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Hermite and Hermite-Fejer Interpolation and Associated Product Integration Rules on the Real Line: The L1 Theory

Published online by Cambridge University Press:  20 November 2018

D. S. Lubinsky  and
P. Rabinowitz
D. S. Lubinsky
Affiliation:
Department of Mathematics University of the WitwatersrandP.O. Wits 2050 Republic of South Africa
P. Rabinowitz
Affiliation:
Department of Applied Mathematics and Computer Science The Weizmann Institute of Science Rehovot76100, Israel
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Abstract

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We investigate convergence in a weighted L1 -norm of Hermite-Fejér and Hermite interpolation at the zeros of orthogonal polynomials associated with weights on the real line. The results are then applied to convergences of product integration rules. From the point of view of orthogonal polynomials, the new feature is that Freud and Erdös weights are treated simultaneously and that relatively few assumptions are placed on the weight. From the point of view of product integration, the rules exhibit convergence for highly oscillatory kernels (for example) and for functions of rapid growth at infinity.

Keywords

41A0565D0565D3042C05Hermite-FejérHermiteinterpolationproduct integrationconvergenceinfinite intervalsFreud weightsErdös weights
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 3 , 01 June 1992 , pp. 561 - 590
Copyright
Copyright © Canadian Mathematical Society 1992

References

1. Alaylioglu, A. and Lubinsky, D.S., A Product Quadrature Algorithm by Hermite Interpolation, J. Comp. Appl. Math., 17 (1987), 237–269.Google Scholar
2. Bauldry, W.C., Estimates of Asymmetric Polynomials, J. Approx. Theory, 63 (1990), 225–237.Google Scholar
3. Bonan, S.S. andClark, D.S., Estimates of the Hermite and Freud Polynomials, J. ApproxJ. Approx. Theory 63 (1990), 210–224.Google Scholar
4. Clunie, J. and Kovari, T., On Integral Functions Having Prescribed Asymptotic Growth II, Can. J. Math. 20 (1968),7–20.Google Scholar
5. Freud, G., Orthogonal Polynomials, Pergamon Press/Akademiai Kiado, Budapest, 1970.Google Scholar
6. Freud, G., On Markov-Bernstein Type Inequalities and Their Applications, J. Approx. Theory 19 (1977), 22- 37.Google Scholar
7. Grünwald, G., On the Theory of Interpolation, Acta Math. 75 (1942), 219–245.Google Scholar
8. Knopfmacher, A. and Lubinsky, D.S., Mean Convergence of Lagrange Interpolation for Freud's Weights with Application to Product Integration Rules, J. Comp. Appl. Math. 17 (1987), 79–103.Google Scholar
9. Levin, A.L. and Lubinsky, D.S., Canonical Products and the Weights exp(—|x|α), α > 1, with Applications, J. Approx. Theory 49 (1987), 170–195.+1,+with+Applications,+J.+Approx.+Theory+49+(1987),+170–195.>Google Scholar
10. Lubinsky, D.S., Strong Asymptotics for Extremal Errors and Polynomials Associated with Erdos-Type Weights, Pitman Research Notes in Math. 202, Longmans, Harlow, (1989).Google Scholar
11. Lubinsky, D.S. and Mthembu, T.Z., Lp Markov-Bernstein Inequalities for Erdôs Weights, J. Approx. Theory 65 (1991), 301–321.Google Scholar
12. Lubinsky, D.S. and Rabinowitz, P., Rates of Convergence of Guassian Quadrature for Singular Integrands, Math. Comp. 43 (1984), 219–242.Google Scholar
13. Lubinsky, D.S. and Saff, E.B., Strong Asymptotics for Extremal Errors and Extremal Polynomials Associated with Weights on (—∞, ∞), Springer Lecture Notes in Math. 1305, Springer, Berlin, (1988).Google Scholar
14. Lubinsky, D.S. and Sidi, A., Convergence of Product Integration Rules for Functions with Interior and Endpoint Singularities over Bounded and Unbounded Intervals, Math. Comp. 46 (1986), 297–313.Google Scholar
15. Mhaskar, H.N., Bounds for Certain Freud-Type Orthogonal Polynomials, J. Approx. Theory 63 (1990), 238–254.Google Scholar
16. Mhaskar, H.N. and Saff, E.B., Where Does the Sup-Norm of a Weighted Polynomial Live?, Constr. Approx. 1 (1985), 71–91.Google Scholar
1. Mhaskar, H.N. and Saff, E.B., Where Does the Lp-Norm of a Weighted Polynomial Live?, Trans. Amer. Math. Soc. 303 (1987), 109–124. (Errata: 308 (1988), 431).Google Scholar
18. Mhaskar, H.N. and Xu, Y., Mean Convergence of Expansions in Freud-Type Orthogonal Polynomials, SIAM J. Math. Anal. 22 (1991), 847–855.Google Scholar
19. Nevai, P., Orthogonal Polynomials, Memoirs of the Amer. Math. Soc. 213 (1979).Google Scholar
20. Nevai, P., Geza Freud: Orthogonal Polynomials and Christoffel Functions, A Case Study, J. Approx. Theory 48 (1986), 3–167.Google Scholar
21. Nevai, P. and Vértesi, P., Mean Convergence ofHermite-Fejér Interpolation, J. Math. Anal. Appl. 105 (1985), 26–58.Google Scholar
22. Nevai, P. and Vértesi, P., Convergence of Hermite-Fejér Interpolation at Zeros of Generalized Jacobi Polynomials, Acta Sci. Math. Szeged 53 (1989), 77–98.Google Scholar
23. Rabinowitz, P., Ignoring the Singularity in Numerical Integration, in Topics in Numerical Analysis III (Miller, J.J.A., éd.), Academic Press, London, 1977. 361–368.Google Scholar
24. Rabinowitz, P., Numerical Integration in the Presence of an Interior Singularity, J. Comp. Appl. Math. 17 (1987), 31–41.Google Scholar
25. Rabinowitz, P., Product Integration Based on Hermite-Fejér Interpolation, J. Comp. Appl. Math. 28 (1989). 85–101.Google Scholar
26. Rabinowitz, P., Product Integration of Singular Integrands using Hermite-Fejér Interpolation, to appear in BIT.Google Scholar
27. Rabinowitz, P. and Vértesi, P., Hermite-Fejér-Related Interpolation and Product Integration, in preparation.Google Scholar
28. Smith, W.E., Sloan, I.H. and Opie, A.H., Product Integration over Infinite Intervals I. Rules Based on the Zeros ofHermite Polynomials, Math. Comp. 40 (1983), 519–536.Google Scholar
29. Szegö, G., Orthogonal Polynomials, Amer. Math. Soc. Colloq. Pub. 23 Amer. Math. Soc., Providence, R.I., 1939.4th Edn., 1975.Google Scholar