# Gram-Schmidt Orthonormal Functions

## 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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DrClaude
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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?

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

haruspex
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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?

DrClaude
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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?

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 in to the integral.

haruspex
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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
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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 in to the integral.
You don't need to guess. Go back to post #2.

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?

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

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

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

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

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