Sinc function as a sum of cosines

[david316]

Hello,

If you plot
y=sin(x)/x
and also plot
y = summation of 0.01*cos(n*x/100) over n = 0 to n =100
you essentially get the same graph. Is there any formal proof that relates the sinc function to a sum of cosines.

Thanks

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

 

[chiro]

Hey david316.

You can prove this by doing a Fourier series decomposition over an interval in which you project the function to the sine and cosine bases functions.

It's basically finding the integral that corresponds to the inner product in that basis. For more see this:

http://en.wikipedia.org/wiki/Fourier_series

 

[greswd]

Hello,

If you plot
y=sin(x)/x
and also plot
y = summation of 0.01*cos(n*x/100) over n = 0 to n =100
you essentially get the same graph. Is there any formal proof that relates the sinc function to a sum of cosines.

Thanks

do you have a graphical plot demonstrating this?

 

[greswd]

nvm, I have it.

Do you have a complete formula that considers a sum to infinity? yours is just an approximation

 

[Ssnow]

I know that $sinc(x)=\prod_{i=1}^{\infty}\cos{\left(\frac{x}{2^{i}}\right)}$ but is not a sum, if you want a sum you can write:

$\log{sinc(x)}=\sum_{i=1}^{\infty}\log{\cos{\left(\frac{x}{2^{i}}\right)}}$ ...