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Hermite and Hermite-Fejér interpolation for Stieltjes polynomials


Author: H. S. Jung
Journal: Math. Comp. 75 (2006), 743-766
MSC (2000): Primary 41A05, 65D05
DOI: https://doi.org/10.1090/S0025-5718-05-01795-3
Published electronically: November 3, 2005
MathSciNet review: 2196990
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Abstract: Let $ w_{\lambda}(x):=(1-x^2)^{\lambda-1/2}$ and $ P_n^{(\lambda)}$ be the ultraspherical polynomials with respect to $ w_{\lambda}(x)$. Then we denote by $ E_{n+1}^{(\lambda)}$ the Stieltjes polynomials with respect to $ w_{\lambda}(x)$ satisfying

\begin{displaymath}\int_{-1}^1 w_{\lambda}(x) P_n^{(\lambda)}(x)E_{n+1}^{(\lambd... ...gin{cases} =0, & 0 \le m < n+1,\\ \neq 0, & m=n+1. \end{cases}\end{displaymath}      

In this paper, we show uniform convergence of the Hermite-Fejér interpolation polynomials $ H_{n+1}[\cdot]$ and $ {\mathcal H}_{2n+1}[\cdot]$ based on the zeros of the Stieltjes polynomials $ E_{n+1}^{(\lambda)}$ and the product $ E_{n+1}^{(\lambda)}P_n^{(\lambda)}$ for $ 0 \le \lambda \le 1$ and $ 0 \le \lambda \le 1/2$, respectively. To prove these results, we prove that the Lebesgue constants of Hermite-Fejér interpolation operators for the Stieltjes polynomials $ E_{n+1}^{(\lambda)}$ and the product $ E_{n+1}^{(\lambda)}P_n^{(\lambda)}$ are optimal, that is, the Lebesgue constants $ \Vert H_{n+1}\Vert _{\infty}(0 \le \lambda \le 1)$ and $ \Vert{\mathcal H}_{2n+1}\Vert _{\infty} (0 \le \lambda \le 1/2)$ have optimal order $ O(1)$. In the case of the Hermite-Fejér interpolation polynomials $ {\mathcal H}_{2n+1}[\cdot]$ for $ 1/2 < \lambda \le 1$, we prove weighted uniform convergence. Moreover, we give some convergence theorems of Hermite-Fejér and Hermite interpolation polynomials for $ 0 \le \lambda \le 1$ in weighted $ L_p$ norms.


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  • 1. P. Barrucand, Intégration Numérique, Abscisses de Kronrod-Patterson et Polyn$ \hat{o}$mes de Szego, C. R. Acad. Sci. Paris, Ser. A, 270 (1970), 147-158. MR 0267755 (42:2657)
  • 2. Z. Ditzian and V. Totik, Moduli of Smoothness, Springer Series in Computational Mathmatics, Springer-Verlag, Berlin, 9 (1987).MR 0914149 (89h:41002)
  • 3. S. Ehrich, Asymptotic properties of Stieltjes polynomials and Gauss-Kronrod quadrature formulae, J. Approx. Theory, 82 (1995), 287-303.MR 1343838 (96i:41028)
  • 4. S. Ehrich, Stieltjes polynomials and the error of Gauss-Kronrod quadrature formulas, in W. Gautschi, G. Golub, G. Opfer (Eds.), Applications and Computation of Orthogonal Polynomials, Proc. Conf. Oberwolfach, International Series Numerical Mathematics, 131, Birkhäuser, Basel, (1999), 57-77.MR 1722715 (2000k:65047)
  • 5. S. Ehrich and G. Mastroianni, On the generalized Stieltjes polynomials and Lagrange interpolation, Approximation Theory and Function Series, Bolyai Soc. Math. Stud., 5 (1996), 187-203. MR 1432668 (97k:41004)
  • 6. S. Ehrich and G. Mastroianni, Stieltjes polynomials and Lagrange interpolation, Math. Comp., 66 (1997), 311-331.MR 1388888 (97j:65013)
  • 7. W. Gautschi, Gauss-Kronrod Quadrature - A survey, in Numerial Methods and Approximation Theory III, G.V. Milovanovic, ed., Nis (1988), 39-66.MR 0960329 (89k:41035)
  • 8. W. Gautschi and S. E. Notaris, An Algebraic and Numerical Study of Gauss-Kronrod Quadrature Formulae for Jacobi Weight Functions, Math. Comp., 51 (1988), 231-248. MR 0942152 (89f:65031)
  • 9. H. S. Jung, Estimates for the first and second derivatives of the Stieltjes polynomials, J. Approx. Theory, 127 (2004), 155-177.MR 2058155 (2005e:42073)
  • 10. D. Leviatan, The behavior of the derivatives of the algebraic polynomials of best approximation, J. Approx. Theory, 35 (1982), 169-176.MR 0662164 (83i:41022)
  • 11. D. S. Lubinsky, A. Máté and P. Nevai, Quadrature sums involving $ p$th powers of polynomials, SIAM J. Math. Anal., 18 (1987), 531-544.MR 0876290 (89h:41058)
  • 12. G. Monegato, A Note on Extended Gaussian Quadrature Rules, Math. Comp., 30 (1976), 812-817.MR 0440878 (55:13746)
  • 13. G. Monegato, Positivity of Weights of Extended Gauss-Legendre Quadrature Rules, Math. Comp., 32 (1978), 243-245. MR 0458809 (56:17009)
  • 14. G. Monegato, An Overview of Results and Questions Related to Kronrod Schemes, in Numerishe Integration, Proc. Conf. Oberwolfach, G. Hämmerlin (Ed.), ISNM 57 Birkhäuser, 1979. MR 0561296 (81m:65033)
  • 15. G. Monegato, Stieltjes Polynomials and Related Quadrature Rules, SIAM Review, 24 (1982), 137-158. MR 0652464 (83d:65067)
  • 16. NAG Fortran Library, Mark 15, NAG Ltd., Oxford, 1991.
  • 17. S. E. Notaris, Gauss-Kronrod quadrature formulae for weight functions of Bernstein-Szego type. II, J. Comput. Appl. Math., 29 (1990), 161-169.MR 1041189 (91b:65030)
  • 18. S. E. Notaris, An overview of results on the existence or nonexistence and the error term of Gauss-Kronrod quadrature formulae, in R. V. M. Zahar (Ed.), Approximation and computation, a Festschrift in Honer of Walter Gautschi, Internat. Ser. Numer. Math., 119, Birkhauser Basel, (1994), 485-496.MR 1333638 (96a:65034)
  • 19. P. Nevai, Mean Convergence of Lagrange Interpolation, III, Trans. Ams. Math. Soc., 282 (1984), 669-698. MR 0732113 (85c:41009)
  • 20. P. Nevai, Hilbert Transform and Lagrange interpolation, J. Approx. Theory, 60 (1990), 360-363. MR 1042655 (91a:42008)
  • 21. F. Peherstorfer, Weight functions admitting repeated positive Kronrod quadrature, BIT, 30 (1990), 145-151. MR 1032847 (91e:65043)
  • 22. F. Peherstorfer, On the Asymptotic Behaviour of Functions of the Second Kind and Stieltjes Polynomials and on the Gauss-Kronrod Quadrature Formulae, J. Approx. Theory, 70 (1992), 156-190. MR 1172017 (93h:42020)
  • 23. F. Peherstorfer, Stieltjes polynomials and functions of the second kind, J. Comput. Appl. Math., 65 (1995), 318-338.MR 1379141 (97c:33013)
  • 24. R. Piessens, E. de Doncker, C. Überhuber and D. K. Kahaner, QUADPACK-A subroutine Package for Automatic Integration, Springer Series in Computational Mathematics 1, Berlin 1983. MR 0712135 (85b:65022)
  • 25. G. Szego, Über gewisse orthogonale polynome, die zu einer oszillierenden Belegungsfunktion gehören, Math. Ann., 110 (1934), 501-513.
  • 26. G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., 23, American Mathematical Society, Providence, RI (1975). MR 0372517 (51:8724)
  • 27. P. Vértesi, Hermite-Fejér interpolations of higher order. I, Acta Math. Hungar., 54(1-2) (1989), 135-152. MR 1015784 (90k:41008)

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Additional Information

H. S. Jung
Affiliation: Division of Applied Mathmatics, KAIST, 373-1 Gusongdong, Yusongku, Taejon 305-701, Korea
Email: hsjung@amath.kaist.ac.kr

DOI: https://doi.org/10.1090/S0025-5718-05-01795-3
Keywords: Generalized Stieltjes polynomial, Hermite--Fej\'er and Hermite interpolations, convergence, Lebesgue constant
Received by editor(s): March 9, 2004
Received by editor(s) in revised form: January 12, 2005
Published electronically: November 3, 2005
Additional Notes: This work was supported by Korea Research Foundation Grant (KRF-2002-050-C00003). The research on this project started when the author visited Professor Sven Ehrich in GSF-IBB. The author thanks Professor Sven Ehrich, Professor G. Mastroianni, and the referees for many kind suggestions and comments.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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