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
MathSciNet review: 0352789
Full-text PDF

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 𝐶^{∞} et analytiques, Ann. Inst. Fourier (Grenoble) 21 (1971), no. 4, 149–173 (French, with English summary). MR 0352790
  • [2] S. N. Bernšteĭn, Sobranie sočinenii. Tom I. Konstruktivnaya teoriya funkciĭ [1905–1930], Izdat. Akad. Nauk SSSR, Moscow, 1952 (Russian). MR 0048360
  • [3] A. Grothendieck, Espaces vectoriels topologiques, Instituto de Matemática Pura e Aplicada, Universidade de São Paulo, São Paulo, 1954 (French). MR 0077884
  • [4] G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York-Chicago, Ill.-Toronto, Ont., 1966. MR 0213785
  • [5] S. N. Mergelyan, Uniform approximations of functions of a complex variable, Uspehi Matem. Nauk (N.S.) 7 (1952), no. 2(48), 31–122 (Russian). MR 0051921
  • [6] J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, Fourth edition. American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I., 1965. MR 0218588
  • [7] Martin Zerner, Développement en séries de polynômes orthonormaux des fonctions indéfiniment différentiables, C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A218–A220 (French). MR 0247451

Similar Articles

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

Retrieve articles in all journals with MSC: 41A10

Additional Information

Keywords: Approximation of real-analytic functions, polynomials, Bernstein inequality
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society