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Explicit Bounds for Hermite Polynomials in the Oscillatory Region

Published online by Cambridge University Press:  01 February 2010

William H. Foster  and
Ilia Krasikov
William H. Foster
Affiliation:
Department of Mathematical Sciences, Brunel University, Kingston Lane, Uxbridge UB8 3PH, London, [email protected]
Ilia Krasikov
Affiliation:
Department of Mathematical Sciences, Brunel University, Kingston Lane, Uxbridge UB8 3PH, London, [email protected]

Abstract

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We apply a method of positive quadratic forms based on polynomial inequalities to establish sharp explicit bounds on the envelope of Hermite polynomials in the oscillatory region |x| < (2k – 3/2)1/2.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 3 , 2000 , pp. 307 - 314
Copyright
Copyright © London Mathematical Society 2000

References

1. Bonan, S. and Clark, D. S., ‘Estimates of the Hermite and the Freud polynomials’, J. Approx. Theory 63 (1990) 210–224.CrossRefGoogle Scholar
2. Deift, P., Kriecherbauer, T., McLaughlin, K., Venakides, S. and Zhou, X., ‘Strong asymptotics of orthogonal polynomials with respect to exponential weights’, Comm. Pure Appl. Math. 52 (1999) 1491–1552.3.0.CO;2-#>CrossRefGoogle Scholar
3. Deift, P., Kriecherbauer, T. and McLaughlin, K., ‘New results for the asymptotics of orthogonal polynomials and related problems via the Lax-Levermore method’, Proc. Symp. Appl. Math. 54 (1998).Google Scholar
4. Deift, P., Kriecherbauer, T., McLaughlin, K., Venakides, S. and Zhou, X., ‘Asymptotics for polynomials orthogonal with respect to varying exponential weights’, Internat. Math. Res. Notices (1997) 759–782.CrossRefGoogle Scholar
5. Foster, W. H. and Krasikov, I., ‘Bounds for the extreme roots of orthogonal polynomials’, Internat. J. Math. Algorithms, to appear.Google Scholar
6. Krasikov, I., ‘Nonnegative quadratic forms and bounds on orthogonal polynomials, Journal Approx. Theory, to appear.Google Scholar
7. Krasikov, I. and Litsyn, S., ‘On the distance distribution of duals of BCH codes’, IEEE Trans. Inform. Theory 45 (1999) 247–250.CrossRefGoogle Scholar
8. Levin, A. L. and Lubinsky, D. S., ‘Bounds for orthogonal polynomials for exponential weights on (−1, 1) and (−∞, ∞)’, Rend. Circ. Mat. Palermo (2) Suppl. 33 (1993) 75–96.Google Scholar
9. Levin, A. L. and Lubinsky, D. S., ‘Asymptotics associated with exponential weights’, Internat. Math. Res. Notices 12 (1999) 673–683.CrossRefGoogle Scholar
10. Levin, A. L. and Lubinsky, D. S., ‘Bounds for orthogonal polynomials for exponential weights’, J. Comput. Appl. Math. 99 (1998) 475–490.CrossRefGoogle Scholar
11. Szegö, G., Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ. 23 (AMS, Providence, RI, 1975).Google Scholar