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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Do discontinuous harmonic functions exist?

A function, $u$, on $\mathbb R^n$ is normally said to be harmonic if $\Delta u=0$, where $\Delta$ is the Laplacian operator $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$. So obviously, according to this definition, $u$ must be twice…
Cheerful Parsnip
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Existence of orthonormal basis for $L^2(G)$ in $C_c(G)$.

Suppose that $G$ is a locally compact (Hausdorff) group endowed with the Haar measure. It is well-known that the compactly supported functions $C_c(G)$ are dense in $L^2(G)$. In the book "Operator algebras, theory of $C^{*}$-algebras and von…
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What does the term "regularity" mean?

When I was an undergraduate, I took a course on regularity theory for nonlinear elliptic systems. This included topics such as the direct method of calculus of variations, mollifiers, integration over distributions, and the Calderón-Zygmund…
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Theorem of Steinhaus

The Steinhaus theorem says that if a set $A \subset \mathbb R^n$ is of positive inner Lebesgue measure then $\operatorname{int}{(A+A)} \neq \emptyset$. Is it true that also $\operatorname{int}{(tA+(1-t)A)} \neq \emptyset$ for $t \in(0,1)$? It is…
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Simple proof that Fourier transform is an isomorphism between $L^p$ spaces for $p \neq 2$?

It is known that the Fourier transform $\mathcal F$ maps $L^2 \to L^2$ as an (isometric) isomorphism and $L^1 \to L^\infty$ as bounded operator. Via Riesz-Thorin this result can be extended to give that $\mathcal F$ also maps $L^p \to L^{p'}$ where…
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Are Sobolev spaces $W^{k,1}(\mathbb R^d)$ and $H^{k,1}(\mathbb R^d)$ the same?

We consider the following spaces $H^{k,p}(\mathbb R^d)$, $k \geq 1$ is integer, $p \geq 1$ (Bessel potential spaces): $$ H^{k,p}(\mathbb R^d) = \bigl\{ f \in L^p(\mathbb R^d) \colon \mathcal F^{-1}[(1+|\xi|^2)^{\frac k 2} \mathcal F f] \in…
Appliqué
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Derivation into dense ideal of Banach algebras

Let $A$ be a Banach algebra and $I$ be an ideal of $A$. A derivation $D\colon A\to I$ is a linear bounded map, with the following property: $$D(ab)=aD(b)+D(a)b,\qquad a,b\in A.$$ Suppose that $I$ is dense in $A$, and any derivation $D\colon A\to I$…
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What is difference between Fourier Transform and Fast Fourier Transform?

If you think about Fourier Transform, in the classical cases, say on the real line, what it is? Just a waded sum. Right? You take a function $f$, and you take it's Fourier Transform at particular frequency k, then it’s going to be something like…
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Rate of divergence for the series $\sum |\sin(n\theta) / n|$

In the following we consider the series $$ S(N;\theta)= \sum_{n = 1}^{N} \left| \frac{\sin n\theta}{n} \right| $$ parametrized by $\theta$. It is well known that this series (taking the limit $N\to\infty$) diverges for any $\theta\in (0,\pi)$, but…
Willie Wong
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Lower Bound on Oscillatory Integral

Let $p,y\in\mathbb{R}^d\setminus\{0\},\beta>0$ be given and fixed and define for all $\alpha>0$, $$I(\alpha) := \int_{x\in\mathbb{R}^d}\exp(\mathrm{i}\alpha p\cdot x-\alpha\beta \|x-y\|^2)f(x)\mathrm{d}x$$ where $f:\mathbb{R}^d\to[0,1]$ is some bump…
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A guess related to Lebesgue differentiation theorem

When I read Lebesgue differentiation theorem, I suddenly have the following conjecture, which I can't prove or find a counterexample. Let $f\in L_{\mathrm{loc}}^1(\mathbb{R}^n)$. If $$ \int_{B_r(x)} f(y)dy=0 $$ holds for any $r\geq 1$ and…
Yuhang
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Given a Schwartz function, is it always possible to write it as a product of two Schwartz function?

Fix any $f\in\mathcal{S}_x(\mathbb{R}^d\to\mathbb{C})$, i.e. $f$ is a Schwartz function from $\mathbb{R}^d$ to $\mathbb{C}$. Is it always possible to find $g,h\in\mathcal{S}_x(\mathbb{R}^d\to\mathbb{C})$ such that $f(x)=g(x)h(x)$ for all…
ydx
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What are the differences and relations of Haar integrals, Lebesgue integrals, Riemann integrals?

Are Riemann integrals special cases of Haar integrals? Why do we need the invariant property under some actions of groups in the definition of Haar integrals? For example, if we have a group of real matrices $G$ and we have an inner product…
user
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Hardy–Littlewood-Sobolev inequality without Marcinkiewicz interpolation?

Here is the statement of the Hardy–Littlewood–Sobolev theorem. Let $0< \alpha< n$, $1 < p < q < \infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then: $$ \left \| \int_{\mathbb{R}^n} \frac{f(y)dy}{|x-y|^{n-\alpha} }…
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Besov spaces---concrete description of spatial inhomogeneity

Some very pedestrian questions about Besov spaces. Just to fix notation: 1.Let $f \in \mathcal{S}'$, the space of tempered distributions. 2.$\Psi, \{ \Phi_n \}_{n \geq 0} \subset \mathcal{S}$ such that their Fourier transforms $\hat{\Psi}, \{…