Bernstein's theorem

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In functional analysis, a branch of mathematics, Bernstein's theorem states that any 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

$(-1)^n{d^n \over dt^n} f(t) \geq 0$

for all nonnegative integers n and for all t ≥ 0.

The "weighted average" statement can be characterized thus: there is a non-negative finite Borel measure on [0, ∞), with cumulative distribution function g, such that

$f(t) = \int_0^\infty e^{-tx} \,dg(x),$

the integral being a Riemann-Stieltjes integral.

In 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.