Abstract
In the application of the Deift-Zhou steepest descent method to the Riemann-Hilbert problem for orthogonal polynomials, a model Riemann-Hilbert problem that appears in the multi-cut case is solved with the use of hyperelliptic theta functions. We present here an alternative approach which uses meromorphic differentials instead of theta functions to construct the solution of the model Riemann-Hilbert problem. By using this representation, we obtain a new and elementary proof for the solvability of the model Riemann-Hilbert problem.
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The first author is supported by FWO-Flanders project G.0427.09, by K.U. Leuven research grant OT/08/33, by the Belgian Interuniversity Attraction Pole P06/02, by the European Science Foundation Program MISGAM, and by grant MTM2008-06689-C02-01 of the Spanish Ministry of Science and Innovation. The second author is supported by the EPSRC grant EP/G019843/1
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Kuijlaars, A.B.J., Mo, M.Y. The Global Parametrix in the Riemann-Hilbert Steepest Descent Analysis for Orthogonal Polynomials. Comput. Methods Funct. Theory 11, 161–178 (2011). https://doi.org/10.1007/BF03321795
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DOI: https://doi.org/10.1007/BF03321795