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Determining f

  1. Feb 5, 2009 #1
    The problem statement, all variables and given/known data
    Suppose f: [0,1] --> R is continuous and

    [tex]\int_0^1 f(x) x^n \, dx = 0[/tex]

    for all n = 0, 1, ... Prove that f(x) = 0.

    The attempt at a solution
    There's a hint that says: Prove that

    [tex]\int_0^1 (f(x))^2 \, dx = 0.[/tex]

    I don't know how to prove this hint and I don't know how that would help in determining what f is. Any tips?
     
  2. jcsd
  3. Feb 5, 2009 #2

    Dick

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    HAH! This is cosmic. Look at this thread.
    https://www.physicsforums.com/showthread.php?p=2064400#post2064400
    I think the course I was outlining for Hitman2-2 will work perfectly for you, and even more simply since you have the condition for all n>=0. Use Stone-Weierstrass. You have integral f(x)=0 since it's true for n=0. Hitman2-2 only gave me n an even natural number. Is zero an even natural number? I would have said, no. And that's the roadblock for that thread.
     
  4. Feb 5, 2009 #3

    Dick

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    Are you SURE you don't know why the integral of f^2 equals zero wouldn't solve the problem?
     
  5. Feb 5, 2009 #4
    The integral of f^2 can only be zero if f is identically zero. If f(x) were greater than zero at some point in the interval, there would be a positive contribution to the integral from that point with no negative contribution to cancel it out (since the square of a real function is non-negative). Think about it.

    I don't know how to prove the assertion. I will have to think about it.
     
  6. Feb 5, 2009 #5
    That thread was really helpful. Now I know why the hint is true. However...

    I don't know why the integral of f^2 equals zero wouldn't solve the problem. You asked Hitman2-2 the same question as well, but he/she didn't respond.
     
  7. Feb 5, 2009 #6

    Dick

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    Brian_C just answered why the integral of f^2 equal zero solves the problem.
     
  8. Feb 5, 2009 #7
    I have an idea how to prove that the integral of f^2 is equal to zero. Try expanding one of the f(x)'s in the integral of f^2 as a taylor series in x. You should end up with an infinite series of integrals of the form f(x) * x^n, which should all vanish (by assumption), thus proving that the integral of f^2 vanishes. The only problem is, I don't know how to prove that a Taylor series will converge on some interval for this particular function.
     
  9. Feb 5, 2009 #8

    Dick

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    The taylor series doesn't have to converge. The function is only continuous, it doesn't have to be differentiable, much less analytic. Check Hitman2-2's thread. You have to use the Weierstrass approximation theorem.
     
    Last edited: Feb 5, 2009
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