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Questions tagged [fejer-kernel]

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

38 questions
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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 us to prove that the periodization of $\mathbf{F}_R(t)$ is…
Lost
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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 $\lim_{n \rightarrow \infty} \int f(t) g(nt) \, dt = \hat{f}(0)\hat{g}(0).$
usere5225321
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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: $\;\lim_{n\to \infty} \int_0^{2\pi} |f(t)-(S_n(f))(t)|^2 dt=0$. We notice that $\;\lim_{n\to…
user652838
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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 $\text{sinc}(x)=\frac{\sin x}{x}$ and the notation $f(T)\sim…
Naomi
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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 show that the series…
Lam18373
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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 $$D_N (x)= \sum_{n=-N}^N e^{inx} $$ I have no ideas of ways…
wondering1123
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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 $$F_N = \sum_{n = -N}^{N} (1 - \frac{|n|}{N})e_n$$ The author of the…
Francisco José Letterio
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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 $\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x-y)K_N(y)dy$ and $K_N$ is the $n$th order…
user780610
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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 $f\in L^1(G)\mapsto f\ast F_\nu\in L^1(G)$ is of finite rank for each…
Pedro Kaufmann
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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 the convolution of the Fejer Kernel and T. might be…
wondering1123
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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 with a period of $2\pi$. Prove: For every…
Um Shmum
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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 is that my example, next Question, has…
user243301
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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 \mathbb{N} : \forall n \ge N : \int_{[-\pi,\pi] \setminus…
StefanH
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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 for Fourier transforms? I mean, is it true that …
tommy1996q
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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 immediately defined as the average of the Dirichlet…
Eric Hansen
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