Riemann-Hilbert Theory without local Parametrix Problems: Applications to Orthogonal Polynomials
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2019
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| _version_ | 1866916084539457536 |
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| author | Piorkowski, Mateusz |
| author_facet | Piorkowski, Mateusz |
| contents | We study whether in the setting of the Deift-Zhou nonlinear steepest descent method one can avoid solving local parametrix problems explicitly, while still obtaining asymptotic results. We show that this can be done, provided an a priori estimate for the exact solution of the Riemann-Hilbert problem is known. This enables us to derive asymptotic results for orthogonal polynomials on $[-1,1]$ with a new class of weight functions. In these cases, the weight functions are too badly behaved to allow a reformulation of a local parametrix problem to a global one with constant jump matrices. Possible implications for edge universality in random matrix theory are also discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1910_00564 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Riemann-Hilbert Theory without local Parametrix Problems: Applications to Orthogonal Polynomials Piorkowski, Mateusz Complex Variables Functional Analysis Primary 42C05, 60B20 Secondary 35Q15, 45E05 We study whether in the setting of the Deift-Zhou nonlinear steepest descent method one can avoid solving local parametrix problems explicitly, while still obtaining asymptotic results. We show that this can be done, provided an a priori estimate for the exact solution of the Riemann-Hilbert problem is known. This enables us to derive asymptotic results for orthogonal polynomials on $[-1,1]$ with a new class of weight functions. In these cases, the weight functions are too badly behaved to allow a reformulation of a local parametrix problem to a global one with constant jump matrices. Possible implications for edge universality in random matrix theory are also discussed. |
| title | Riemann-Hilbert Theory without local Parametrix Problems: Applications to Orthogonal Polynomials |
| topic | Complex Variables Functional Analysis Primary 42C05, 60B20 Secondary 35Q15, 45E05 |
| url | https://arxiv.org/abs/1910.00564 |