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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 $\mathbb{R}^n$.

2797 questions
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Fourier transform of a a function in the space

Which is the Fourier transform (in the sense of distributions) of the function $f(x)=x/\|x\|^n $, where $x$ belongs to the Euclidean space $ R^n$?
user47005
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Uncertainty principle in harmonic analysis

Given a function $f$ (in some suitable (EDIT: nice) function space) supported on a ball $B\subset\mathbb{R}^n$ of radius $R>0$, I have commonly heard people, who understand these things, say stuff like "the Fourier transform $\hat{f}$ is morally…
Anonymous999
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Prove that the Fourier transform of a function agreeing with $1/\log(|x|)$ for $|x| \geq 2$ is integrable.

Let $f \in C^\infty(\mathbf{R})$ be an even, smooth function such that for $|x| \geq 2$, $f(x) = 1/\text{log}|x|$. I am trying to show that $\widehat{f}$ is integrable. This must use the fact that the function is even, because odd functions agreeing…
Jacob Denson
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Approximating a Lipschitz function by spherical harmonics

Let $f:\mathbb{S}^n\to \mathbb{R}$ be a Lipschitz function (i.e. so $$ \Vert f\Vert_{L}=\sup_{x\in \mathbb{S}^n} |f(x)|+\sup_{x\neq y\in \mathbb{S}^n} \frac{|f(x)-f(y)|}{d_{\mathbb{S}^n}(x,y)}<\infty. $$ Can one find a sequence $S_N(f)$ consisting…
RBega2
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Uniqueness of Hilbert transform as an operators from Schwartz class to bounded continuous functions on $R$. (Terry tao notes exercise)

Let $T : S(R) \rightarrow C_b(R)$ be a continuous linear operator which maps Schwartz functions to bounded continuous, and which commutes with translations and dilations. Show that T is a linear multiple of the Hilbert transform and the identity.…
user95731
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What kind of $\mathcal{L}^1$ function has bounded Hilbert transform?

It is known that if a function $f \in \mathcal{L}^1 $, then its Hilbert transform may not map it to the $\mathcal{L}^1$ space. So my question is under what kind of condition, the Hilbert transform of $f$ can still be in the $\mathcal{L}^1$ space?
Claire Fan
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Bound for the quasinorm of sum of functions

Let $\|\cdot \|$ be a quasinorm on functions, thus $\|cf\|=|c|\|f\|$ for scalar $c$, $\|f\|=0$ if and only if $f=0$, and we have the quasitriangle inequality $$\|f+g\|\lesssim\|f\|+\|g\|\ \ \ \ \ (1)$$ for all functions $f,g$. Let $f_n$,…
Xiang Yu
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Question on Schwarz space why is it $\sup_{x\in \mathbb R^n}|(1+|x|^N)\partial _x^\alpha f(x)|<\infty $

Why Schwarz space is given by the set of $f\in \mathcal C^\infty (\mathbb R^n)$ s.t. $$\sup_{x\in \mathbb R^n}|(1+|x|^N)\partial _x^\alpha f(x)|<\infty ,$$ where $\alpha \in \mathbb N^n$ and $N\in\mathbb N$. To have $$\sup_{x\in \mathbb…
MSE
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$L^2$ norm for difference of translation of disjoint interval

I just need some hint since I've been stuck in this for several hours.. Let $I_1,I_2,\ldots,I_K$ bne disjoint intervals in $[-1/2,1/2)$,and $f(x)=\sum_{j=1}^K\chi_{I_j}$, where $\chi_I(x)$ is the character function that takes $1$ when $x\in I$ and…
Golbez
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Left and Right Discrete Maximal Functions

Define the uncentered maximal function $$\widetilde{M}f(n)=\sup_{s,r\in\mathbb{Z}^{+}}\dfrac{1}{s+r+1}\sum_{k=-r}^{k=s}\left|f(n+k)\right|,$$ where $\mathbb{Z}^{+}=\left\{0,1,2,\ldots\right\}$. Define the left and right maximal functions $M_{L}f$…
Matt Rosenzweig
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Determine an operator is weak type (1,1) by its Fourier multiplier

I have an interest on associated Fourier multipler $m$ for a given operator $T$ defined for $f\in L^p(R^d)$, $1\le p<\infty$ by $$ \hat{Tf}(\xi)=m(\xi)\tilde{f}(\xi),\ \ \xi\in R^d $$ where $\hat{f}$ is the Fourier transform of $f$. There are some…
linrr
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Show that two series are equal

In my harmonic analysis class I have to prove that for all $a>0$ the following equality holds: $$\sum_{n\in \mathbb{Z}}e^{-n^2\pi a}=\frac{1}{\sqrt{a}}\sum_{k\in \mathbb{Z}}e^{-k^2 \pi / a}.$$ I'd appreciate any help on how to approach this problem…
Ludolila
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Definition of Left Translation of a function on a topological group

In Folland's A Course in Abstract Harmonic Analysis, he defines for a function $f$ on a topological group $G$, and $y\in G$, $$L_{y}f:x\to f(y^{-1}x)\text{ and }R_{y}f:x\to f(xy)$$ He then remarks that the $y$ is inverted so that the map $y\mapsto…
roo
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A question about BMO

In $\mathbb{R}^n$, suppose $f \in \mathrm{BMO}$ and $\phi \in \mathrm{C}_0^\infty$, then can we show that the convolution $f * \phi \in \mathrm{BMO}$?
van abel
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Grafakos Classical Fourier Analysis - Explanation of Theorem 2.3.20

In one of the steps for the proof of Theorem 2.3.20, Grafakos shows that you can pass a continuous, linear functional through an integral by considering the convergence (in the topology of the Schwartz space on $\mathbb{R}^n$) of the Riemann sum of…
opaladozen
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