From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci)
Subject: Re: positive expansion of functions
Date: 16 Jul 1999 14:41:18 GMT
Newsgroups: sci.math.num-analysis
Keywords: best approximation of a positive function by positive functions

In article <7mlu4a$q87$1@gannett.math.niu.edu>,
 rusin@vesuvius.math.niu.edu (Dave Rusin) writes:
|> In article <378A2032.3084429A@tu-bs.de>,
|> Andreas Keese  <a.keese@tu-bs.de> wrote:
|> 
|> >When you expand a positive function as a Taylor-series, usually a
|> >non-positive function results.
|> >
|> >Can anyone point me to references on how to project a positive function
|> >- e.g. the exponential function - to the space of polynomials in a way
........ snip
what you have here is in general terms a semiinfinite optimization problem:
given f say [0,1]->R and some family of functions phi_i, find a_i such that
  ||f-\sum_i a_i \phi_i|| = min _{a_i} 
subject to the constraints
   \sum_i a_i \phi_(x) >= 0 for all x in [0,1]
the form of the norm ||.|| decides whether this is rather simple or not. 
there are special methods for semiinfinite optimization, indeed a vast 
literature, but for your task a look at Hettich&Zencke: methods of
semiinfinite optimization (Teubner) should suffice. Indeed you can make it 
easier:
as far as I remember correctly the set of functions 
  \sum_i a_i exp(s_i*x)/(1+exp(s_i*x)) , a_i>=0  
is dense in the space of positive continuous functions on [0,1]
hence, if you restrict yourself to the l2-norm, a bound constrained 
nonlinear least squares problem remains (optimize with respect to s_i and a_i ,
with a_i>=0). there are ready to use codes for this task. see
http://plato.la.asu.edu/guide.html
If you restrictyourself to a spline approximation by piecewise quadratic
C1-splines, positivity of the approxiamtion can also easily be expressed as
a finite set of inequalities for the coefficients (using a B-spline basis)
and you are left with a finite nonlinear optimization problem.
for methods: see the URL above 


there is also a lot of theory available: e.g. make an electronic search in 
Zentralblat fuer Mathematik.

hope this helps
peter