Abstract
We prove that every meromorphic function on the closure of an analytic Jordan domain that is sufficiently well-behaved on the boundary is conformally equivalent to a rational map whose degree is smallest possible. We also show that the minimality of the degree fails in general without the boundary assumptions. As an application, we generalize a theorem of Ebenfelt, Khavinson, and Shapiro by characterizing fingerprints of polynomial pseudo-lemniscates.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ebenfelt, P., Khavinson, D., Shapiro, H.S.: Two-dimensional shapes and lemniscates. In: Complex Analysis and Dynamical Systems IV. Part 1, vol. 553, pp. 45–59 (2011)
Fortier Bourque, M., Younsi, M.: Rational Ahlfors functions. Constr. Approx. 41, 157–183 (2015)
Hamilton, D.H.: Conformal welding. Handb. Complex Anal. Geom. Funct. Theory 1, 137–146 (2002)
Jeong, M., Taniguchi, M.: Bell representations of finitely connected planar domains. Proc. Am. Math. Soc. 131, 2325–2328 (2003)
Lehto, O., Virtanen, K.I.: Quasiconformal Mappings in the Plane. Springer, New York (1973)
Marshall, D.E.: Conformal welding for finitely connected regions. Comput. Methods Funct. Theory 11, 655–669 (2011)
Pfluger, A.: Ueber die Konstruktion Riemannscher Flächen durch Verheftung. J. Indian Math. Soc. 24, 401–412 (1961)
Richards, T.: Conjecture: Every analytic function on the closed disk is conformally a polynomial, Math.stackexchange question. http://math.stackexchange.com/questions/437598 (2013)
Richards, T.: Level curve configurations and conformal equivalence of meromorphic functions. Comput. Methods Funct. Theory 15, 323–371 (2015)
Sharon, E., Mumford, D.: 2d-shape analysis using conformal mapping. In: 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 2, pp. 350–357 (2004)
Younsi, M.: Shapes, fingerprints and rational lemniscates. Proc. Am. Math. Soc. 144, 1087–1093 (2016)
Acknowledgments
The authors thank the anonymous referees for helpful suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Edward B. Saff.
Malik Younsi supported by NSERC.
Rights and permissions
About this article
Cite this article
Richards, T., Younsi, M. Conformal Models and Fingerprints of Pseudo-lemniscates. Constr Approx 45, 129–141 (2017). https://doi.org/10.1007/s00365-016-9348-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00365-016-9348-0