Orthonormal polynomials with generalized Freud-type weights

  • Authors:

  • T. Kasuga;R. Sakai

  • Affiliations:

  • Department of Mathematics, Kumamoto National College of Technology, Nishigousi-machi, Kikuchi-gun, Kumamoto 861-1102, Japan;Department of Mathematics, Asuke Senior High School, Kawahara 5, Yagami, Asuke-cho, Higashikamogun, Aichi 444-2451, Japan

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2003

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Abstract

We consider a certain generalized Freud-type weight WrQ2(x) = |x|2r exp(-2Q(x)), where r -½ and Q: R → R is even and continuous, Q' is continuous, Q' 0 in (0, ∞), and Q'' is continuous in (0, ∞). Furthermore, Q satisfies further conditions. Recently, Levin and Lubinsky have studied the sequence of orthonormal polynomials {Pn(WQ2;x)}n=0∞ with the Freud weight WQ2(x) = exp(-2Q(x)) on R, and then they have obtained many interesting properties of Pn(WQ2;x) [LL1]. We investigate the properties of Pn(WrQ2;x), which contain extensions of Levin and Lubinsky's results and improvements of Bauldry's results [Ba1,LL1].