Question

Use Gram-Schmidt Orthogonalization process to find first 4 orthogonal polynomials.
1) (-inf,inf) sigma(x)=e^(-x^2)
Note: sigma(x) A.K.A. weight function

Answer 1

I also know that this is the basis for the Hermite Polynomials. I am having a hard time rectifying that H0(x)=1 based off the G-S Ortho. Process. As it would be
phi(0)(x)= f(0)-(<f0,phi(-1)>/<phi(-1),(phi(-1)>^2)*phi(-1)

Answer 2

which would give
phi_0(x)=x^0-(<x^0,x^-1>/<x^-1,x^-1>^2)(x^-1)
where <x^0,x^-1>=intg(-inf,inf)[(1)(x^-1)(e^(-x^2))dx]
and <x^-1,x^-1>=(intg(-inf,inf)[(x^-1)(x^-1)(e^(-x^2))dx])^2

Answer 3

Aren't you supposed to start with 1 as the first polynomial?
I can't add much to this, but here's wolfram's write-up:
http://mathworld.wolfram.com/Gram-SchmidtOrthonormalization.html

Answer 4

Yeah. I just don't understand how it starts with one. Other than x^0 is one. But, if you use the actual G-S equation it isn't 1... I think the problem arises in the fact that n>0 for G-S and I am plugging in a n=-1 which is apparently a problem. I am interested in knowing whether or not that boundary condition can be worked around.

Answer 5

Well to be more specifc j=0...(n-1) and if I plug in n=0, then j=-1 and that becomes a problem with the inner product.

Answer 6

I think the idea is to start with the sequence of polynomials 1,x,x^2,x^3 and use G-S to make them orthogonal (as determined by your weighting function).