Show that c_1,c_2 Exist for Dirichlet Kernel Integral

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In summary, the problem is to show that there exist positive constants c_1 and c_2 such that the Dirichlet kernel satisfies the inequality c_1 \log n \le \int\limits_{ - \pi }^\pi {\left| {D_n \left( t \right)} \right|dt} \le c_2 \log n for n=2,3,4,..., with the current progress being an upper bound of (1/π)(1/2 + n). Suggestions are needed to find a better approximation for |\sum_{N=1}^{n}\cos(Nt)| using negative terms in the Taylor expansions of \cos(Nt).
  • #1
Zaare
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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:
[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)
[/tex]
Which is not good enough.
Any suggestions would be appreciated.

Edit:
By "log" I mean the natural logarithm.
 
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  • #2
The integrals over ##\cos(Nt)|## are all constantly ##4##, hence ##\int_{-\pi}^{\pi}|D_n(t)|\leq \dfrac{1}{\pi}\left(\dfrac{1}{2}+4n\right)## by the triangle inequality. So the real problem is to find a qualitatively better approximation for ##|\sum_{N=1}^{n}\cos(Nt)|##, i.e. using the negative terms in the Taylor expansions of ##\cos(Nt)##.
 

Related to Show that c_1,c_2 Exist for Dirichlet Kernel Integral

What is the Dirichlet kernel integral?

The Dirichlet kernel integral is a mathematical concept used in Fourier analysis to represent a periodic function as a combination of sine and cosine functions. It is defined as the integral of the Dirichlet kernel, which is a periodic function that takes the value of 1 at 0 and decreases as the argument increases.

Why is it important to show the existence of c1 and c2 for the Dirichlet kernel integral?

The existence of c1 and c2 is important because they are the coefficients that allow us to represent a function as a combination of sine and cosine functions. Without these coefficients, we would not be able to accurately represent a function using Fourier analysis.

How do you show the existence of c1 and c2 for the Dirichlet kernel integral?

The existence of c1 and c2 can be shown by solving the integral using mathematical techniques such as integration by parts or substitution. This will result in a formula for c1 and c2 that proves their existence.

What happens if c1 or c2 do not exist for the Dirichlet kernel integral?

If c1 or c2 do not exist, it means that the function cannot be accurately represented using Fourier analysis. This could be due to the function being non-periodic or not meeting certain mathematical criteria.

Are there any real-world applications of the Dirichlet kernel integral and the existence of c1 and c2?

Yes, the Dirichlet kernel integral and the existence of c1 and c2 have many real-world applications in fields such as signal processing, image compression, and data analysis. They are also used in the study of periodic phenomena in physics and engineering.

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