A Can the Fejér Kernel Be Approximated by Polynomials?

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The Fejér kernel can be approximated by polynomials through methods such as rewriting it as a sum of exponentials, which are easily approximated using their series expansions. Additionally, the Stone-Weierstrass theorem provides a framework for polynomial approximation of continuous functions. These approaches confirm that polynomial approximations of the Fejér kernel are feasible. Understanding these methods can enhance the study of the kernel's properties. Overall, polynomial approximation is a valid technique for analyzing the Fejér kernel.
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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!
 
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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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