Can the Fejér Kernel Be Approximated by Polynomials?

Vannel · Jun 3, 2019

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!

member 587159 · Jun 3, 2019

Take a look here:

https://en.wikipedia.org/wiki/Fejér_kernel
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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