Strong Asymptotics for a 3x3 Riemann-Hilbert Problem in a Regular Hard-Soft Two-Edge Regime
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915780018307072 |
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| author | Kandaian, Artur |
| author_facet | Kandaian, Artur |
| contents | We develop a complete Deift-Zhou steepest descent analysis for a 3x3 matrix Riemann-Hilbert problem arising in quadratic Hermite-Pade approximation and multiple orthogonality. We focus on a regular two-edge regime with a hard edge at 0 and a soft edge at x0. Under natural geometric and analytic assumptions ensuring a nondegenerate sign structure of the associated phase functions, the standard lens-opening mechanism applies. The analysis is organized as a reusable scheme: once the equilibrium/sign-chart input is verified (assumptions R1-R7), the remaining steps are purely analytic. As a result, the solution is described in terms of a reduced outer parametrix with permutation-type jumps, complemented by Bessel- and Airy-type local parametrices at the endpoints. We obtain uniform strong asymptotics for the top-left entry, with an explicit error bound of order 1/n outside the endpoint neighborhoods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06690 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Strong Asymptotics for a 3x3 Riemann-Hilbert Problem in a Regular Hard-Soft Two-Edge Regime Kandaian, Artur Classical Analysis and ODEs 30E10, 42C05, 41A21, 41A60, 33C45 We develop a complete Deift-Zhou steepest descent analysis for a 3x3 matrix Riemann-Hilbert problem arising in quadratic Hermite-Pade approximation and multiple orthogonality. We focus on a regular two-edge regime with a hard edge at 0 and a soft edge at x0. Under natural geometric and analytic assumptions ensuring a nondegenerate sign structure of the associated phase functions, the standard lens-opening mechanism applies. The analysis is organized as a reusable scheme: once the equilibrium/sign-chart input is verified (assumptions R1-R7), the remaining steps are purely analytic. As a result, the solution is described in terms of a reduced outer parametrix with permutation-type jumps, complemented by Bessel- and Airy-type local parametrices at the endpoints. We obtain uniform strong asymptotics for the top-left entry, with an explicit error bound of order 1/n outside the endpoint neighborhoods. |
| title | Strong Asymptotics for a 3x3 Riemann-Hilbert Problem in a Regular Hard-Soft Two-Edge Regime |
| topic | Classical Analysis and ODEs 30E10, 42C05, 41A21, 41A60, 33C45 |
| url | https://arxiv.org/abs/2602.06690 |