SpringerLink
Log in

Nevanlinna Domains and Uniform Approximation by Polyanalytic Polynomial Modules

  • Chapter
  • First Online:
Function Spaces, Theory and Applications

Part of the book series: Fields Institute Communications ((FIC,volume 87))

Abstract

We consider the concept of a Nevanlinna domain and its modifications: the concept of a locally Nevanlinna domain and the concept of g-Nevanlinna domain. All these concepts are closely and naturally related with problems on approximation by polyanalytic polynomials and by elements of polynomial modules of polyanalytic type on compact sets in the complex plane. In particular we obtain new criterion for uniform approximation of functions by elements of polyanalytic polynomial modules generated by entire antiholomorphic functions g. We also discuss the relationships between Nevanlinna and g-Nevanlinna domains.

The author was partially supported by the Theoretical Physics and Mathematics Advancement Foundation “BASIS.” The results of Section 4 were obtained in frameworks of the project supported by the Russian Science Foundation, grant no. 22-11-00071 (https://rscf.ru/en/project/22-11-00071/).

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (Brazil)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (Brazil)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 159.99
Price excludes VAT (Brazil)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. A. Baranov, Yu. Belov, A. Borichev, K. Fedorovskiy, Univalent functions in model spaces: revisited, ar**v:1705.05930 [math.CV].

    Google Scholar 

  2. A. Baranov, J. Carmona, K. Fedorovskiy, Density of certain polynomial modules, J. Approx. Theory 206 (2016) 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Baranov, K. Fedorovskiy, Boundary regularity of Nevanlinna domains and univalent functions in model subspaces, Sb. Math. 202 (2011) 1723–1740.

    Article  MathSciNet  MATH  Google Scholar 

  4. Yu. Belov, A. Borichev, K. Fedorovskiy, Nevanlinna domains with large boundaries, J. Funct. Anal. 277 (2019) 2617–2643.

    Article  MathSciNet  MATH  Google Scholar 

  5. Yu. Belov, K. Fedorovskiy, Model spaces containing univalent functions, Russian Math. Surv. 73 (2018) 172–174.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Boivin, P. Gauthier, P. Paramonov, On uniform approximation by n-analytic functions on closed sets in\(\mathbb C\), Izv. Math. 68 (2004) 447–459.

    Google Scholar 

  7. E. Borovik, K. Fedorovskiy, On the relationship between Nevanlinna and quadrature domains, Math. Notes 99 (2016) 460–464.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Carmona, A necessary and sufficient condition for uniform approximation by certain rational modules, Proc. Amer. Math. Soc. 86 (1982) 487–490.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Carmona, Mergelyan’s approximation theorem for rational modules, J. Approx. Theory 44 (1985) 113–126.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Carmona, P. Paramonov, K. Fedorovskiy, On uniform approximation by polyanalytic polynomials and the Dirichlet problem for bianalytic functions, Sb. Math. 193 (2002) 1469–1492.

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Carmona, K. Fedorovskiy, Conformal maps and uniform approximation by polyanalytic functions, Selected Topics in Complex Analysis, Oper. Theor. Adv. Appl. 158, Birkhäuser, Basel, 2005, pp. 109–130.

    Google Scholar 

  12. D. N. Clark, One dimensional perturbations of restricted shifts, J. Anal. Math. 25 (1972) 169–191.

    Article  MathSciNet  MATH  Google Scholar 

  13. P. Davis, The Schwarz function and its applications, Carus Math. Monogr. 17, Math. Ass. of America, Buffalo, NY 1974.

    Google Scholar 

  14. R. G. Douglas, H. S. Shapiro and A. L. Shields, Cyclic vectors and invariant subspaces for the backward shift operator, Annales de l’institut Fourier, 20 (1970), 37–76.

    Article  MathSciNet  MATH  Google Scholar 

  15. K. Dyakonov, D. Khavinson, Smooth functions in star-invariant subspaces, Recent advances in operator-related function theory, Contemp. Math. 393, Amer. Math. Soc., Providence, RI 2006, pp. 59–66.

    Google Scholar 

  16. K. Fedorovskiy, On uniform approximations of functions by n-analytic polynomials on rectifiable contours in\(\mathbb C\), Math. Notes 59 (1996) 435–439.

    Google Scholar 

  17. K. Fedorovskiy, On some properties and examples of Nevanlinna domains, Proc. Steklov Inst. Math. 253 (2006), no. 2, 186–194.

    Article  MathSciNet  Google Scholar 

  18. P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Adv. Math. 44, Cambridge University Press, Cambridge 1995.

    Google Scholar 

  19. M. Mazalov, An example of a nonconstant bianalytic function vanishing everywhere on a nowhere analytic boundary, Math. Notes 62 (1997) 524–526.

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Mazalov, On uniform approximations by bi-analytic functions on arbitrary compact sets in\(\mathbb C\), Sb. Math. 195 (2004) 687–709.

    Google Scholar 

  21. M. Mazalov, A criterion for uniform approximability on arbitrary compact sets for solutions of elliptic equations, Sb. Math. 199 (2008) 13–44.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Mazalov, An example of a nonrectifiable Nevanlinna contour, St. Petersburg Math. J. 27 (2016) 625–630.

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Mazalov, On Nevanlinna domains with fractal boundaries, St. Petersburg Math. J. 29 (2018) 777–791.

    Article  MathSciNet  MATH  Google Scholar 

  24. N. Nikolskiı̆, Treatise on the shift operator, Springer–Verlag, Berlin 1986.

    Google Scholar 

  25. A. G. O’Farrell, Annihilators of rational modules, J. Funct. Anal. 19 (1975) 373–389.

    Article  MathSciNet  MATH  Google Scholar 

  26. Ch. Pommerenke, Boundary behaviour of conformal maps, Springer–Verlag, Berlin 1992.

    Book  MATH  Google Scholar 

  27. Ch. Pommerenke, Conformal maps at the boundary, Handbook of Complex analysis: Geometric Function Theory. Volume 1, 2002.

    Google Scholar 

  28. M. Sakai, Regularity of a boundary having a Schwarz function, Acta Math. 166 (1991) 263–297.

    Article  MathSciNet  MATH  Google Scholar 

  29. H. Shapiro, The Schwarz function and its generalization to higher dimensions, University of Arkansas Lecture Notes in the Mathematical Sciences 9, John Wiley & Sons, Inc., New York 1992.

    Google Scholar 

  30. T. Trent, J. L.-M. Wang, Uniform approximation by rational modules on nowhere dense sets, Proc. Amer. Math. Soc. 81 (1981) 62–64.

    Article  MathSciNet  MATH  Google Scholar 

  31. T. Trent, J. L.-M. Wang, The uniform closure of rational modules, Bull. London Math. Soc. 13 (1981) 415–420.

    Article  MathSciNet  MATH  Google Scholar 

  32. D. Vardakis, A. Volberg, Free boundary problems in the spirit of Sakai’s theorem, Comptes Rendus Mathematique 359 (2021) 1233–1238.

    Article  MathSciNet  MATH  Google Scholar 

  33. J. Verdera, Approximation by rational modules in Sobolev and Lipschitz norms, J. Funct. Anal. 58 (1984) 267–290.

    Article  MathSciNet  MATH  Google Scholar 

  34. J. Verdera, On the uniform approximation problem for the square of the Cauchy-Riemann operator, Pacific J. Math. 159 (1993) 379–396.

    Article  MathSciNet  MATH  Google Scholar 

  35. J. L.-M. Wang, A localization operator for rational modules, Rocky Mountain J. Math. 19 (1989) 999–1002.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

    Konstantin Fedorovskiy

  2. Moscow Center for Fundamental and Applied Mathematics, Lomonosov Moscow State University, Moscow, Russia

    Konstantin Fedorovskiy

  3. St. Petersburg State University, St. Petersburg, Russia

    Konstantin Fedorovskiy

Authors
  1. Konstantin Fedorovskiy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Konstantin Fedorovskiy .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, ON, Canada

    Ilia Binder

  2. Department of Mathematics and Statistics, Laval University, Québec City, QC, Canada

    Damir Kinzebulatov

  3. Department of Mathematics and Statistics, Laval University, Québec City, QC, Canada

    Javad Mashreghi

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Fedorovskiy, K. (2023). Nevanlinna Domains and Uniform Approximation by Polyanalytic Polynomial Modules. In: Binder, I., Kinzebulatov, D., Mashreghi, J. (eds) Function Spaces, Theory and Applications. Fields Institute Communications, vol 87. Springer, Cham. https://doi.org/10.1007/978-3-031-39270-2_6

Download citation

Publish with us

Policies and ethics

Search

Navigation