Solving Riemann-Hilbert problems with meromorphic functions

Dan Kucerovsky; Aydin Sarraf

doi:10.2478/ausm-2019-0010

Abstract

In this paper, we introduce the use of a powerful tool from theoretical complex analysis, the Blaschke product, for the solution of Riemann-Hilbert problems. Classically, Riemann-Hilbert problems are considered for analytic functions. We give a factorization theorem for meromorphic functions over simply connected nonempty proper open subsets of the complex plane and use this theorem to solve Riemann-Hilbert problems where the given data consists of a meromorphic function.

References

[1] Mark J. Ablowitz and Peter A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, volume 149. Cambridge Univ. Press, 1991.10.1017/CBO9780511623998
Search in Google Scholar Back to article
[2] Norair Unanovich Arakelian, Uniform approximation on closed sets by entire functions, Izv. Ross. Akad. Nauk Ser. Mat., 28 (5) (1964), 1187–1206.
Search in Google Scholar Back to article
[3] Joseph Bak and Donald J. Newman, Complex analysis, undergraduate texts in mathematics, 1997.
Search in Google Scholar Back to article
[4] Richard Beals, Advanced mathematical analysis: periodic functions and distributions, complex analysis, Laplace transform and applications, volume 12. Springer Science & Business Media, 2013.
Search in Google Scholar Back to article
[5] Ilia Binder, Mark Braverman, and Michael Yampolsky, On the computational complexity of the Riemann mapping, Ark. Mat., 45 (2) (2007), 221–239.10.1007/s11512-007-0045-x
Search in Google Scholar Back to article
[6] Ilia Binder, Cristobal Rojas, and Michael Yampolsky, Computable caratheodory theory, Adv. Math., 265 (2014), 280–312.10.1016/j.aim.2014.07.039
Search in Google Scholar Back to article
[7] Christopher J. Bishop, A fast approximation to the Riemann map, preprint, 420, 2003.
Search in Google Scholar Back to article
[8] Christopher J. Bishop, Conformal mapping in linear time, Discrete Comput. Geom., 44 (2) (2010), 330–42810.1007/s00454-010-9269-9
Search in Google Scholar Back to article
[9] Ronald R. Coifman and Stefan Steinerberger, Nonlinear phase unwinding of functions, J. Fourier Anal. Appl., 23 (4) (2017), 778–809.10.1007/s00041-016-9489-3
Search in Google Scholar Back to article
[10] Fedor Dmitrievich Gakhov, Boundary value problems, volume 85, Pergamon Press, 1966.10.1016/B978-0-08-010067-8.50007-4
Search in Google Scholar Back to article
[11] Stephan Ramon Garcia, Javad Mashreghi, and William T Ross, Finite Blaschke products: a survey, arXiv preprint arXiv:1512.05444, 2015.
Search in Google Scholar Back to article
[12] Paul M. Gauthier, Lectures on several complex variables, Springer, 2014.10.1007/978-3-319-11511-5
Search in Google Scholar Back to article
[13] Peter Hertling, An effective Riemann mapping theorem, Theoret. Comput. Sci., 219 (1-2) (1999), 225–265.10.1016/S0304-3975(98)00290-4
Search in Google Scholar Back to article
[14] Kenneth Ho man, Banach spaces of analytic functions, Courier Corporation, 2007.
Search in Google Scholar Back to article
[15] Daniel Huybrechts, Complex geometry: an introduction, Springer Science & Business Media, 2006.
Search in Google Scholar Back to article
[16] Dan Kučerovský, Amir TP Najafabadi, and Aydin Sarraf, On the riemann-hilbert factorization problem for positive definite functions, Positivity, 20 (3) (2016), 743–754.10.1007/s11117-015-0384-y
Search in Google Scholar Back to article
[17] Serge Lang, Complex analysis, volume 103, Springer Science & Business Media, 2013.
Search in Google Scholar Back to article
[18] Javad Mashreghi and Emmanuel Fricain, Blaschke products and their applications, Springer, 2013.10.1007/978-1-4614-5341-3
Search in Google Scholar Back to article
[19] S. N. Mergelian, On the representation of functions by series of polynomials on closed sets, Number 85. Amer. Math. Soc., 1953.
Search in Google Scholar Back to article
[20] Carle Runge, Zur Theorie der eindeutigen analytischen Functionen, Acta Math., 6 (1) (1885), 229–244.10.1007/BF02400416
Search in Google Scholar Back to article
[21] Joel L. Schi, The Laplace transform: theory and applications, Springer Science & Business Media, 2013.
Search in Google Scholar Back to article
[22] Elias M. Stein and Rami Shakarchi, Complex analysis, Princeton lectures in analysis, ii, 2003.
Search in Google Scholar Back to article
[23] Anatoliy Georgievich Vitushkin, Uniform approximations by holomorphic functions, J. Funct. Anal., 20 (2) (1975), 149–157.10.1016/0022-1236(75)90047-6
Search in Google Scholar Back to article
[24] J. L. Walsh, Note on the location of zeros of extremal polynomials in the non-euclidean plane, Acad. Serbe Sci. Publ. Inst. Math, 4 (1952), 157–160.
Search in Google Scholar Back to article
[25] Carl Weierstrass, Zur Theorie der eindeutigen analytischen Functionen, na, 1877.
Search in Google Scholar Back to article

Solving Riemann-Hilbert problems with meromorphic functions

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