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If [tex]D_n[/tex] is the Dirichlet kernel, show that there exist positive constants [tex]c_1[/tex] and [tex]c_2[/tex] such that

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c_1 \log n \le \int\limits_{ - \pi }^\pi {\left| {D_n \left( t \right)} \right|dt} \le c_2 \log n ,

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for [tex]n=2,3,4,...[/tex]. By [tex]\log[/tex] they mean the natural logarithm.c_1 \log n \le \int\limits_{ - \pi }^\pi {\left| {D_n \left( t \right)} \right|dt} \le c_2 \log n ,

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I know that

[tex]

D_n \left( t \right) = \frac{1}{\pi }\left( {\frac{1}{2} + \sum\limits_{N = 1}^n {\cos \left( {Nt} \right)} } \right) = \frac{1}{{2\pi }}\sum\limits_{N = - n}^n {e^{iNt} } = \frac{{\sin \left( {nt + \frac{t}{2}} \right)}}{{2\pi \sin \left( {\frac{t}{2}} \right)}}

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And it's easy to see thatD_n \left( t \right) = \frac{1}{\pi }\left( {\frac{1}{2} + \sum\limits_{N = 1}^n {\cos \left( {Nt} \right)} } \right) = \frac{1}{{2\pi }}\sum\limits_{N = - n}^n {e^{iNt} } = \frac{{\sin \left( {nt + \frac{t}{2}} \right)}}{{2\pi \sin \left( {\frac{t}{2}} \right)}}

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[tex]

\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)

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But that is all I can do. I have no idea on how [tex]\log[/tex] comes into the picture. Any help, suggestions or tips would be much appreciated.\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)

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