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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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How to recover the strong-type estimate from a weak-type estimate?

First, I should explain some notation. $A\lesssim B$ denotes the estimate $A\leq C_{\varepsilon}\delta^{-\varepsilon}B.$ If $A$ has polynomial size if $\delta^C\lesssim|A|\lesssim\delta^{-C}$ for some constant $C$. According to author, If I want to…
zyy
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Prove that Hardy-Littlewood maximal functions are lower semi-continuous.

When I was reading Singular Integrals and Related Topics by ShanHen Lu, Yong Ding and Duncan yan(ISBN-13:978-981-270-623-2), I noticed that in page 2, after the author proved that several Hardy-Littlewood maximal functions are pointsise equivalent…
xdyy
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$L_2$ space in harmonic analysis

Let ${v_n}_{n=1}^\infty$ be a closed orthonormal system in $L^2[0,1]$. We want to prove that for every $a\in[0,1]$, we have $$a=\sum_{n=1}^\infty \left|\int_0^a \overline{v_n(x)}dx\right|^2$$ where $\overline{v_n(x)}$ denotes the complex conjugate…
mayyas haj
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Two basic questions on Nyquist-Shannon Theorem for sinusoide

Consider the following sinusoid signal with frequency $F_0$: $$ x(t)=a\sin(2\pi F_0t) $$ If we sample $x(t)$ every $T$ seconds, we get the following discrete signal: $$ x[nT]=a\sin(2\pi F_0nT) $$ (a) Does Nyquist–Shannon sampling theorem claims that…
boaz
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convergence in $C^*(A)$ implies uniform convergence of the fourier transform

Let $A$ be a locally compact abelian group. Let $f_n$ be a sequence in $L^1(A)$ conveging to $f\in C^*(A)$ with the operator norm why does the fourier transform $\hat{f_n}$ converge to $\hat{f}$ uniformly in $\hat{A}$? that is, why do we have…
Maclio
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Hilbert transform and maximal function

What is the relation between the Hilbert transform (in $\mathbb{R}$) and maximal functions?
Anonymous999
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Non non probability Haar measure on locally compact group

Suppose that $G$ is a locally compact group, which is not compact. I know that in general there is no way to "normalize" the Haar measure on $G$. But can we say, in general, that the Haar measure is not a probability measure on $G$? Are there…
esteban
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How is $\frac{|B(x,2\delta)|}{|B(y,\delta)|} = 2^n$?

In an answer to this question a user writes, $\frac{|B(x,2\delta)|}{|B(y,\delta)|} = 2^n$. I am not sure how the user arrived at this. Can someone kindly help me ? Thanks
gaufler
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A_2 muckenhoupt weights

My question is easy. Does A_2 muckenhoupt condition hold even if the weight $\omega$ is zero on some sets? What is the condition in this case? I did not find a clear reference or explanation about this...
C94
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A question about the limit of a sequence of uniformly bounded armonic functions

Given an open set $\Omega \subseteq \mathbb{R}^n$, a sequence of armonic functions $\{u_n\} \subseteq \Omega$ that are uniformly bounded on every compact set of $\Omega$ and that are supposed to reach almost everywhere a limit function $u$, what can…
Gabrielek
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canonical orthonormal basis of a finite dimensional vector space

currently I am reading Ole Christensen's An Introduction to Frames and Riesz Bases. what is canonical orthonormal basis of a finite dimensional vector space and how to calculate . thanks in advance.
Shankhadeep Mondal
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Sine wave harmonics, sawtooth waveform modified

I am trying to find the formula to generate the waveform below. By using harmonics on standard sine waves and then combining the outcomes, I have managed to generate, triangle, sawtooth and square waveforms but I am having great difficulty…
user9731
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Boundedness of Poisson Integral

I'm going crazy with this: $\mathbb{H}$ is the upper half plane. $f\in L^p(\mathbb{R})$, $u(z)$ is the poisson integral of $f$. $u(z)=(P_y*f)(x)$ with $z=(x,y)\in\mathbb{H}$ and $P_y(x)=\frac{y}{\pi(x^2+y^2)}$ the poisson kernel. $\mu$ a positive…
Davide Vecchi
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Which spherical harmonic function will correspond to such a representation?

On Wiki there's a figure displaying "visual representations of the first few spherical harmonics." I was wondering exactly which spherical harmonic function will generate a representation like this (red means positive, green means negative; I don't…
Mengliu
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Understanding the proof of the Hausdorff-Young theorem

I'm having some trouble understanding Terence Tao's proof of the Hausdorff-Young theorem in his lectures notes 1. The theorem states Proposition 3.1. Let $B$ denote the open unit ball in $\mathbb R^n$ and let $1\leq p,q\leq\infty$ such that $\|\hat…
Cubi73
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