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Application of the Krein-Milman theorem to completely monotonic functions

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1757)

Abstract

A real valued function f on (0, ∞) is said to be completely monotonic if f has derivatives f (0) = f, f (1), f (2),... of all orders and if (-1)n f (n) ≧ 0 for n = 0, 1, 2,... Thus, f is nonnegative and non-increasing, as is each of the functions (-1)n f n. [[Some examples: x and eαx (α ≧ 0).] S. Bernstein proved a fundamental representation theorem for such functions. (See [82] for several proofs and much related material). We will prove the theorem only for bounded functions; the extension to unbounded functions (with infinite repre-senting measures) follows from this by classical arguments [82]. We denote the one-point compactification of [0, ∞) by [0, ∞].

Keywords

  • Extreme Point
  • Monotonic Function
  • Borel Probability Measure
  • Classical Argument
  • Lebesgue Dominate Convergence Theorem

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 2001 Springer-Verlag Berlin Heidelberg

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(2001). Application of the Krein-Milman theorem to completely monotonic functions. In: Phelps, R.R. (eds) Lectures on Choquet’s Theorem. Lecture Notes in Mathematics, vol 1757. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48719-0_2

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  • DOI: https://doi.org/10.1007/3-540-48719-0_2

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41834-4

  • Online ISBN: 978-3-540-48719-7

  • eBook Packages: Springer Book Archive