The function f(x) = xe-3x2 is expressed as a linear combination of the basis functions un(x), which are orthogonal and normalised from minus infinity to infinity.
It is expressed by xe-3x2 = ∑anun(x)
the un(x)'s are even functions of x for n = 0,2,4 and are odd functions of x for n = 1,3,5
Calculate a0, a1 and a2 given that:
u1(x) = (4sqrt(2)/sqrt(pi))1/2xe-x2
I'm also told that the integral from -infinity to infinity of x2e-a2x2dx = sqrt(pi)/2a3
I'm aware of Gram Schmidt orthogonalisation but am at a bit of a loss as to how to do it from the second term and how to guess the next terms. I get the process but don't you need a set of linearly independent functions to do it to?
The Attempt at a Solution
It's asking for the various coefficients and not the u(x) terms, but do I have to work those out with G-S orthogonalisation? If so I need to find the set of LI functions to orthogonalise.