Date: Tue, 19 Dec 95 22:35:15 CST
From: rusin (Dave Rusin)
To: A4411DAL@helios.edvz.univie.ac.at
Subject: Re: Class of functions
Newsgroups: sci.math.research

In article <1746FB0AF.A4411DAL@awiuni11.edvz.univie.ac.at> you write:
>I am looking for a classification of the class of all integrable functions
>f defined on the real interval (a,b) which have the following property:
>the integral of exp(-c x)f(x) over the interval (a,b) is non-positive for
>all positive real numbers c.

I would imagine that there is no "classification". Let  g  be any integrable
function on  (a,b)  which is non-positive everywhere. Then the function
f(x)=g(x) + M  lies in your class iff 
	M <= inf c/(exp(-ca)-exp(-cb)) * integral_a^b  exp(-c x) |g(x)|,
the infimum taken over all positive numbers  c.  Note that on the right side
we are taking the infimum of positive numbers only, so that the inf.
does in fact exist (for any  g).

Conversely, if  f  is any function in your class then for some  M  we
have  g(x) = f(x)-M  always non-positive, as long as  f  is a bounded
function (this is true if  f  is continuous, and tends to be true
whenever  f  is integrable, although of course unbounded integrable
functions are easy to construct).

So except for this boundedness issue, we see that the shapes of the graphs
of the functions in your class are no different from those of 
functions which are everywhere negative.

dave