Bernstein's theorem on monotone functions

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. Another convention puts the opposite inequality in the above definition. Completely monotone functions are also called Bernstein functions.

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.

Further, any Bernstein function can be written in the following form:

$f(t)=a+bt+\int _{0}^{\infty }(1-e^{-tx})\mu (dx)$

where $a,b\geq 0$ and $\mu$ is a measure on the positive real half-line such that $\int _{0}^{\infty }(1\wedge x)\mu (dx)<\infty .$

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.

References

• S.N. Bernstein (1928). "Sur les fonctions absolument monotones". Acta Mathematica. 52: 1–66. doi:10.1007/BF02592679.
• D. Widder (1941). The Laplace Transform. Princeton University Press.

External links

• MathWorld page on completely monotonic functions