Riemann-Hilbert analysis for Jacobi polynomials orthogonal on a single contour
Classical Jacobi polynomials $P_{n}^{(\alpha,\beta)}$, with $\alpha, \beta>-1$, have a number of well-known properties, in particular the location of their zeros in the open interval $(-1,1)$. This property is no longer valid for other values of the parameters; in general, zeros are complex. In t...
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| Format: | info:eu-repo/semantics/article |
| Language: | English |
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2012
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| Online Access: | http://hdl.handle.net/10835/1641 |
| _version_ | 1789406363651145728 |
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| author | Martínez-Finkelshtein, Andrei Orive, R. |
| author_facet | Martínez-Finkelshtein, Andrei Orive, R. |
| author_sort | Martínez-Finkelshtein, Andrei |
| collection | DSpace |
| description | Classical Jacobi polynomials $P_{n}^{(\alpha,\beta)}$, with $\alpha, \beta>-1$, have a number of well-known properties, in particular the location of their zeros in the open interval $(-1,1)$. This property is no longer valid for other values of the parameters; in general, zeros are complex. In this paper we study the strong asymptotics of Jacobi polynomials where the real parameters $\alpha_n,\beta_n$ depend on $n$ in such a way that $$ \lim_{n\to\infty}\frac{\alpha_{n}}{n}=A, \quad \lim_{n\to\infty}\frac{\beta_{n}}{n}=B, $$ with $A,B \in \mathbb{R}$. We restrict our attention to the case where the limits $A,B$ are not both positive and take values outside of the triangle bounded by the straight lines A=0, B=0 and $A+B+2=0$. As a corollary, we show that in the limit the zeros distribute along certain curves that constitute trajectories of a quadratic differential. The non-hermitian orthogonality relations for Jacobi polynomials with varying parameters lie in the core of our approach; in the cases we consider, these relations hold on a single contour of the complex plane. The asymptotic analysis is performed using the Deift-Zhou steepest descent method based on the Riemann-Hilbert reformulation of Jacobi polynomials. |
| format | info:eu-repo/semantics/article |
| id | oai:repositorio.ual.es:10835-1641 |
| institution | Universidad de Cuenca |
| language | English |
| publishDate | 2012 |
| record_format | dspace |
| spelling | oai:repositorio.ual.es:10835-16412023-04-12T19:36:51Z Riemann-Hilbert analysis for Jacobi polynomials orthogonal on a single contour Martínez-Finkelshtein, Andrei Orive, R. Análisis asintótico Ortogonalidad no hermitiana Caracterización de Riemann-Hilbert Asymptotic analysis Non-hermitian orthogonality Steepest descent method Riemann–Hilbert characterization Classical Jacobi polynomials $P_{n}^{(\alpha,\beta)}$, with $\alpha, \beta>-1$, have a number of well-known properties, in particular the location of their zeros in the open interval $(-1,1)$. This property is no longer valid for other values of the parameters; in general, zeros are complex. In this paper we study the strong asymptotics of Jacobi polynomials where the real parameters $\alpha_n,\beta_n$ depend on $n$ in such a way that $$ \lim_{n\to\infty}\frac{\alpha_{n}}{n}=A, \quad \lim_{n\to\infty}\frac{\beta_{n}}{n}=B, $$ with $A,B \in \mathbb{R}$. We restrict our attention to the case where the limits $A,B$ are not both positive and take values outside of the triangle bounded by the straight lines A=0, B=0 and $A+B+2=0$. As a corollary, we show that in the limit the zeros distribute along certain curves that constitute trajectories of a quadratic differential. The non-hermitian orthogonality relations for Jacobi polynomials with varying parameters lie in the core of our approach; in the cases we consider, these relations hold on a single contour of the complex plane. The asymptotic analysis is performed using the Deift-Zhou steepest descent method based on the Riemann-Hilbert reformulation of Jacobi polynomials. 2012-08-03T10:08:32Z 2012-08-03T10:08:32Z 2005 info:eu-repo/semantics/article http://hdl.handle.net/10835/1641 en info:eu-repo/semantics/openAccess Journal of approximation theory Vol. 134, nº 2 (2005) |
| spellingShingle | Análisis asintótico Ortogonalidad no hermitiana Caracterización de Riemann-Hilbert Asymptotic analysis Non-hermitian orthogonality Steepest descent method Riemann–Hilbert characterization Martínez-Finkelshtein, Andrei Orive, R. Riemann-Hilbert analysis for Jacobi polynomials orthogonal on a single contour |
| title | Riemann-Hilbert analysis for Jacobi polynomials orthogonal on a single contour |
| title_full | Riemann-Hilbert analysis for Jacobi polynomials orthogonal on a single contour |
| title_fullStr | Riemann-Hilbert analysis for Jacobi polynomials orthogonal on a single contour |
| title_full_unstemmed | Riemann-Hilbert analysis for Jacobi polynomials orthogonal on a single contour |
| title_short | Riemann-Hilbert analysis for Jacobi polynomials orthogonal on a single contour |
| title_sort | riemann-hilbert analysis for jacobi polynomials orthogonal on a single contour |
| topic | Análisis asintótico Ortogonalidad no hermitiana Caracterización de Riemann-Hilbert Asymptotic analysis Non-hermitian orthogonality Steepest descent method Riemann–Hilbert characterization |
| url | http://hdl.handle.net/10835/1641 |
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