Gram-Schmidt Orthonormal Functions

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Homework Help Overview

The problem involves expressing the function f(x) = xe-3x2 as a linear combination of orthogonal basis functions un(x). The basis functions are categorized as even for n = 0, 2, 4 and odd for n = 1, 3, 5. The task includes calculating specific coefficients a0, a1, and a2 using the Gram-Schmidt process.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the need for a set of linearly independent functions for the Gram-Schmidt process and question how to identify these functions. There are inquiries about the relationship between the function f(x) and the basis functions, particularly regarding their even and odd properties. Some participants suggest using integrals to find coefficients and explore the implications of f(x) being odd.

Discussion Status

The discussion is active, with participants exploring various interpretations of the problem and clarifying the roles of even and odd functions in the context of the coefficients. Some have provided insights into the integration process and its relation to the orthogonality of the basis functions, while others are confirming their understanding of the implications of f(x)'s properties.

Contextual Notes

There is a mention of constraints regarding the nature of the function f(x) being odd and how this affects the coefficients a0 and a2. Participants are also considering the implications of integrating odd functions over symmetric limits.

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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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