# Questions tagged [fejer-kernel]

A Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series.

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### Fourier transform/coefficients of Fejer kernel

Consider the Fejer kernel given by $$F_n(x) := \frac{1}{n}\left(\frac{1-\cos(n x)}{1 - \cos(x)}\right).$$ for $n \in \mathbb{N}$, which is a $2\pi$-periodic function. What is the Fourier transform (...
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### Some identities regarding fourier transform

Let $f\in L^1(R)$. a. Assume that $\hat{f}\geq 0$. Show that $\hat{f}\in L^1(R)$ and $Lim_{\lambda\to \infty} K_{\lambda} *f= \frac{1}{2\pi} (\hat{f}\check)$ uniformly. Wher $K_{\lambda}$ is the fejer ...
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### Showing that a given sequence is an approximate identity!

We know that Fejer Kernel: $(K_n)_{n=0}^{\infty}$ is an approximate identity of $L^1(T)$. $K_n=\sum_{k=-n}^{n} \left(1-\frac{|k|}{n+1}\right)e_k$ , ($n\in Z_+$). I am trying to use this in order to ...
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### summation methode of Fejér

let's define $f:\mathbb{R} \to \mathbb{C}$ a $2\pi$ periodic continuous function. Let's say $s_{n}(x)=\sum_{k=-n}^{n} f\hat (k) e^{ikx}$. Let's say $t_{n}(x)=\frac{1}{n+1}\sum_{k=0}^{n} s_{k}(x)$. ...
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### Third condition in definition of summability kernel?

https://en.wikipedia.org/wiki/Summability_kernel The third condition of the definition involving $\delta$, I don't understand what it is trying to say. Why would the integral go to 0, when condition ...
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### Poisson summation formula clarification regarding Fejer kernel

Define $$\mathbf{F}_R(t) = \begin{cases} R \left(\dfrac{\sin(\pi R t)}{\pi R t}\right)^2 & t \neq 0\\[10pt] R & t = 0 \end{cases}$$ A problem in Stein's Fourier Analysis asks ...
Statement: Given the Fejer Kernel $F_n(x) = \frac{1}{n}\bigg(\frac{\sin(\frac{nx}{2})}{\sin(\frac{x}{2})}\bigg)^2$. Show that $F_n(x)$ is unbounded for $x=0$ as $n\rightarrow \infty$