On the Asymptotics of Orthogonal Polynomials on Multiple Intervals with Non-Analytic Weights
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910003826262016 |
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| author | Trogdon, Thomas |
| author_facet | Trogdon, Thomas |
| contents | We consider the asymptotics of orthogonal polynomials for measures that are differentiable, but not necessarily analytic, multiplicative perturbations of Jacobi-like measures supported on disjoint intervals. We analyze the Fokas-Its-Kitaev Riemann-Hilbert problem using the Deift-Zhou method of nonlinear steepest descent and its $\overline{\partial}$ extension due to Miller and McLaughlin. Our results extend that of Yattselev in the case of Chebyshev-like measures with error bounds that give similar rates while allowing less regular perturbations. For the general Jacobi-like case, we present, what appears to be the first result for asymptotics when the perturbation of the measure is only assumed to be differentiable with bounded second derivative. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_18656 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Asymptotics of Orthogonal Polynomials on Multiple Intervals with Non-Analytic Weights Trogdon, Thomas Classical Analysis and ODEs Complex Variables 42C05, 33C47 We consider the asymptotics of orthogonal polynomials for measures that are differentiable, but not necessarily analytic, multiplicative perturbations of Jacobi-like measures supported on disjoint intervals. We analyze the Fokas-Its-Kitaev Riemann-Hilbert problem using the Deift-Zhou method of nonlinear steepest descent and its $\overline{\partial}$ extension due to Miller and McLaughlin. Our results extend that of Yattselev in the case of Chebyshev-like measures with error bounds that give similar rates while allowing less regular perturbations. For the general Jacobi-like case, we present, what appears to be the first result for asymptotics when the perturbation of the measure is only assumed to be differentiable with bounded second derivative. |
| title | On the Asymptotics of Orthogonal Polynomials on Multiple Intervals with Non-Analytic Weights |
| topic | Classical Analysis and ODEs Complex Variables 42C05, 33C47 |
| url | https://arxiv.org/abs/2412.18656 |