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Approximation of functions and all derivatives on compact sets

Abstract

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.

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References

  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 

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Correspondence to Giorgos Kotsovolis.

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

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Communicated by Adrian Constantin.

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Armeniakos, S., Kotsovolis, G. & Nestoridis, V. Approximation of functions and all derivatives on compact sets. Monatsh Math (2021). https://doi.org/10.1007/s00605-021-01604-x

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Keywords

  • 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