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Questions tagged [harmonic-analysis]

Harmonic analysis is the generalisation of Fourier analysis. Use this tag for analysis on locally compact groups (e.g. Pontryagin duality), eigenvalues of the Laplacian on compact manifolds or graphs, and the abstract study of Fourier transform on Euclidean spaces (singular integrals, Littlewood-Paley theory, etc). Use the (wavelets) tag for questions on wavelets, and the (fourier-analysis) for more elementary topics in Fourier theory.

Harmonic analysis can be considered a generalisation of Fourier analysis that has interactions with fields as diverse as isoperimetric inequalities, manifolds and number theory.

Abstract harmonic analysis is concerned with the study of harmonic analysis on topological groups, whereas Euclidean harmonic analysis deals with the study of the properties of the Fourier transform on $\mathbb{R}^n$.

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anton deitmar harmonic analysis exercise

Let P be the group of upper triangular matrices in SL2(R). The injection η : P → GL2(C) can be viewed as a representation on V = C2. Show that η is not the sum of irreducible representations.
salome saberi
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If $u$ subharmonic, why $U_s=\{x\mid u(x)=s\}$ is open?

Let $U$ a domain and $u\in \mathcal C^2(U)\cap \mathcal C^1(\bar U)$ and let $s=\sup_{U}u$. Let $U_s=\{x\mid u(x)=s\}$. I want to prove that $U$ is clopen. I proved that it's closed (because $U=u^{-1}\{s\}$ and $U$ continuous), but I don't…
user657607
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Is it possible to find a superharmonic function with such conditions

$\lim_{z\rightarrow 0}\psi(z) = 0$, $\lim_{z\rightarrow w}\psi(z) = 1$ for $\{w: |w|=r, w\neq r\}$ and $\psi$ is superharmonic on $B(0;r)\setminus [0,\infty)$? In the context I use, a superharmonic function has to have super mean value property.…
MonkeyKing
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How to prove $g(x,y) = \int_{0}^{2\pi} u(e^{i\theta}(x+iy)) sin(\theta)d\theta$ is a polynomial if $u$ is harmonic?

I don't know any test which would prove $g$ is a polynomial. I assume $u$ to be a holomorphic function, wrote the power series expansion, and separated out the $z$ term, and I am left with $\sum_{1}^{\infty}a_mz^m (\int_{0}^{2\pi}…
ManishKumar Singh
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Convergence of BMO norm

I am studying Stein's Functional Analysis. I worked for a long time but cannot check the following proposition: Let $f$ be a real valued BMO function, define $f^k$, the truncation of $f$ by $$f^k(x)= \begin{cases} f(x) &\text{if }|f|
J.Y Lee
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Littlewood-Paley decomposition in Tao's note

I am reading Tao's lecture note on Littlewood-Paley decomposition. Let $\phi(\xi)$ be a bump function supported on $\{\xi\in\mathbb{R}^n:|\xi|\leq 2\}$ and equals $1$ on $\{\xi\in\mathbb{R}^n:|\xi|\leq 1\}$. Let $\varphi(\xi)=\phi(\xi)-\phi(2\xi)$,…
89085731
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Does every continous function in L1 is a fourier transform?

Let $f: \mathbb{R}\rightarrow\mathbb{C}$ denote a continous function in $L_1(\mathbb{R})$. Does this imply that there exists a function $G: \mathbb{R}\rightarrow\mathbb{C}$ in $L_1(\mathbb{R})$ such that $f$ is $G$'s fourier transform?
Gal
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Muckenhoupt weights and maximal function

It is well known that $w \in A_p(\mathbb{R}^n)$ (the Muckenhoupt weight class) if and only $$\int_{\mathbb{R}^n} \mathcal{M} (|f|)^p w(x) \ dx \leq C\int_{\mathbb{R}^n} |f|^p w(x) \ dx.$$ My question is if one takes $M_{0$, the…
Adi
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A topological group and inner invariant group

A topological group G is called inner invariant group if there is a compact neighborhood $U$ of $e$ with ‎$‎ xUx‎^{-1} ‎\subseteq ‎U‎$ ‎for ‎‎$‎x\in G‎$‎. show that discrete groups, compact group, and abelian group are inner invariant group.
user356319
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Image under Hilbert Transform in $L^1$

I have a question concerning the proof of following proposition: Proposition: Let $\phi\in S(\mathbb{R})$ be given. Then $H\phi \in L^1(\mathbb{R})$ if and only if $\int_{\mathbb{R}}\phi(x)dx=0$. Where H is the Hilbert Transform…
MorganeMaPh
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What does $f$ continue on $\mathbb S^1$ mean?

Let $\mathbb S^1:=\mathbb R/\mathbb Z$. What does $f$ continuous on $\mathbb S^1$ mean ? That it's continuous over $[0,1)$ or $[0,1]$ ? I would say $[0,1)$ but I have doubt since we sometimes take the norm $\|f\|_{L^\infty }$ and if $\lim_{x\to…
user349449
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If $f\in S(\mathbb R^n)$ (schwarz space), why $f\in L^p(\mathbb R^n)$?

Let $$\mathcal S(\mathbb R^n)=\left\{f\in \mathcal C^\infty (\mathbb R^n)\mid \forall N\in\mathbb N,\forall \alpha \in\mathbb N^n, \sup_{x\in\mathbb R^n}|(1+|x|^N)\partial _x^\alpha f(x)|<\infty \right\},$$ the schwarz space where $\partial…
idm
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Find $f\in C^0(S^1)$ that satisfy $\limsup_{n\to \infty }\|S_nf -f\|_{L^\infty }>0$

I have to construct a continuous function $f\in \mathcal C^0(S^1)$ (where $S^1=\mathbb R/\mathbb Z$) that satisfy $$\limsup_{n\to \infty }\|S_n f-f\|>0$$ where $S_nf$ is the $n-$th Fourier partial series. I absolutely don't know how to do (I in fact…
user301068
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problem on finding representations

i am currently pursuing a course in basic harmonic analysis.i have gone through a lot of texts but i am really finding difficulty to proceed in the following problem: (1) find a representation of $\Bbb Z_2$ on $\Bbb R^2$ i have no clue how to…
Abhishek Shrivastava
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The support of $f(x)= \cos(x)$

The support of a function is the closure of the set of points where the function has non zero values. The function $f(x)=\cos(x)$ is zero only at the points $x=\frac{(2k+1)\pi}{2}$, $k \in \mathbb{Z}$ So the support of $f$ is the set $\{ x \in…
Al jabra
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