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I'm trying to use orthogonal polynomials to approximate the value
function of a dynamic programming problem.  I am having problems
choosing the right polynomial.

I'm trying to approximate a function f(x), and x generally takes
values from roughly -10 to +200.  But the theory of this problem says
that x is unbounded in the +ve direction.

I plan to evaluate the fourier coefficients of the approximation rule
using quadrature, so if I use N points in the rule, that gives me the
N values of x at which I need to evaluate f(x).

1. It still seems like black magic to me that out of N values of f(x),
I am getting a approximation rule which claims to know f(x)
everywhere.  What are the caveats of such a claim?

2. What weight function and orthogonal polynomial should I use?  The
approximation theory says that the error of the truncated summation is
minimised in a least squares sense with respect to the weight
function.  Because I may never need to calculate f(x) for x outside
[-10, 200], is it theoretically correct to just use a Legendre
polynomial or a Chebyshev polynomial, using a simple scaling to
convert [-1, 1] into [-10, 200]?

3. How does one choose between Legendre and Chebyshev?  Legendre has a
weight function w(x) = 1, which makes sense to my approximation needs.
Chebyshev has the attractive property of lower computational cost and
smaller error.

4. What are the pitfalls of extracting derivatives out of the
approximation?

5. If I must use the Laguerre polynomial, which is defined on 
[0, infinity], then where do I find out the recurrence relation 
for evaluating it?

Thanks,

        -ans.


-- 
Ajay Shah, (213)749-8133, ajayshah@usc.edu