Bernstein's theorem on monotone functions, named after Sergei Bernstein, states that every real-valued function on the half-line [0, infinity) that is completely monotone can be written as a mixture of exponential functions, equivalently as the Laplace transform of a positive Borel measure on that half-line. In the special case where the mixture is a weighted average, it corresponds to an expected value, and the result is also known as the Bernstein-Widder theorem or the Hausdorff-Bernstein-Widder theorem.
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Statement FormCharacterization Theorem 1 Bernstein's Theorem on Monotone Functions
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Source Bernstein's Theorem on Monotone Functions (Wikipedia)
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Source Bernstein's Theorem on Monotone Functions (Wikipedia)
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Source Bernstein's Theorem on Monotone Functions (Wikipedia)
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1. Bernstein's Theorem on Monotone Functions (Wikipedia)
Lead section
In real analysis, a branch of mathematics, Bernstein's theorem, named after Sergei Bernstein, states that every real-valued function on the half-line [0, ∞) that is completely monotone is a mixture of exponential functions or in more abstract language, that it is the Laplace transform of a positive Borel measure on [0, ∞).
History
The result was first proved by Bernstein in 1928
Principal Result Of: Sergei Bernstein, Lead sentence of the article on the result
In real analysis, a branch of mathematics, Bernstein's theorem, named after Sergei Bernstein, states that every real-valued function on the half-line [0, ∞) that is completely monotone is a mixture of exponential functions or in more abstract language, that it is the Laplace transform of a positive Borel measure on [0, ∞).
In Branch: Real Analysis, Lead sentence
In real analysis, a branch of mathematics, Bernstein's theorem, named after Sergei Bernstein, states that every real-valued functi
Proved By: Sergei Bernstein, Lead
In real analysis, a branch of mathematics, Bernstein's theorem, named after Sergei Bernstein, states that every real-valued function on the half-line [0, ∞) that is completely monotone is a mixture of exponential functions or in more abstract language, that it is the Laplace transform of a positive Borel measure on [0, ∞)
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