Questions tagged [fejer-kernel]

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Integrating powers of the Fejer kernel

In attempting to solve a combinatorics problem (number of ways of rolling $2n$ $k$-sided dice and getting the median sum), I reduced things to finding the constant term the Laurent series for $(f(z)f(...
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36 views

Why does the Dirichlet Kernel $D_N$, the sum of exponentials gives $\frac{1}{2\pi}\int_{-\pi}^{\pi}D_N(x)~dx =1$?

Let the Dirichlet kernel be defined by $$D_N(x) = \sum_{n=-N}^{N}e^{inx}$$ I found a proposition that says that for an integer $j$, $$\int_{0}^{1}e^{i2\pi jx}dx = \begin{cases} 1, ~~~j=0\\0,~~~j\...
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43 views

Show that the Fejér Kernel $F_n$ is a good kernel and $\int_{0}^{1}F_n(x)dx = 1$

Given that the Fejér Kernel is defined as the Cesaro sum of the $k$-th Dirichlet kernels from wikipedia, it also notes that the closed form can be written as $$ F_n(x) = \frac{1}{n}\left(\frac{\sin(...
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40 views

If $f$ is Riemann-integrable, $\int_{-\pi}^{\pi}|\sigma_Nf(x)-f(x)|dx \rightarrow 0$ as $N\rightarrow\infty$

Let $f$ be Riemann-integrable (not necessarily continuous). Prove $\int_{-\pi}^{\pi}|\sigma_Nf(x)-f(x)|dx \rightarrow 0$ as $N\rightarrow\infty$, where $\sigma_Nf(x)$ is $\frac{1}{2\pi}\int_{-\pi}^{\...
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1answer
33 views

A question about Fejer kernel

I'm reading something about the Fejer Kernel on the space $\mathbb{T}$. Now yesterday I find this affirmation: If $f\in\mathcal{L^1(\mathbb{T})}$ and $g\in\mathcal{L^\infty ({\mathbb{T})}}$ than: $...
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About the notion of summability kernel in $L^1(\mathbb{T}^2)$

We say that a sequence $(k_n)\subset L^1(\mathbb{T})$ is a summability kernel if the following properties are satisfied. $\bullet$ $\displaystyle\int_{-\pi}^\pi k_n(t) dt=1$. $\bullet$ $\...
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1answer
33 views

Convergence of Cesaro sums on $L^p$

Let $K_N=\frac{1}{N}\sum_{n=0}^{N-1}D_n(x)$ be the Fejer kernel and let $\sigma_N(f)=\frac{1}{N}\sum_{n=0}^{N-1}S_N(f)$ where $S_n(f)=\frac{1}{\pi}\int_{-\pi}^{\pi}f(\tau)D_n(t-\tau)d\tau=f*D_n.$ With ...
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22 views

Convergence on locally compact groups with an additional condition

This question concerns locally compact groups equipped with Haar measure, $(G,\lambda)$. For a class of such groups, there exists an approximate identity $F_\nu$ such that the map $f\in L^1(G)\mapsto ...
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1answer
112 views

showing property for the derivative $ \partial_x T$ of a trigonometric polynomial

Let be $$T= \sum_{n=0}^N \hat{T} (n)e^{inx} $$ a trigonometric polynomial of grade $N$ without negative frequencies. I wanna show that $$ \partial_x T= -iN(F_N \ast T-T) $$ Where $F_N \ast T$ meas ...
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38 views

Sequence of arithmetic means of Dirichlet kernels

Let be $F_N$ the sequence of the arithmetic means of Dirichlet kernels $D_N (x)$ defined by $$ F_N := \frac{1}{N+1} (D_0 (x) +D_1 (x)+..+D_N(x)) $$ Where the Dirichlet kernel is defined by $$D_N (x)= ...
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1answer
48 views

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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1answer
58 views

Inversion theorem for Fourier Series

In section 8.4 of Folland's Real Analysis it's stated that if $f \in L^1(\mathbb{T}^n)$ has a Fourier series $\widehat f \in l^1(\mathbb{Z}^n)$, then the Fourier series $\sum_{\kappa} \widehat f(\...
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1answer
118 views

Why does this identity hold for Fejér Kernels?

I'm trying to read a proof for the existence of an $(\epsilon , \delta)$ approximation to the identity that is a trigonometric polynomial. For this, the Fejér Kernel is defined as $$F_N = \sum_{n = -N}...
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41 views

A question bout Fourier coefficients

Let $\{a_n\}_n\in\mathbb{Z}$ be a sequence of complex numbers and let $$p_n(e^{it})=\sum_{k=-n}^{n}(1-\frac{|k|}{n+1})a_ke^{ikt},\;\;\;\;(e^{it}\in\mathbb{T}).$$ Let $1\leq p\leq\infty$. Suppose that ...
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1answer
112 views

Regarding convolution of Fejér kernel with a Lipschitz function

Let $\mathbb{T}$ be the unit circle in the complex plane. The Lipschitz class of functions on $\mathbb{T}$ is defined here. And the Fejér kernel is defined here. If $f\in Lip_{\alpha}(\mathbb{T}),\, 0&...
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227 views

Prove Weierstrass Theorem using Fejer Theorem

Using the following theorem: The trigonometric polynomials ($\mathbb{C} \to \mathbb{C}$) are uniformly dense in $C(\mathbb{T})$ (functions $\mathbb{C} \to \mathbb{C}$ that are continuous and periodic ...
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1answer
63 views

Evaluation of convolutions by Fejér kernels have finite-dimentional range?

Consider $\mathbb{R}^N = (\mathbb{R}^N,\|\cdot\|_\infty)$ with $N\in\mathbb{N}$. Let $\mathrm{Lip}_0(\mathbb{R}^N, \mathbb{R})$, or just $\mathrm{Lip}_0(\mathbb{R}^N)$, be the Banach space of real-...
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503 views

Proof of Fejer's lemma

How does ont prove Fejer's lemma: If $f \in L^1(\mathbb{T})$ and $g\in L^\infty(\mathbb{T})$, then $\lim_{n \rightarrow \infty} \int f(t) g(nt) \, dt = \hat{f}(0)\hat{g}(0).$
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102 views

Real analysis and convergence versus $\int_0^1\sum_{n=1}^\infty x^2\mu(n)\frac{1-\cos(nx)}{n(1-\cos(x))}dx$, where $\mu(n)$ is the Möbius function

This morning I was playing (using Wolfram Alpha online calculator) with series involving the Möbius function $\mu(n)$ and the so-called Fejér kernel, see if you need it this Wikipedia. My conclusion ...
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354 views

Fejer's theorem for Fourier transforms of $L^1(\mathbb{R})$ functions

I know there is a version of Fejer's theorem stating:"If $f$ is a function in $L^1(\mathbb{(- \pi, \pi)})$ then its Fejer's sums converge to $f$ in $L^1$ norm". The question is: is this still true ...
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84 views

Is there a closed form for the “positive summands” of the fejer kernel?

I've come across the Fejer kernel $K_N(t) = \sum_{k = -N}^N (1- \frac{\vert k \vert}{N+1}) e^{ikt}$. Now, in my application, I only need the positive part, that is $K_n^{+}(t) =\sum_{k=0}^N (1- \frac{...
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2answers
645 views

How do I prove my Fejer kernel definition is equivalent?

In my notes, the definition of the Fejer kernel is $$ F_{n} = \sum_{j=-N}^{N} \left(1 - \frac{|j|}{N+1}\right) e^{ijt}. $$ But in most of the reference material I come across online, it is ...
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1answer
382 views

Bounding Derivative of Fejer Kernel

On pages $116-117$ of Stein and Shakarchi's Fourier Analysis book, there is a claim about the Fejer Kernel $$F_N(t) = \frac{1}{N} \frac{\sin^2(Nt/2)}{\sin^2(t/2)}$$ and the claim is $$|F_N'(t)| \...
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1answer
164 views

Prove an equality involving Fejer Kernel

Define $$V_n(\theta) = \sum_{k=-n-1}^{n+1} e^{in\theta} + \sum_{k=n+2}^{2n+1}\frac{2n+2-k}{n+1}(e^{ik\theta}+e^{-ik\theta})$$ I want to show that $$V_n(\theta) = 2K_{2n+1}(\theta) - K_n(\theta)$$ ...
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1answer
1k views

Properties of the Fejer kernel

The Fejer kernel $k_m : \mathbb{R} \to \mathbb{C}$ is defined by $k_m (t) = \frac{1}{2\pi (m+1)} \sum^m_{n=0} \sum^n _{k=-n} e^{ikt}$ One of the properties of the Fejer kernel is For any $\delta \...
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863 views

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 ...
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169 views

Fejer Kernel is Unbounded

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 $n\rightarrow \infty$
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849 views

Property of Fejer kernel

Let $$ F_n(x) = \frac{1}{n} \left( \frac{ \sin(\frac{1}{2} n x ) } { \sin(\frac{1}{2} x ) } \right)^2 $$ be the n-th Fejer-Kernel. Then $$ \forall \epsilon > 0, r < \pi : \exists N \in \...
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1answer
204 views

Fejer kernel applied to a measure

Let $\mu$ be a positive finite measure on $\mathbb R$. Is it true that $$\int_{\mathbb R} T \text{sinc}^2(Tx) d\mu(x) \sim\frac{\mu([-1/T,1/T])}{1/T}, \text{ as } T\to\infty?$$ Here $\text{sinc}(x)=\...