Gram-Schmidt Orthonormal Functions

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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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Answers and Replies

  • #2
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?
 
  • #3
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I'm not sure what you mean. I know that u0 and u2 will be even functions?
 
  • #4
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But there are so many even functions! Which ones do I choose?
 
  • #5
DrClaude
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Can you write an equation for a coefficient ##a_n## in terms of ##u_n## and ##f(x)##?
 
  • #6
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?
 
  • #7
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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?
 
  • #8
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?
Forget about G-S, think about the scalar product instead. So your second sentence there is correct.
 
  • #9
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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.
 
  • #10
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?
 
  • #11
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.
 
  • #12
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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
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f(x) is odd, since it's a product of an odd function and an even function.
 
  • #14
DrClaude
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So what happens when you integrate an odd function for ##-\infty## to ##\infty##?
 
  • #15
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Does that imply that it can be made with just odd functions so a0 and a2 are zero?
 
  • #16
DrClaude
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Does that imply that it can be made with just odd functions so a0 and a2 are zero?
Exact!
 
  • #17
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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
haruspex
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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
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Great, I got it done. Pretty straightforward and not nearly as daunting as it first seemed! Thanks for your help :)
 

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