10.1016/S0377-0427(99)00043-6
De Andrade, E.X., McCabe, J.H.: On the two point Padé table for a distribution. Rocky Mountain J. Math. 33(2), 545–566 (2003). https://doi.org/10.1216/rmjm/1181069966
McCabe, J.H.: A formal extension of the Padé table to include two-point Padé quotients. J. Inst. Math. Appl. 15, 363–372 (1975). https://doi.org/10.1093/imamat/15.3.363
Saff, E.B., Varga, R.S.: Padé and Rational Approximation: Theory and Applications. Academic Press, London (1977)
Draux, A.: Polynômes Orthogonaux Formels. Applications. ISNM, Vol. 974. Springer, Berlin (1983)
Jones, W.B., Njåstad, O., Thron, W.B.: Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle. Bull. London Math. Soc. 21, 113–152 (1989). https://doi.org/10.1112/blms/21.2.113
Bultheel, A., Cruz-Barroso, R., Díaz Mendoza, C.: Zeros of quasi-paraorthogonal polynomials and positive quadrature. J. Comput. Appl. Math. 407, 14039 (2022). https://doi.org/10.1016/j.cam.2021.114039
González-Vera, P., Santos León, J.C., Njåstad, O.: Some results about numerical quadrature on the unit circle. Adv. Comput. Math. 5, 297–328 (1996). https://doi.org/10.1007/BF02124749
Bultheel, A., González-Vera, P., Hendriksen, E., Njåstad, O.: On the convergence of multipoint Padé-type approximants and quadrature formulas associated with the unit circle. Numer. Algorithms 13, 321–344 (1996). https://doi.org/10.1007/BF02207699
De La Calle Ysern, B., Lopez Lagomasino, G., Reichel, L.: Stieltjes-type polynomials on the unit circle. Math. Comput. 78(266), 969–997 (2009). https://doi.org/10.1090/S0025-5718-08-02195-9
De La Calle Ysern, B.: Optimal extension of the szegő quadrature. IMA J. Numer. Anal. 35, 722–748 (2015)
Gonchar, A.A.: On the convergence of generalized Padé approximants of meromorphic functions. Sb. Math 27, 503–514 (1975)
Acknowledgements
The authors are indebted to the inspiring work of Claude Brezinski, to whom they want to dedicate this paper on the occasion of his eightieth birthday. They also want to thank the referees for careful reading of an earlier version of the paper which allowed to improve the presentation and which triggered a generalization of several theorems from the 2PTA to the 2PPA case.
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Bultheel, A., Mendoza, C.D. Two-point Padé approximation to Herglotz-Riesz transforms. Numer Algor 92, 269–299 (2023). https://doi.org/10.1007/s11075-022-01467-9
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DOI: https://doi.org/10.1007/s11075-022-01467-9
Keywords
- Padé approximation
- Herglotz-Riesz transform
- Carathéodory function
- Stieltjes polynomial
- Quasi-paraorthogonal
- Unit circle