2022-11-15 21:46 pm

Type :Note Tags : Analysis

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Fejer Kernel

Motivation

Dirichlet Kernels fail to be good kernels, but their averages are better behaved and are good kernels.

Definition

The N-th Fejer Kernel is defined by: $F_N(x)=\frac{D_0(x)+D_1(x)+\cdots D_{N-1}(x)}{N}$

• Since $S_n(f)=f\ast D_n$, we can say that ${\sigma }_N(f)(x)=(f\ast F_N)(x)$ where ${\sigma }_N(f)(x)=\frac{S_0(x)+S_1(x)+\cdots +S_{N-1}(x)}{N}$

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Proposition

We have a closed form for the Fejer Kernel, closely related to that of the Dirichlet Kernels. $F_N(x)=\frac{1}{N}\frac{{\sin }^2(Nx/2)}{{\sin }^2(x/2)}$

Proof:

• Use $D_n(x)=\sum _{j=-n}^n{\omega }^j=\frac{{\omega }^{-n}-{\omega }^{n+1}}{1-\omega }$where $\omega =e^{ix}$.
• This gives F_N(x) = \dfrac{1}{N}\sum\limits_{n=0}^{N-1}\dfrac{\omega^{-n}-\omega^{n+1}}{1-\omega}$$$$ \implies F_N(x) = \dfrac{1}{N(1-\omega)}\sum\limits_{n=0}^{N-1}(\omega^{-n}-\omega^{n+1}) $⟹F_N(x)=\frac{1}{N(1-\omega {)}^2}({\omega }^{1-N}-2\omega +{\omega }^{N+1})$ $⟹F_N(x)=\frac{1}{N({\omega }^{-1/2}-{\omega }^{1/2}{)}^2}({\omega }^{-N}-2+{\omega }^N)=\frac{1}{N({\omega }^{-1/2}-{\omega }^{1/2}{)}^2}({\omega }^{-N/2}-{\omega }^{N/2}{)}^2$ This gives the desired result.

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Theorem

The Fejer Kernel is a good kernel.

Proof:

• The first property of good kernels can be verified, since it also holds for dirichlet kernels.
• The second property obviously holds because the first property holds and $F_N(x)$ is always positive.
• $si{n}^2(x/2)\ge c_{\delta }>0$ for all $\delta \le |x|\le \pi$ and so, $F_N(x)\le 1/N{c}_{\delta }$ for such $x$. Which gives that $\underset{\delta \le |x|\le \pi }{\int }{F}_N(x)\to 0$ as $N\to \infty$.

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Theorem

If $f$ is integrable on the circle then the fourier series of $f$ is cesaro summable to $f$ at all points of continuity of $f$. If $f$ is continuous, then it is uniformly cesaro summable to $f$.

Proof:

This is a direct application of property (1) in Good Kernels.

Corollary

Continuous functions on the circle can be approximated by trigonometric polynomials.

• This is closely related to the Weierstrass Approximation Theorem.

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Related Problems

Dirichlet Kernels

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References

Good Kernels Cesaro Summability Dirichlet Kernels Weierstrass Approximation Theorem

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