Orthogonal polynomials with respect to a family of Sobolev inner products on the unit circle
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- by A. Sri Ranga
- Proc. Amer. Math. Soc. 144 (2016), 1129-1143
- DOI: https://doi.org/10.1090/proc12766
- Published electronically: July 1, 2015
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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 \[ \langle f,g\rangle _{S^{(b,s,t)}} = (1-t) \langle f,g\rangle _{\mu ^{(b)}} + t \overline {f(1)} g(1) + s \langle f^{\prime },g^{\prime }\rangle _{\mu ^{(b+1)}}, \] where $\langle f,g\rangle _{\mu ^{(b)}} = \frac {\tau (b)}{2\pi } \int _{0}^{2\pi }\overline {f(e^{i\theta })} g(e^{i\theta }) (e^{\pi -\theta })^{\mathcal {I}m(b)} (\sin ^{2}(\theta /2))^{\mathcal {R}e(b)} d\theta .$ 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.References
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Bibliographic Information
- A. Sri Ranga
- Affiliation: Departamento de Matemática Aplicada, IBILCE, UNESP - Universidade Estadual Paulista, 15054-000, São José do Rio Preto, SP, Brazil
- MR Author ID: 238837
- Email: ranga@ibilce.unesp.br
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 1129-1143
- MSC (2010): Primary 42C05, 33C47; Secondary 33C45
- DOI: https://doi.org/10.1090/proc12766
- MathSciNet review: 3447666