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Orthogonal polynomials with respect to a family of Sobolev inner products on the unit circle


Author: A. Sri Ranga
Journal: Proc. Amer. Math. Soc. 144 (2016), 1129-1143
MSC (2010): Primary 42C05, 33C47; Secondary 33C45
DOI: https://doi.org/10.1090/proc12766
Published electronically: July 1, 2015
MathSciNet review: 3447666
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Abstract: The principal objective here is to look at some algebraic properties of the orthogonal polynomials $ \Psi _n^{(b,s,t)}$ with respect to the Sobolev inner product on the unit circle

$\displaystyle \langle f,g\rangle _{S^{(b,s,t)}} = (1-t)\, \langle f,g\rangle _{... ...ine {f(1)}\,g(1) + s\, \langle f^{\prime },g^{\prime }\rangle _{\mu ^{(b+1)}}, $

where $ \langle f,g\rangle _{\mu ^{(b)}} = \frac {\tau (b)}{2\pi } \int _{0}^{2\pi }\o... ...heta })^{\mathcal {I}m(b)} (\sin ^{2}(\theta /2))^{\mathcal {R}e(b)} d\theta . $ Here, $ \mathcal {R}e(b) > -1/2$, $ 0 \leq t < 1$, $ s > 0$ and $ \tau (b)$ is taken to be such that $ \langle 1,1\rangle _{\mu ^{(b)}} = 1$. We show that, for example, the monic Sobolev orthogonal polynomials $ \Psi _n^{(b,s,t)}$ satisfy the recurrence $ \Psi _n^{(b,s,t)}(z) - \beta _n^{(b,s,t)} \Psi _{n-1}^{(b,s,t)}(z) = \Phi _n^{(b,t)}(z), $ $ n \geq 1$, where $ \Phi _n^{(b,t)}$ are the monic orthogonal polynomials with respect to the inner product $ \langle f,g\rangle _{\mu ^{(b,t)}} = (1-t)\, \langle f,g\rangle _{\mu ^{(b)}} + t\, \overline {f(1)}\,g(1)$. Some related bounds and asymptotic properties are also given.

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Additional Information

A. Sri Ranga
Affiliation: Departamento de Matemática Aplicada, IBILCE, UNESP - Universidade Estadual Paulista, 15054-000, São José do Rio Preto, SP, Brazil
Email: ranga@ibilce.unesp.br

DOI: https://doi.org/10.1090/proc12766
Keywords: Orthogonal polynomials on the unit circle, Sobolev orthogonal polynomials on the unit circle, para-orthogonal polynomials, positive chain sequences
Received by editor(s): October 31, 2014
Received by editor(s) in revised form: November 1, 2014, January 28, 2015, and February 13, 2015
Published electronically: July 1, 2015
Additional Notes: This work received support from the funding bodies CNPq (grant No. 475502/2013-2) and FAPESP (Grant No. 2009/13832-9) of Brazil
Communicated by: Walter Van Assche
Article copyright: © Copyright 2015 American Mathematical Society