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Dirichlet kernel

  1. May 12, 2005 #1
    First the problem:
    If [tex]D_n[/tex] is the Dirichlet kernel, I need to show that there exist positive constants [tex]c_1[/tex] and [tex]c_2[/tex] such that
    [tex]c_1 \log n \le \int\limits_{ - \pi }^\pi {\left| {D_n \left( t \right)} \right|dt} \le c_2 \log n[/tex]
    for [tex]n=2,3,4,...[/tex].

    The only thing I have been able to do is this:
    \left| {D_n \left( t \right)} \right| = \frac{1}{\pi }\left( {\frac{1}{2} + \left| {\sum\limits_{N = 1}^n {\cos \left( {Nt} \right)} } \right|} \right) \le \frac{1}{\pi }\left( {\frac{1}{2} + n} \right)
    Which is not good enough.
    Any suggestions would be appreciated.

    By "log" I mean the natural logarithm.
    Last edited: May 13, 2005
  2. jcsd
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