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- Bernstein's_theorem_on_monotone_functions abstract "In real analysis, a branch of mathematics, Bernstein's theorem states that every real-valued function on the half-line [0, ∞) that is totally monotone is a mixture of exponential functions. In one important special case the mixture is a weighted average, or expected value.Total monotonicity (sometimes also complete monotonicity) of a function f means that f is continuous on [0, ∞), infinitely differentiable on (0, ∞), and satisfiesfor all nonnegative integers n and for all t > 0. Another convention puts the opposite inequality in the above definition. The "weighted average" statement can be characterized thus: there is a non-negative finite Borel measure on [0, ∞), with cumulative distribution function g, such thatthe integral being a Riemann–Stieltjes integral.Nonnegative functions whose derivative is completely monotone are called Bernstein functions. Every Bernstein function has the Lévy-Khintchine representation:where and is a measure on the positive real half-line such thatIn more abstract language, the theorem characterises Laplace transforms of positive Borel measures on [0,∞). In this form it is known as the Bernstein–Widder theorem, or Hausdorff–Bernstein–Widder theorem. Felix Hausdorff had earlier characterised completely monotone sequences. These are the sequences occurring in the Hausdorff moment problem.".
- Bernstein's_theorem_on_monotone_functions wikiPageExternalLink CompletelyMonotonicFunction.html.
- Bernstein's_theorem_on_monotone_functions wikiPageID "3526626".
- Bernstein's_theorem_on_monotone_functions wikiPageRevisionID "587727813".
- Bernstein's_theorem_on_monotone_functions hasPhotoCollection Bernstein's_theorem_on_monotone_functions.
- Bernstein's_theorem_on_monotone_functions subject Category:Theorems_in_measure_theory.
- Bernstein's_theorem_on_monotone_functions subject Category:Theorems_in_real_analysis.
- Bernstein's_theorem_on_monotone_functions type Abstraction100002137.
- Bernstein's_theorem_on_monotone_functions type Communication100033020.
- Bernstein's_theorem_on_monotone_functions type Message106598915.
- Bernstein's_theorem_on_monotone_functions type Proposition106750804.
- Bernstein's_theorem_on_monotone_functions type Statement106722453.
- Bernstein's_theorem_on_monotone_functions type Theorem106752293.
- Bernstein's_theorem_on_monotone_functions type TheoremsInMeasureTheory.
- Bernstein's_theorem_on_monotone_functions type TheoremsInRealAnalysis.
- Bernstein's_theorem_on_monotone_functions comment "In real analysis, a branch of mathematics, Bernstein's theorem states that every real-valued function on the half-line [0, ∞) that is totally monotone is a mixture of exponential functions. In one important special case the mixture is a weighted average, or expected value.Total monotonicity (sometimes also complete monotonicity) of a function f means that f is continuous on [0, ∞), infinitely differentiable on (0, ∞), and satisfiesfor all nonnegative integers n and for all t > 0.".
- Bernstein's_theorem_on_monotone_functions label "Bernstein's theorem on monotone functions".
- Bernstein's_theorem_on_monotone_functions label "Théorème de Bernstein sur les fonctions monotones".
- Bernstein's_theorem_on_monotone_functions sameAs Théorème_de_Bernstein_sur_les_fonctions_monotones.
- Bernstein's_theorem_on_monotone_functions sameAs m.09jhzd.
- Bernstein's_theorem_on_monotone_functions sameAs Q3527015.
- Bernstein's_theorem_on_monotone_functions sameAs Q3527015.
- Bernstein's_theorem_on_monotone_functions sameAs Bernstein's_theorem_on_monotone_functions.
- Bernstein's_theorem_on_monotone_functions wasDerivedFrom Bernstein's_theorem_on_monotone_functions?oldid=587727813.
- Bernstein's_theorem_on_monotone_functions isPrimaryTopicOf Bernstein's_theorem_on_monotone_functions.