Skip to main content
Log in

Approximation by Proper Holomorphic Maps and Tropical Power Series

  • Published:
Constructive Approximation Aims and scope

Abstract

Let \(w\) be an unbounded radial weight on the complex plane. We study the following approximation problem: find a proper holomorphic map \(f: \mathbb {C}\rightarrow \mathbb {C}^n\) such that |f| is equivalent to \(w\). We give several characterizations of those \(w\) for which the problem is solvable. In particular, a constructive characterization is given in terms of tropical power series. Moreover, the following natural objects and properties are involved: essential weights on the complex plane, approximation by power series with positive coefficients, and approximation by the maximum of a holomorphic function modulus. Extensions to several complex variables and approximation by harmonic maps are also considered.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abakumov, E., Doubtsov, E.: Reverse estimates in growth spaces. Math. Z. 271(1–2), 399–413 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abakumov, E., Doubtsov, E.: Moduli of holomorphic functions and logarithmically convex radial weights. Bull. Lond. Math. Soc. 47(3), 519–532 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abakumov, E., Doubtsov, E.: Growth of proper holomorphic maps and tropical power series. C. R. Math. Acad. Sci. Paris 354(5), 465–469 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Abakumov, E., Doubtsov, E.: Volterra type operators on growth Fock spaces. Arch. Math. 108(4), 383–393 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Aleksandrov, A.B.: Proper holomorphic mappings from the ball to the polydisk. Dokl. Akad. Nauk SSSR 286(1), 11–15 (Russian); English transl.: Soviet Math. Dokl. 33(1), 1–5 (1986)

  6. Alexander, H.: Explicit imbedding of the (punctured) disc into \({\mathbf{C}}^{2}\). Comment. Math. Helv. 52(4), 539–544 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bierstedt, K.D., Bonet, J., Taskinen, J.: Associated weights and spaces of holomorphic functions. Stud. Math. 127(2), 137–168 (1998)

    MathSciNet  MATH  Google Scholar 

  8. Bonet, J., Domański, P., Lindström, M.: Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions. Can. Math. Bull. 42(2), 139–148 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Borichev, A.: The polynomial approximation property in Fock-type spaces. Math. Scand. 82(2), 256–264 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Domar, Y.: Closed primary ideals in a class of Banach algebras. Math. Scand. 7, 109–125 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  11. Doubtsov, E.: Harmonic approximation by finite sums of moduli. J. Math. Anal. Appl. 430(2), 685–694 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Eliashberg, Y., Gromov, M.: Embeddings of Stein manifolds of dimension \(n\) into the affine space of dimension \(3n/2+1\). Ann. Math. (2) 136(1), 123–135 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Erdös, P., Kövári, T.: On the maximum modulus of entire functions. Acta Math. Acad. Sci. Hungar. 7, 305–317 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  14. Forstnerič, F., Wold, E.F.: Embeddings of infinitely connected planar domains into \(\mathbb{C}^2\). Anal. PDE 6(2), 499–514 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Globevnik, J.: On growth of holomorphic embeddings into \(\mathbb{C}^2\). Proc. R. Soc. Edinb. Sect. A 132(4), 879–889 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Globevnik, J., Stensønes, B.: Holomorphic embeddings of planar domains into \(\mathbf{C}^{2}\). Math. Ann. 303(4), 579–597 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Itenberg, I., Mikhalkin, G., Shustin, E.: Tropical Algebraic Geometry, Oberwolfach Seminars, vol. 35, 2nd edn. Birkhäuser, Basel (2009)

    Book  MATH  Google Scholar 

  18. Kasahara, K., Nishono, T.: As announced in Math Reviews 38, #4721 (1969)

  19. Kiselman, C.O.: Croissance des fonctions plurisousharmoniques en dimension infinie. Ann. Inst. Fourier (Grenoble) 34(1), 155–183 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kiselman, C.O.: Questions inspired by Mikael Passare’s mathematics. Afr. Mat. 25(2), 271–288 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Koosis, P.: Sur l’approximation pondérée par des polynomes et par des sommes d’exponentielles imaginaires. Ann. Sci. École Norm. Sup. (3) 81, 387–408 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  22. Laufer, H.B.: Imbedding annuli in \(\mathbf{C}^{2}\). J. Anal. Math. 26, 187–215 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lubinsky, D.S.: Strong Asymptotics for Extremal Errors and Polynomials Associated with Erdős-Type Weights, Pitman Research Notes in Mathematics Series, vol. 202. Longman Scientific & Technical, Harlow (1989)

    Google Scholar 

  24. Mergelyan, S.N.: Weighted approximations by polynomials. In: American Mathematical Society Translations, Ser. 2, vol. 10, pp. 59–106. American Mathematical Society, Providence (1958)

  25. Ryll, J., Wojtaszczyk, P.: On homogeneous polynomials on a complex ball. Trans. Am. Math. Soc. 276(1), 107–116 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  26. Schmid, U.: On the approximation of positive functions by power series. J. Approx. Theory 83(3), 342–346 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  27. Schmid, U.: On the approximation of positive functions by power series. II. J. Approx. Theory 92(3), 486–501 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Stehlé, J.L.: Plongements du disque dans \(C^{2}\). In: Séminaire Pierre Lelong (Analyse), Année 1970–1971. Lecture Notes in Math., vol. 275, pp. 119–130. Springer, Berlin (1972)

Download references

Acknowledgements

The authors are grateful to José Bonet and Jari Taskinen for useful discussions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Evgueni Doubtsov.

Additional information

Communicated by Doron S. Lubinsky.

Evgueni Doubtsov was supported by the Russian Science Foundation (Grant No. 14-41-00010).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abakumov, E., Doubtsov, E. Approximation by Proper Holomorphic Maps and Tropical Power Series. Constr Approx 47, 321–338 (2018). https://doi.org/10.1007/s00365-017-9375-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00365-017-9375-5

Keywords

Mathematics Subject Classification

Navigation