A Riemann--Hilbert approach to computing the inverse spectral map for measures supported on disjoint intervals
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929211404451840 |
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| author | Ballew, Cade Trogdon, Thomas |
| author_facet | Ballew, Cade Trogdon, Thomas |
| contents | We develop a numerical method for computing with orthogonal polynomials that are orthogonal on multiple, disjoint intervals for which analytical formulae are currently unknown. Our approach exploits the Fokas--Its--Kitaev Riemann--Hilbert representation of the orthogonal polynomials to produce an $\mathrm{O}(N)$ method to compute the first $N$ recurrence coefficients. The method can also be used for pointwise evaluation of the polynomials and their Cauchy transforms throughout the complex plane. The method encodes the singularity behavior of weight functions using weighted Cauchy integrals of Chebyshev polynomials. This greatly improves the efficiency of the method, outperforming other available techniques. We demonstrate the fast convergence of our method and present applications to integrable systems and approximation theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_12930 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Riemann--Hilbert approach to computing the inverse spectral map for measures supported on disjoint intervals Ballew, Cade Trogdon, Thomas Numerical Analysis Complex Variables 42C05, 65E05, 33C47 We develop a numerical method for computing with orthogonal polynomials that are orthogonal on multiple, disjoint intervals for which analytical formulae are currently unknown. Our approach exploits the Fokas--Its--Kitaev Riemann--Hilbert representation of the orthogonal polynomials to produce an $\mathrm{O}(N)$ method to compute the first $N$ recurrence coefficients. The method can also be used for pointwise evaluation of the polynomials and their Cauchy transforms throughout the complex plane. The method encodes the singularity behavior of weight functions using weighted Cauchy integrals of Chebyshev polynomials. This greatly improves the efficiency of the method, outperforming other available techniques. We demonstrate the fast convergence of our method and present applications to integrable systems and approximation theory. |
| title | A Riemann--Hilbert approach to computing the inverse spectral map for measures supported on disjoint intervals |
| topic | Numerical Analysis Complex Variables 42C05, 65E05, 33C47 |
| url | https://arxiv.org/abs/2302.12930 |