From: hougen@focus.csl.uiuc.edu (Darrell Roy Hougen)
Newsgroups: sci.math
Subject: Re: Approximation and Estimation (HELP!!)
Date: 23 Nov 1992 21:27:21 GMT
Organization: Center for Reliable and High-Performance Computing, University of Illinois at Urbana-Champaign
NNTP-Posting-Host: focus.csl.uiuc.edu

baez@guitar.ucr.edu (john baez) writes:

% In article <1ejbebINNber@roundup.crhc.uiuc.edu> hougen@focus.csl.uiuc.edu (Darrell Roy Hougen) writes:
%% I need to estimate a 'hill' function between -pi/2 and pi/2.  It must
%% be 0 at -pi/2 and pi/2 and it must be 1 at 0.  It must increase
%% monotonically from 0 to 1 on [-pi/2,0] and decrease monotonically from
%% 1 to 0 on [0,pi/2].  It must be smooth, ie., differentiable everywhere
%% in (-pi/2,pi/2), and have a derivative of 0 at 0.
%% 
%% My first question is, does there exist a basis for functions of the
%% above form?

% Sure, infinitely many.  It's best to say what *kind* of functions  In
% other words, smooth, continuous, or L^2 functions.  But in any of the case
% I just mentioned, there is a countable basis.
	
First, thanks for the information.  Perhaps I wasn't clear, but I
would prefer to have smooth, continuous functions, ie., the first
derivative must exist and be continuous on (-pi/2,pi/2).  Right now I
don't care about the norm too much, as long as it is reasonable.

%% One possible basis that I've been thinking about is (cos(x))^k with k
%% ranging from -inf to +inf.  All the members of this set satisfy the
%% conditions given above and it appears that any weighted sum of such
%% functions will also although it would be nice to prove that.

% Nope, it's not true, because all the functions you listed are "even"
% (satisfy f(x) = f(-x)), so you can't approximate anything but even 
% functions.    Also for k negative the functions you list are unbounded,
% which is technically a pain in the butt.  

Let me simplify things a little by asking for even functions, ie.,
assume that the function to be estimated is even.  Also, I made an
error when I allowed k to be negative.  Instead, I should have defined
my potential basis to be (cos(x))^k for k = 1,2,3,... and
(cos(x))(1/j) for j = 1,2,3,...

% The best basis to use is  sin(2kx) and cos(2kx), taken together.  
% This basis is used in the Fourier transform and is well-known to
% be a complete basis.  I recommend this basis because a huge amount
% is known about it and there is a huge amount of software written to
% deal with it.

The problem with this basis, a basis of which I am well aware and that
several people have suggested in email, is that the estimate of the
function might be negative in places or have the the wrong slope in
places.  Let me stress that any estimate, g, of the function, f,
**MUST** satisfy the following conditions:
	g(-pi/2) = g(pi/2) = 0
	g(x) > 0  for x in (-pi/2,pi/2)
	g'(x) > 0 for x in (-pi/2,0)
	g'(x) = 0 for x = 0
	g'(x) < 0 for x in (0,pi/2)
 Corresponding to what I said above, it is also exceptable if g(x) =
g(-x).  This reduces the problem to finding a function on [0,pi/2]
that satisfies the above conditions.  

It is the above list of conditions that make the problem hard.  In
particular, it appears that every basis function in the set of basis
functions must satisfy the above conditions and that all of the
coefficients must be nonnegative.  If there is a different set of
conditions that yield the same result, it would be nice to hear about
it.  For example, if there is some way to formulate the above problem
as a constrained optimization, it would be nice to know.  My current
proposal for solving the problem is to use my amended set of basis
functions listed above with nonnegative coefficients, ie.,
	g(x) = sum_i a_i * g_i(x) 
where all of the coefficients are forced to satisfy a_i >= 0.

Can anyone tell whether this set of basis functions is sufficient for
the problem as I have now stated it, ie., is it a complete basis?  If
not, is there another basis which is?

Darrell R. Hougen
Xref: sparky sci.math:15387 sci.physics:19443