Sum of Exponentials: Describing F(t) with Bernstein's Theorem

[anJos]

Hello! Let's see if you can give me some advice on this:

I want to describe a function with a sum of real exponentials

$$F(t)=\sum a(n) \cdot exp(-k(n) \cdot t)$$

Now, I don't have to calculate the coefficents (an or kn). The only thing I have to do is to make sure that F(t) is a function that can be rewritten like this. How do I know this?

I found something called "Bernstein's theorem" which stated that a function which is totally monotone is a mixture of exponentials. I checked the definition and some of the functions I'm struggling with does not fulfill this. Yet, it seems like these functions can still be described as a sum of exponentials (I used fminsearch in Matlab to check it).

Is it enough if F(t) is strictly decreasing?

For example, I have the following function:

$$F(t)=(1-at)({2 \over 3}+{1 \over 3}cos(2 \pi at))+{1 \over 2 \pi}sin(2 \pi at)$$

... and want to show that it can be rewritten as a sum of exponentials.

 

[Dr. Lady]

http://en.wikipedia.org/wiki/Total_monotonicity

If a function is strictly decreasing, then it's monotonically decreasing. Which then means that it's a total monotone by the definition in the link above.