Are you interested for a specific data partition, ie equidistant for example? Or are you looking for a theoretical functional analytic approach?

To Prof. Christopoulos: A theoretical functional analytic approach would be very good, there is no constraints of any kind, on the approximating function, I just want it to be completely monotonous.

Dear Professor Henk Smid,

The considered function $f(t)=1/(1+t^2)$ is not completely monotonous on

$([0,+\infty)$, and I need to find a completely monotonous function $g$ (or a linear combination of decreasing exponentials with positive coefficients), which is as close as possible to $f$ with respect to the $L^1$ norm.

It makes sense to look for the best completely monotone approximation from the subset of decreasing exponential sums with positive coefficients.

However, according to my calculations, no finite sum of decreasing exponentials with positive coefficients can be a best L1 approximation to f(x) = 1/(1+x^2).

I'm curious as to your interest in approximating 1/(1+x^2). Could you please give us some background as to the origin of this problem?

Dear Omran, keep in mind that your function is not analytic beyond the x=+1 singularity (its range of convergence is R=1 and the interval of convergence does not contain +1). So, you have a difficult task, since you are interested for $([0,+\infty)$...

In fact, I am trying to show that $a=\sqrt{\pi/2}$ is the best constant

in the inequality:

||f(t)||_1

This is a partial answer. Consider the space of all continuous functions f  on [0, infinity) such that lim f(t) exists in R, when t converges to infinity. Define f(infinity)=limf(t), when t converges to infinity. One obtains the algebra of continuous real functions defined on the compact [0, infinity], endowed with Alexandrov's topology.. The subspace Span{exp(-nt; n in N, t>=0}, where each function is extended by continuity to the whole interval [0,infinity] is a dense sub algebra, by  Stone - Weierstrass theorem. In particular, it follows that each compactly supported function, with the support contained in [0,infinity) can be uniformly approximated by elements from  Span{exp(-nt; n in N, t>=0}. On the other hand, the subspace of compactly supported functions is dense in L^1([0,infinity)) (Walter Rudin, "Real and complex analysis"). So, our function can be approximated by differences of completely monotonic functions, in L^1 norm. Of course, this does not prove that it can be approximated by completely monotonic functions. However, it might be a starting point, because our function is positive and decreasing. Notice that the convex set of all completely monotonic functions g with g(0)=1 is compact in a topology which is different with respect to L^1 topology. On the other hand, our function is not convex on [0, infinity), so that it is not completely monotonic. The problem remains open.