Gram-Schmidt Orthonormal Functions

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The discussion revolves around expressing the function f(x) = xe^(-3x^2) as a linear combination of orthonormal basis functions un(x). Participants clarify that the coefficients a0 and a2 are zero because f(x) is an odd function, which means it can only be represented using odd basis functions. The integration of the product of f(x) and the basis functions is emphasized as the method to find the coefficients. The conversation highlights the importance of understanding the properties of odd and even functions in the context of Gram-Schmidt orthogonalization. Ultimately, the problem is resolved with the realization that only odd functions contribute to the representation of f(x).
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Homework Statement


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

Homework Equations



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.
 
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Robsta said:
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
Can't you use that by itself?
 
I'm not sure what you mean. I know that u0 and u2 will be even functions?
 
But there are so many even functions! Which ones do I choose?
 
Can you write an equation for a coefficient ##a_n## in terms of ##u_n## and ##f(x)##?
 
Either I'm misreading it or u1 is a constant multiple of f. Doesn't that make it somewhat trivial?
 
Sorry haruspex I typed the question wrong. I've now rewritten u1 as it should be.

Dr Claude, I don't think I can. I could write a particular phase of G-S orthogonalisation but not a general one in terms of n. If I integrated the product of f(x) and un(x) would I get an?
 
Robsta said:
Dr Claude, I don't think I can. I could write a particular phase of G-S orthogonalisation but not a general one in terms of n. If I integrated the product of f(x) and un(x) would I get an?
Forget about G-S, think about the scalar product instead. So your second sentence there is correct.
 
Yep, okay. So if you integrate the product of a function and one of it's basis functions, it gives you the amount that it projects along that basis function (using the vector analogy). So to find a0 I have to integrate f(x)u0(x) from minus infinity to infinity. So how do I guess what u(x) is so that I can put it into the integral.
 
  • #10
Robsta said:
Sorry haruspex I typed the question wrong. I've now rewritten u1 as it should be.
OK. Is f odd, even or neither? What does that suggest about the contributions from the basis functions?
 
  • #11
Robsta said:
Yep, okay. So if you integrate the product of a function and one of it's basis functions, it gives you the amount that it projects along that basis function (using the vector analogy). So to find a0 I have to integrate f(x)u0(x) from minus infinity to infinity. So how do I guess what u(x) is so that I can put it into the integral.
You don't need to guess. Go back to post #2.
 
  • #12
So I know that I have to integrate f(x)u0(x)dx and that u0(x) is even. Is there some kind of dimensional grounds that I can work out what sort of even function it is?
 
  • #13
f(x) is odd, since it's a product of an odd function and an even function.
 
  • #14
So what happens when you integrate an odd function for ##-\infty## to ##\infty##?
 
  • #15
Does that imply that it can be made with just odd functions so a0 and a2 are zero?
 
  • #16
Robsta said:
Does that imply that it can be made with just odd functions so a0 and a2 are zero?
Exact!
 
  • #17
Fantastic! Thanks for your help, both of you. I'm just going to compute the integral to find a1 then. Will reply to let you know of success.
 
  • #18
Robsta said:
Does that imply that it can be made with just odd functions so a0 and a2 are zero?
It's a bit stronger than that - it can only be made from the odd functions. If you imagine summing the odd and even basis functions separately into F, G respectively, you have f = F+G. But to make f odd, G must be zero. Since the even functions are orthonormal, their coefficients must all be zero.
 
  • #19
Great, I got it done. Pretty straightforward and not nearly as daunting as it first seemed! Thanks for your help :)
 

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