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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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The equivalence of the two definitions of fractional Laplacian

Using the Fourier transform we can easily define the fractional Laplacian by $$(-\Delta)^{s/2}f(x)=(|\xi|^s\hat f(\xi))^\vee(x), \ \ f\in C_0^\infty. $$ However, I learned that there is another definition using the principal value of singular…
Mr.right
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A Schwartz function problem

Let f be a strictly positive Schwartz function on $\mathbb R$. Does it imply $\sqrt f$ is a Schwartz function on $\mathbb R$?
users31526
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Is every integrable function on the real line with compact support also square integrable?

I wonder that whether every integrable function on the real line with compact support is also square integrable ? In other words, is $L^1_c(\mathbb R)\subseteq L^2(\mathbb R)$ holds true? Thanks in advance for any hints.
linrr
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Left Haar Measure on the Borel subgroup of the general linear group

If we consider the group of upper triangular matrices $B=\bigl(\begin{smallmatrix} a&b\\ 0&a^{-1} \end{smallmatrix} \bigr)$ where $a$ and $b$ are either real or complex and $a\neq1$, then the left Haar measure is given by $a^{-2}\,da \,db$. While I…
user34748
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non tangential maximal function and Hardy-Littlewood maximal function

I'm studying harmonic analysis and found that we can bound non-tangential maximal function by Hardy-Littlewood maximal function. Most books don't give the proof of it. How can I see that? Is there a proof of it using dyadic decomposition? It…
Danny
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question regarding Fourier restriction estimates

Thanks for reading my post. I am trying to prove the following claim: If we have \begin{equation*} \left\|\hat{f}\right\|_{L^q(N_{1/R}(S))}\lesssim R^{\alpha-1/q}\left\|f\right\|_{L^p(B(0,R))} \end{equation*} then we have \begin{equation*} …
Peng
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BMO and truncation

I'm trying to solve Exercise 7.1.4 of Grafakos' book $\textit{Modern Fourier Analysis}$: consider two real numbers $K
David Tewodrose
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Dual group of $\mathbb Z$

We know $\hat{\mathbb Z}=\mathbb T$ and the map $\alpha\longmapsto\chi_{\alpha}$ is an isomorphism of $\mathbb T$ on to the character group of $\mathbb Z$, but I can't prove this map is continuous? I don't know what topology on $\hat{\mathbb Z}$ …
user244115
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Help on understanding Schwartz space

Can someone give an example of Schwartz space function that doesn't decay exponentially?
SOM
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The extension of Fourier transform from $L^1\cap L^2$ to $L^2$

In Grafakos's book, in order to the define Fourier transform on $L^2$, he claims that the Fourier transform is an isometry on $L^1\cap L^2$. Since Fourier transform is an isometry, thus $\hat{f}\in L^2$. However, I couldn't figure out why…
89085731
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The comparison of two Hardy-Littlewood maximal function

The uncentered Hardy-Littlewood maximal function of $f$ is $$\tilde{M}(f)=\sup\limits_{|y-x|<\delta}\frac{1}{|B(y,\delta)|}\int_{B(y,\delta)}|f|$$ If we denote the original Hardy-Littlewood maximal function as $M(f)$. In Grafakos's book, he states…
89085731
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Computation in harmonic analysis

I have a very precise question concerning p. 82-83 of Stein's book "Singular integrals and differentiability properties of functions". Actually it is a calculation problem. For $f \in L^{2}(\mathbb{R}^n)$, denote by $u(x,y)$ the Poisson integral of…
David Tewodrose
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the convolution of integrable functions is continuous?

I just look through a webpage from "mathoverfolw", on https://mathoverflow.net/questions/136681/the-convolution-of-integrable-functions-is-continuous Let me remark that it is sufficient that one of the functions is bounded (the convolution of an…
David Chan
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convolution of $f$ and $g$ is in $L^p$ where $f$ has compact support.

Suppose 1≤p≤∞ and $f∈L^1(G)$ and $g∈L^p(G)$. I know this is true $‖f*g‖_p ≤‖f‖_1 ‖g‖_q$ , Where ∗ is convolution of f and g. I want to prove that If G is not unimodular, we still have g∗f is in LP when f has compact support. proof: By definition…
user147972
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Laplace operator and Fourier transform

Let $f$ be a function with compact support in $D\subset \mathbf{R}^2$. Is is known that $$-\Delta f=F^{-1}|x|^2 F f,$$ where $F$ is the Fourier transform. Which is the $n-$dimensional analogous formula?
user47005
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