Thread: Least Squares with Weight Function

Use the Laguerre polys
L1=x-1, L2=x^2-4x+2, L3=x^3+9x^2+18x-6
to compute least squares polys on (0, infinity) with respect to the weight fn w(x)=e^-x
for the following functions:

f(x)=x^2 , f(x)=e^-x, f(x)=x^3, f(x)=e^-2x

Originally Posted by monomoco

Use the Laguerre polys
L1=x-1, L2=x^2-4x+2, L3=x^3+9x^2+18x-6
to compute least squares polys on (0, infinity) with respect to the weight fn w(x)=e^-x
for the following functions:

f(x)=x^2 , f(x)=e^-x, f(x)=x^3, f(x)=e^-2x

The Laguerre polynomials are orthogonal with respect to your weight on the given interval. So what do you know about expansion in terms of orthogonal functions?

CB

I am very confused about all this stuff, but I think when the functions are orthogonal we can use the following to calcualte coefficients in the polynomial:

*Integral from a to b of

Sorry, I don't really know Latex

Originally Posted by monomoco

I am very confused about all this stuff, but I think when the functions are orthogonal we can use the following to calcualte coefficients in the polynomial:

*Integral from a to b of

Sorry, I don't really know Latex

Define the inner product:



on the space of real square integrable functions over with respect to the weight . Then if is a complete basis of orthogonal functions for this space then:



and the least squares (with respect to ) approximation to in the subspace spanned by is:



CB

Not so bad after all. Thanks very much!