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 x ∈ Δ. In other words, ||f–Pє|| < 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 ||f–p 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
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© 1989 Springer-Verlag Berlin Heidelberg
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Zaanen, A.C. (1989). Theorems of Korovkin and Stone-Weierstrass. In: Continuity, Integration and Fourier Theory. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-73885-2_2
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DOI: https://doi.org/10.1007/978-3-642-73885-2_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-50017-9
Online ISBN: 978-3-642-73885-2
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