Asymptotics for Jacobi-Sobolev orthogonal polynomials associated with non-coherent pairs of measures

  • Authors:

  • Eliana X. L. de Andrade;Cleonice F. Bracciali;Laura Castaño-García;Juan J. Moreno-Balcázar

  • Affiliations:

  • DCCE, IBILCE, UNESP-Univ Estadual Paulista, Rua Cristóvão Colombo, 2265, 15054-000 São José do Rio Preto, SP, Brazil;DCCE, IBILCE, UNESP-Univ Estadual Paulista, Rua Cristóvão Colombo, 2265, 15054-000 São José do Rio Preto, SP, Brazil;Dpto de Estadística y Matemática Aplicada, Universidad de Almería, La Cañada de San Urbano s/n, 04120, Almería, Spain;Dpto de Estadística y Matemática Aplicada, Universidad de Almería, La Cañada de San Urbano s/n, 04120, Almería, Spain and Instituto Carlos I de Física Teórica y ...

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2010

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Abstract

We consider the Sobolev inner product =@!"-"1^1f(x)g(x)d@j^(^@a^,^@b^)(x)+@!f^'(x)g^'(x)d@j(x), where d@j^(^@a^,^@b^)(x)=(1-x)^@a(1+x)^@bdx with @a,@b-1, and @j is a measure involving a rational modification of a Jacobi weight and with a mass point outside the interval (-1,1). We study the asymptotic behaviour of the polynomials which are orthogonal with respect to this inner product on different regions of the complex plane. In fact, we obtain the outer and inner strong asymptotics for these polynomials as well as the Mehler-Heine asymptotics which allow us to obtain the asymptotics of the largest zeros of these polynomials. We also show that in a certain sense the above inner product is also equilibrated.