A Approximate Fejér kernel

  • Thread starter Vannel
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Summary
Methods to approximate Fejér kernel
Hello, I'm currently studying the Fejér kernel, which has the form of
F_{n}(x)={\frac  {1}{n}}\left({\frac  {\sin {\frac  {nx}{2}}}{\sin {\frac  {x}{2}}}}\right)^{2}={\frac  {1}{n}}\left({\frac  {1-\cos(nx)}{1-\cos x}}\right)
. I want to know whether there are some methods to approximate this function into polynomials.

Thanks a lot for the help!
 

Math_QED

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Take a look here:


We can rewrite the kernel as a sum of exponentials, and exponentials are easily approximated by polynomials (take some terms of their series expansion). Thus we can approximate the kernel as well.

Alternatively, you can use results like Stone-Weierstrass.
 

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