Strong asymptotics of extremal polynomials on the segment in the presence of denumerable set of mass points

Rabah Khaldi; Ahcene Boucenna

doi:10.33993/jnaat401-949

Authors

Rabah Khaldi University of Annaba, Algeria
Ahcene Boucenna University of Annaba, Algeria

DOI:

https://doi.org/10.33993/jnaat401-949

Keywords:

orthogonal polynomials, asymptotic behavior

Abstract views: 241

Abstract

The strong asymptotics of the monic extremal polynomials with respect to a \(L_{p}(\sigma )\) norm are studied. The measure \(\sigma \) is concentrated on the segment \([-1,1]\) plus a denumerable set of mass points which accumulate at the boundary points of \([-1,1]\) only. Under the assumptions that the mass points satisfy Blaschke's condition and that the absolutely continuous part of \(\sigma \) satisfies Szeg?'s condition.

Downloads

Download data is not yet available.

References

Geronimus, L., On some extremal problem in Lσ(p) spaces, Math. Sbornik, 31, pp. 3-23, 1952 (in Russian).

Kaliaguine, V.A., On Asymptotics of L_{p} extremal polynomials on a complex curve (0

https://doi.org/10.1006/jath.1993.1063

DOI: https://doi.org/10.1006/jath.1993.1063

Kaliaguine, V.A., A note on the asymptotics of orthogonal polynomials on a complex arc: the case of a measure with a discrete part, J. Approx. Theory, 80, pp. 138-145 1995. https://doi.org/10.1006/jath.1995.1009 DOI: https://doi.org/10.1006/jath.1995.1009

Khaldi, R., Strong asymptotics for Lp extremal polynomials off a complex curve, J. Appl. Math., 2004, no 5, pp. 371-378, 2004. https://doi.org/10.1155/s1110757x0430906x DOI: https://doi.org/10.1155/S1110757X0430906X

Khaldi, R., Szegő asymptotics of extremal polynomials on the segment [-1,+1]: the case of a measure with finite discrete part, Georgian Math. J., 14, no. 4, pp. 673-680, 2007. DOI: https://doi.org/10.1515/GMJ.2007.673

Li, X. and Pan, K., Asymptotics of Lp extremal polynomials on the unit circle, J. Approx. Theory, 67, pp. 270-283, 1991.https://doi.org/10.1016/0021-9045(91)90003-s DOI: https://doi.org/10.1016/0021-9045(91)90003-S

Lubinsky, D.S. and Saff, E.B., Strong asymptotics for Lp extremal polynomials (1

https://doi.org/10.1007/bfb0078899

DOI: https://doi.org/10.1007/BFb0078899

Peherstorfer, F. and Yudiskii, P., Asymptotics of orthogonal polynomials in the presence of a denumerable set of mass points, Proc. Amer. Math. Soc., 129, pp. 3213-3220, 2001. https://doi.org/10.1090/s0002-9939-01-06205-0 DOI: https://doi.org/10.1090/S0002-9939-01-06205-0

Rudin, W., Real and Complex Analysis, Mc Graw-Hill, New York, 1968.

Szegö, G., Orthogonal plynomials, 4th ed. Amer. Math. Soc. Colloquium Publ., 23, Amer. Math. Soc. Providence, RI, 1975.

Widom, H., Extremal polynomials associated with a system of curves and arcs in the complex plane, Adv. Math., 3, pp. 127-232, 1969. https://doi.org/10.1016/0001-8708(69)90005-x DOI: https://doi.org/10.1016/0001-8708(69)90005-X