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Fourier Analasys - Inner Product Spaces

  1. Aug 30, 2009 #1
    1. The problem statement, all variables and given/known data
    I have two assignments I have some problems with.

    The first one:

    For n > 0, let

    fn(t) = {1, 0 [tex]\leq[/tex] t [tex]\leq[/tex] 1/n
    0, otherwise

    Show that fn [tex]\rightarrow[/tex] 0 in L2[0,1]. Show that fn does NOT converge to zero uniformly on [0,1]


    The second one:
    Find the L2[-[tex]\pi[/tex], [tex]\pi[/tex]] orthogonal projection of the function f(x) = x2 onto the space Vn [tex]\subset[/tex] L2[-[tex]\pi[/tex], [tex]\pi[/tex]] spanned by

    {1, sin(jx)/sqrt([tex]\pi[/tex]), cos(jx)/sqrt([tex]\pi[/tex]); j =1,...,n}
    for n = 1. Repeat this exercise for n= 2 and n = 3.



    2. Relevant equations



    3. The attempt at a solution
    For the first one, I will take the integral:
    |fn|2 = int(0 to 1/n) (1)^2 dt = 1/n - and therefor it converges to 0 for n -> infinite.

    It's the second part of this I'm not sure about. I'm not sure how to explain it. Any help ?


    For the second one, I first test if the basis' is orthonormal, but first taking the inner product of each of them <1, sin(jx)/sqrt([tex]\pi[/tex])>, <1, cos(jx)/sqrt([tex]\pi[/tex])> and <sin(jx)/sqrt([tex]\pi[/tex]), cos(jx)/sqrt([tex]\pi[/tex])> to find out they are all = 0.

    And then I take the inner product of every basis <1,1>, <sin(jx)/sqrt([tex]\pi[/tex]), sin(jx)/sqrt([tex]\pi[/tex])> and so on, to find out that 1 is not 1, but 2*pi. So I normalize that one by dividing it with the 2*pi, and then I have an orthonormal basis, since the other to are have the length 1.

    And then to find the orthogonal projection I say:
    <x2, 1/(2*pi)>1/2*pi + <x2, sin(jx)/sqrt([tex]\pi[/tex])>sin(jx)/sqrt([tex]\pi[/tex]) + <x2, cos(jx)/sqrt([tex]\pi[/tex])>cos(jx)/sqrt([tex]\pi[/tex])

    And since I do it for j, and not a 1, 2 or 3, I guess I really don't need to show it for 1, 2 or 3, since it's only some different integrals that make the difference +


    Well, I hope you can help me a bit.


    Regards
     
  2. jcsd
  3. Aug 31, 2009 #2
    For problem 1, showing that the convergence is not uniform is very easy. I don't say that to make you feel bad, but rather as a hint. What is the definition of uniform convergence?

    For problem 2, the answer in the n=1 case will be a sum of 3 terms, in the n=2 case a sum of 5 terms, and in the n=3 case a sum of 7 terms. I couldn't tell whether or not you realized that.

    Perhaps someone else can comment about whether your use of 1/2π instead of 1 is correct, because I always use the convention 2π=1 (just kidding a little, but not completely). Maybe you should use 1/√(2π).
     
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