Skip to main content

Approximation of functions and all derivatives on compact sets


In Mergelyan type approximation we uniformly approximate functions on compact sets K by polynomials or rational functions or holomorphic functions on varying open sets containing K. In the present paper we consider analogous approximation, where uniform convergence on K is replaced by uniform approximation on K of all order derivatives.

This is a preview of subscription content, access via your institution.


  1. 1.

    Diamantopoulos, E., Mouratides, Ch., Tsirivas, N.: Universal Taylor series on unbounded open sets. Anal. Int. Math. J. Anal. Its Appl. 26(3), 323–336 (2006)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Kariofillis, C., Konstadilaki, C., Nestoridis, V.: Smooth universal Taylor series. Monatsheft. Math. 147(3), 249–257 (2006)

  3. 3.

    Rudin, R.: Real and Complex Analysis. Tata McGraw-hill Education, London (2006)

    MATH  Google Scholar 

  4. 4.

    Tarkhanov, N.N.: The Cauchy problem for solutions of elliptic equations, vol. 7. Vch Pub (1995)

  5. 5.

    Anatoliy Georgievich Vitushkin: Analytic capacity of sets in problems of approximation theory uspehi mat. Nauk 22, 141–199 (1967)

    MathSciNet  Google Scholar 

  6. 6.

    Anatoliy Georgievich Vitushkin: Uniform approximations by holomorphic functions. J. Funct. Anal. 20(2), 149–157 (1975)

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Giorgos Kotsovolis.

Additional information

Dedicated to Professor Paul M. Gauthier on the occasion of his 80th birthday.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Adrian Constantin.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Armeniakos, S., Kotsovolis, G. & Nestoridis, V. Approximation of functions and all derivatives on compact sets. Monatsh Math (2021).

Download citation


  • Mergelyan theorem
  • Uniform approximation of all order derivatives
  • \(C^\infty \)
  • \(A^\infty \)
  • A(K)
  • C(K)
  • Holomorphic functions
  • Completion of a metric space

Mathematics Subject Classification

  • 30E10
  • 30H50