Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Approximation of analytic functions on compact sets and Bernstein's inequality


Authors: M. S. Baouendi and C. Goulaouic
Journal: Trans. Amer. Math. Soc. 189 (1974), 251-261
MSC: Primary 41A10
DOI: https://doi.org/10.1090/S0002-9947-1974-0352789-7
MathSciNet review: 0352789
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The characterization of analytic functions defined on a compact set K in $ {{\mathbf{R}}_N}$ by their polynomial approximation is possible if and only if K satisfies some ``Bernstein type inequality", estimating any polynomial P in some neighborhood of K using the supremum of P on K. Some criterions and examples are given. Approximation by more general sets of analytic functions is also discussed.


References [Enhancements On Off] (What's this?)

  • [1] M. S. Baouendi and C. Goulaouic, Approximation polynomiale de fonctions $ {\mathcal{C}^\infty }$ et analytiques, Ann. Inst. Fourier (Grenoble) 21 (1971), 149-173. MR 0352790 (50:5276)
  • [2] S. Bernstein, Collected works, Izdat. Akad. Nauk SSSR, Moscow, 1952. (Russian) MR 14, 2. MR 0048360 (14:2c)
  • [3] A Grothendieck, Espaces vectoriels topologiques, Instituto de Mathemática Pura e Aplicada, Universidade de São Paulo, São Paulo, 1954. MR 17, 1110. MR 0077884 (17:1110a)
  • [4] G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York, 1966. MR 35 #4642. MR 0213785 (35:4642)
  • [5] S. N. Mergelian, Uniform approximations to functions of a complex variable, Uspehi Mat. Nauk 7 (1952), no. 2 (48), 31-122; English transl., Amer. Math. Soc. Transl. (1) 3 (1962), 294-391. MR 14, 547. MR 0051921 (14:547e)
  • [6] J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, 4th ed., Amer. Math. Soc. Colloq. Publ., vol. 20, Amer. Math. Soc., Providence, R.I., 1965. MR 36 # 1672b. MR 0218588 (36:1672b)
  • [7] M. Zerner, Développement en série de polynômes orthogonaux des fonctions indéfiniment différentiables, C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A218-A220. MR 40 #717. MR 0247451 (40:717)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 41A10

Retrieve articles in all journals with MSC: 41A10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0352789-7
Keywords: Approximation of real-analytic functions, polynomials, Bernstein inequality
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society