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Orthonormalizing functions

  1. May 11, 2010 #1
    In class we worked out the following

    [tex]

    \int e^{ik(x-X)}dk=\frac{e^{ik(x-X)}}{i(x-X)}\approx \frac{sin[k(x-X)]}{x-X}

    [/tex]

    by taking the real part of the solution. My teacher wants us to graph the following functions

    [tex]

    \psi_{1} \sim \frac{sin(x)}{x}

    [/tex]

    [tex]

    \psi_{2} \sim \frac{sin(x)}{x}-\frac{1}{2}\frac{sin(2x)}{2x}

    [/tex]

    The second function, though, has a little dip in it at 0 that shouldn't be there. He says that's due to the fact that the functions aren't normalized and that we should be doing Gram-Scmidt or some other procedure to obtain the proper results. However, when I do Gram-Schmidt I get something nasty that can't possibly be correct (and most importantly, doesn't correct the problem). So what am I doing wrong?
     
    Last edited: May 11, 2010
  2. jcsd
  3. May 11, 2010 #2
    For Gram-Schmidt I take interval as (0, 2π). Then

    [tex]

    \phi _{0} = \frac{sin(x)}{x\sqrt {Si(4 \pi)}}

    [/tex]

    [tex]

    \psi _{1} = \left \{ \frac{sin(x)}{x}-\frac{1}{2}\frac{sin(2x)}{(2x)}\right \}-\left [ \int_{0}^{2\pi}\left \{ \frac{sin(x)}{x}-\frac{1}{2}\frac{sin(2x)}{(2x)}\right \}\frac{sin(x)}{x\sqrt {Si(4 \pi)}}dx \right ]\frac{sin(x)}{x\sqrt {Si(4 \pi)}}

    [/tex]

    [tex]

    \psi _{1} \sim \frac{0.2627sin(x)}{x}-\frac{1}{2}\frac{sin(2x)}{2x}

    [/tex]

    [tex]

    \phi _{1} \sim 3.3631\left \{ \frac{0.2627sin(x)}{x}-\frac{1}{2}\frac{sin(2x)}{2x} \right \}

    [/tex]

    using the procedure for Gram-Schmidt outlined here: http://mathworld.wolfram.com/Gram-SchmidtOrthonormalization.html

    and with a weighting function of 1.
     
  4. May 11, 2010 #3
    Any suggestions?
     
    Last edited: May 11, 2010
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