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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What is the Fourier transform of spherical harmonics?

What is the definition (or some sources) of the Fourier transform of spherical harmonics?
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Suppose $\phi$ is a weak solution of $\Delta \phi = f \in \mathcal{H}^1$. Then $\phi\in W^{2,1}$

I'm trying to prove the statement in the title in as simple a way as possible. It is Theorem 3.2.9 in Helein's book "Harmonic maps, conservation laws, and moving frames", although it is not proved there. The statement is as follows. Suppose…
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Plancherel formula for compact groups from Peter-Weyl Theorem

I'm trying to derive the following Plancherel formula: $$\|f\|^{2}=\sum_{\xi\in\widehat{G}}{\dim(V_{\xi})\|\widehat{f}(\xi)\|^{2}}$$ from the statement of the Peter-Weyl Theorem as given by Terence Tao here: Let $G$ be a compact group. Then the…
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Prove that $|k(x)|\le C|x|^{-n}$ under suitable hypothesis on $k\in\mathcal{C}^1(\Bbb R^n\setminus\{0\})$

DON'T BE AFRAID FROM THE +500 BOUNTY: it doesn't matter that I KNOW this problem is really hard, I put it only because I need to solve the problem really URGENTLY! Let $n\ge2$; given a kernel $k\in\mathcal{C}^1(\Bbb R^n\setminus\{0\})$ such…
Joe
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norm of a singular integral operator

My question is from Harmonic Analysis, about the study of singular kernels (in the Calderon Zygmund sense.) Suppose that a kernel $K$ is a singular kernel, extending to a bounded operator on $L^p$, for any $1\leq p<\infty$. On $L^2$ it is easy to…
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Extension of character in Banach algebras

Let $A$ be a Banach algebra. The continuous linear functional $\phi:A\to\Bbb{C}$ is called character if it is non-zero multiplicative function i.e., for every $a,b\in A$ we have $\phi(ab)=\phi(a)\phi(b)$. The set of all character is shown by…
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Stone-Weierstrass implies Fourier expansion

To prove the existence of Fourier expansion, I have to solve the following exercise, which supposedly follows from the Stone-Weierstrass theorem: Let $G$ be a compact abelian topological group with Haar measure $m$. Let $\hat G$ be the dual. The…
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Wiener's theorem in $\mathbb{R}^n$

Reading Stein's "Singular integrals and differentiability properties of functions" I came across the following statement (this is in the proof of Lemma 3.2, pages 133-134): We now invoke the $n$-dimensional version of Wiener's theorem, to wit: If…
i like xkcd
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About the Gibbs phenomenon for the Legendre polynomials as an orthogonal base of $L^2(-1,1)$

In a recent question, I proved that the Fourier-Legendre expansion of the function $f(x)=\text{sign}(x)$ over $(-1,1)$ is given by: $$2\sum_{m\geq 0}\frac{4m+3}{4m+4}\cdot\frac{(-1)^m}{4^m}\binom{2m}{m}\cdot P_{2m+1}(x)\tag{1}$$ with $P_n(x)$ being…
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Why unitary characters for the dual group in Pontryagin duality if $G$ is not compact?

In harmonic analysis, for any locally compact abelian group, one constructs the dual group as the group of homomorphisms into the unit circle with the compact open topology. In other words, unitary linear characters of the group. We know from…
ziggurism
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What are the "right" spaces for the Laplace transform

There are for example several canonical spaces to define the Fourier transform (i.e. Schwartz's space). Is there also a particularly suitable space to define the Laplace transform, so that the Laplace transform is at least bijective?
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Kakeya Needle problem video

I'm intruiged by the Kakeya Needle problem, described here on Wikipedia. Wikipedia has a nice animation of a needle turning through a hypo-cycloid: What I'm searching for is a visualisation of the same process but on a Besicovitch set…
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Mean Value Property of Harmonic Function on a Square

A friend of mine presented me the following problem a couple days ago: Let $S$ in $\mathbb{R}^2$ be a square and $u$ a continuous harmonic function on the closure of $S$. Show that the average of $u$ over the perimeter of $S$ is equal to the…
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Prove the following beta integral identity

I am looking at the following exercise from Grafakos' classical fourier analysis. For $0 < \alpha_1, \alpha_2 < n, \alpha_1 + \alpha_2 > n$, I want to prove that $$\int_{\mathbb{R}^n} \frac{1}{\left| x-t \right|^{\alpha_1} \left| y -t…
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