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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Fourier coefficients and uniform/pointwise convergence
Let $0\neq p\in R$ and let $f:R\to C$ a $2\pi$ period function given by $f(t)=e^{pt}$ for all $t\in (-\pi,\pi]$.
A. Find fourier coefficients of $f$.
B. Does the series of functions $(S_n(f))_{n=0}^{\...
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1answer
275 views
Fourier series: proving that the limit is zero
Let $f: \mathbb{R}\to \mathbb{C}$ be a $2\pi$ periodic function that satisfies:
$f(t)=\frac{1}{t^{\frac{1}{3}}}$ for every $t\in (0,2\pi]$.
Show that:
$\;\lim_{n\to \infty} \int_0^{2\pi} |f(t)-(S_n(f))...
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1answer
253 views
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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A question about approximate unit and Convolutions
Let $n\in Z_+$.
Define $U_n:=2K_{2n+1}-K_n$ where $(K_n)$ is Fejer kernel in $L^1(T)$.
a. Prove that $(U_n)_{n=0}^{\infty}$ is an approximate unit of $L^1(T)$.
b. Prove that $\hat {U_n}(k)=1$ for all $...
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Fourier coefficients of Cesaro means of partial fourier series
Let $S_nf(x) = \sum_{m = -n}^{n} \hat{f}(x)e^{2\pi imx}$ be the partial Fourier series of $f$. Let $\sigma_Nf(x) = \frac{1}{N}\sum_{n = 0}^{N-1} S_nf(x)$ be the Cesaro means of the $S_nf$.
What is $\...
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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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Prove that $ \sigma(f)(x_0)\rightarrow f(x_0)\ ( n\rightarrow +\infty )$
$f$ is a periodic function with period $2\pi$ and is integrable in $[-\pi, \pi]$. Suppose f is continuous at the point $x_0$. Prove that
$$
\sigma(f)(x_0)\rightarrow f(x_0), \quad n\rightarrow +\infty
...
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Integrating powers of the Fejer kernel
In attempting to solve a combinatorics problem (number of ways of rolling $2n$ $k$-sided dice and getting the median sum), I reduced things to finding the constant term the Laurent series for $(f(z)f(...
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Why does the Dirichlet Kernel $D_N$, the sum of exponentials gives $\frac{1}{2\pi}\int_{-\pi}^{\pi}D_N(x)~dx =1$?
Let the Dirichlet kernel be defined by
$$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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2answers
175 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
$$ F_n(x) = \frac{1}{n}\left(\frac{\sin(...
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1answer
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If $f$ is Riemann-integrable, $\int_{-\pi}^{\pi}|\sigma_Nf(x)-f(x)|dx \rightarrow 0$ as $N\rightarrow\infty$
Let $f$ be Riemann-integrable (not necessarily continuous). Prove $\int_{-\pi}^{\pi}|\sigma_Nf(x)-f(x)|dx \rightarrow 0$ as $N\rightarrow\infty$, where $\sigma_Nf(x)$ is $\frac{1}{2\pi}\int_{-\pi}^{\...
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1answer
66 views
A question about Fejer kernel
I'm reading something about the Fejer Kernel on the space $\mathbb{T}$.
Now yesterday I find this affirmation:
If $f\in\mathcal{L^1(\mathbb{T})}$ and $g\in\mathcal{L^\infty ({\mathbb{T})}}$ than: $...
1
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1answer
47 views
Convergence of Cesaro sums on $L^p$
Let $K_N=\frac{1}{N}\sum_{n=0}^{N-1}D_n(x)$ be the Fejer kernel and let $\sigma_N(f)=\frac{1}{N}\sum_{n=0}^{N-1}S_N(f)$ where $S_n(f)=\frac{1}{\pi}\int_{-\pi}^{\pi}f(\tau)D_n(t-\tau)d\tau=f*D_n.$ With ...
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Convergence on locally compact groups with an additional condition
This question concerns locally compact groups equipped with Haar measure, $(G,\lambda)$. For a class of such groups, there exists an approximate identity $F_\nu$ such that the map $f\in L^1(G)\mapsto ...
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1answer
138 views
showing property for the derivative $ \partial_x T$ of a trigonometric polynomial
Let be $$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) $$
Where $F_N \ast T$ meas ...
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1answer
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Sequence of arithmetic means of Dirichlet kernels
Let be $F_N$ the sequence of the arithmetic means of Dirichlet kernels $D_N (x)$ defined by
$$ 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
59 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
96 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(\...
3
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1answer
325 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 $$F_N = \sum_{n = -N}...
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1answer
42 views
A question bout Fourier coefficients
Let $\{a_n\}_n\in\mathbb{Z}$ be a sequence of complex numbers and let
$$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
134 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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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
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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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2answers
667 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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1answer
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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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1answer
489 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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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
752 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}.
$$
But in most of the reference material I come across online, it is ...
2
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1answer
483 views
Bounding Derivative of Fejer Kernel
On pages $116-117$ of Stein and Shakarchi's Fourier Analysis book, there is a claim about the Fejer Kernel
$$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
179 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
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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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2answers
981 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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2answers
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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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2answers
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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
$$
\forall \epsilon > 0, r < \pi : \exists N \in \...
5
votes
1answer
226 views
Fejer kernel applied to a measure
Let $\mu$ be a positive finite measure on $\mathbb R$. Is it true that
$$\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)=\...
