Bernstein's theorem on monotone functions

In functional analysis, a branch of mathematics, Bernstein's theorem, named after Sergei Natanovich Bernstein, states that any real-valued function on the half-line [0, ∞) that is totally monotone and whose integral from 0 to ∞ is 1 is a weighted average (or expected value) of exponential functions.

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 cumulative probability distribution function g on [0, ∞) 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. F. Hausdorff had earlier characterised completely monotone sequences. These are the sequences occurring in the Hausdorff moment problem.

References

• S. N. Bernstein, Sur les fonctions absolument monotones, Acta Mathematica 1928 pp.1-66
• D. Widder (1941) The Laplace Transform

External link

• MathWorld page on completely monotonic functions