Gram-Schmidt Orthonormal Functions

[Robsta]

Homework Statement

The function f(x) = xe^-3x2 is expressed as a linear combination of the basis functions u_n(x), which are orthogonal and normalised from minus infinity to infinity.

It is expressed by xe^-3x2 = ∑a_nu_n(x)

the u_n(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.

 

[DrClaude]

the u_n(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?

 

[Robsta]

I'm not sure what you mean. I know that u₀ and u₂ will be even functions?

 

[Robsta]

But there are so many even functions! Which ones do I choose?

 

[DrClaude]

Can you write an equation for a coefficient $a_n$ in terms of $u_n$ and $f(x)$?

 

[haruspex]

Either I'm misreading it or u1 is a constant multiple of f. Doesn't that make it somewhat trivial?

 

[Robsta]

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 u_n(x) would I get a_n?

 

[DrClaude]

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 u_n(x) would I get a_n?

Forget about G-S, think about the scalar product instead. So your second sentence there is correct.

 

[Robsta]

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.

 

[haruspex]

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?

 

[DrClaude]

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.

 

[Robsta]

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?

 

[Robsta]

f(x) is odd, since it's a product of an odd function and an even function.

 

[DrClaude]

So what happens when you integrate an odd function for $-\infty$ to $\infty$?

 

[Robsta]

Does that imply that it can be made with just odd functions so a₀ and a₂ are zero?

 

[DrClaude]

Does that imply that it can be made with just odd functions so a₀ and a₂ are zero?

Exact!

 

[Robsta]

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.

 

[haruspex]

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.

 

[Robsta]

Great, I got it done. Pretty straightforward and not nearly as daunting as it first seemed! Thanks for your help :)