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Theorems of Korovkin and Stone-Weierstrass

  • Adriaan C. Zaanen
Part of the Universitext book series (UTX)

Abstract

Let Δ = [a, b] be a closed interval in ℝ. The classical approximation theorem of Weierstrass (1885) asserts that any f ∈ C(Δ) can be approximated uniformly by polynomials, i.e., for any є > 0 there exists a polynomial Pє such that |f(x)–Pє (x) | < є holds for all xΔ. In other words, ||fPє|| < є, where || • || denotes the uniform norm in C(Δ). Equivaiently, we may say that there exists a sequence (p n : n = 1,2,…) of polynomials such that ||fp n || → 0 as n → ∞. Is it possible to denote explicitly a sequence (p n ) satisfying this condition? The answer is affirmative. For Δ = [0,1] we may choose for p n the n-th Bernstein polynomial B n (f), defined on [0,1] by
$${B_n}(f) = \sum\limits_{m = o}^n {\left( {_m^n} \right)f\left( {\frac{m}{n}} \right)} {x^m}{(1 - x)^{n - m}}.$$
(1)

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Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Adriaan C. Zaanen
    • 1
  1. 1.Department of MathematicsUniversity of LeidenLeidenThe Netherlands

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