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-...

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2
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2answers
38 views

Extend the Fourier transform over $L^2(\mathbb R^n)$

Using Plancherel theorem, we have that the Fourier transform is an isometry over $L^2(\mathbb R^n)$. But anyway. In my course it's written that Plancherel theorem is extremely important since it allow ...
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0answers
8 views

Irreducible representations of locally compact semigroups

What class of semigroups have finite dimensional irreducible representations. For example, If we have a compact groups $G$ then every continuous irreducible representation of $G$ is finite ...
5
votes
1answer
61 views

Is $\Delta C_c^\infty$ a dense subset of $L^p(\mathbb{R}^d)$?

I'm struggling to obtain some density result. It is well known that $C^\infty_c(\mathbb{R}^d)$ is dense in $L^p (\mathbb{R}^d)$ for $1\leq p<\infty$. It is well known that for $\lambda>0$, $(\...
4
votes
1answer
62 views

Finding $f \in L^p(\mathbb{R}^n)$ such that $\hat{f} \notin L^p(\mathbb{R}^n)$

I heard there were functions $f \in L^p(\mathbb{R}^n)$ such that $\hat{f} \notin L^p(\mathbb{R}^n)$. Is there a concrete example of such functions ? Thanks in advance !
0
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0answers
22 views

The discrete Laplacian

I am working on the $d$-dimensional integer lattice. Let $S$ be a random walk with increment distribution $p$. Given the distribution $p$ we can define the discrete Laplacian just as in Wikipedia is ...
4
votes
0answers
26 views

On sharp bounds of some dyadic operators

I am now having interest in finding the sharp bounds of some kinds of dyadic operators which map $L^p(\mathbb{R})$ to $L^p(\mathbb{R})$. For example, the martingale transform $T_{\sigma}$ which ...
0
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0answers
36 views

Proof of iff condition of harmonic map

How to compute the equation above the red line in the picture below ? Below picture is from the Harmonic maps and their heat flows. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...
0
votes
1answer
30 views

A specific maximal function of of a potential function

Let $$f(x)=\frac 1{(1+|x|)^2},$$ Then what's the maximal function of $f$ ? By definition $$Mf(x)=\sup_{r>0}\frac 1{|B_r(x)|}\int_{B_r(x)}\frac 1{(1+|y|)^2}dy,$$ If one can prove that the average ...
1
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0answers
16 views

Singular integral operator on a decay Hölder space

Denote $$E^{k,m}=\left\{f\in C^m(\mathbb{R}^3)\mid\sup_{x\in\mathbb{R}^3}(1+|x|)^{k+|\alpha|}|\partial^\alpha_xf(x)|<\infty\right\}.$$ Now given $f\in E^{2,m}$ for any fixed $m\in \mathbb{Z}^+$, ...
1
vote
0answers
18 views

What are the applications of quantum shuffle multiplication?

I am writing a research paper on quantum shuffle multiplication, and there is just one piece that I'm missing: I want to give one or several examples of its applications, and so far I have not been ...
-1
votes
0answers
45 views

Boundedness of the Fourier transform of a Battle-Lemarie scaling function

Could anyone please give a short and simple proof of the following proposition: the Fourier transform $\hat\phi$ of Battle-Lemarie scaling function (of arbitrary order) is bounded on $\mathbb{R}$, ...
1
vote
0answers
14 views

Finite dimensional irreducible representations of Sp(2).

I am looking for an explicit description of the finite dimensional irreducible representations of the classical Lie group $\text{Sp}(2) = \{A\in M_2(\mathbb{H})\,|\,A\overline{A}^T = I\}$. I can ...
2
votes
2answers
52 views

Morrey's inequality for Sobolev spaces of fractional order

Let $H^s(\mathbb T)$, where $s\in\mathbb R$, be the space of $2\pi$-periodic functions, $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx}$, such that $$ \|u\|_{H^s}^2=\sum_{k\in\mathbb Z}(1+k^2)^{...
0
votes
0answers
28 views

Semigroups and Finite Dimensional Representaions

It is well known that if a group $G$ is compact then every irreducible continuous representation of $G$ is finite dimensional. As far as I know the semigroup equivalent of this statement is not true. ...
2
votes
0answers
28 views

Classification of irreducible (g,K)-modules for other g than sl2

Harish Chandra showed how to associate to an admissible representation $(\pi,V)$ of a real semisimple Lie group $G$ the so-called Harish-Chandra module $V_K$ of $K$-finite vectors in $V$. This is a $(\...
1
vote
0answers
43 views

Absolute maximum

I´m trying to find the absolute maximum of $(2N-1)$ partial sum of the Fourier´s series of signum function on $[0,\pi]$, I have: $S_{2N-1}[f](x)=\frac{4}{\pi}\displaystyle\sum_{k=0}^{N-1}{\frac{sen((...
2
votes
1answer
50 views

Problem on the pointwise boundedness of the partial sums of the j-series in Tuomas Hytonen's paper

Recently, I have read Tuomas Hytonen's paper On Petermichl's Dyadic Shift And The Hilbert Transform and got into trouble in a certain part of his article. In the first place, we should have some ...
4
votes
2answers
107 views

Why does the coordinate transformation from Cartesian coordinates leads to an additional term in the biharmonic operator in spherical coordinates

I am trying to solve a problem in physics where the biharmonic operator is involved. I think that the bihahmonic operator can be obtained by taking twice the Laplace operator, such that $\nabla^4 f = \...
0
votes
0answers
8 views

Can you find shearlets in CNNs?

I read that you can find wavelets in the first layers of a CNN. Can you also find shearlets, and if not, why? Edit (further explanations): I read that, when you train a CNN the filters in the first ...
0
votes
0answers
31 views

convolution of Schwartz functions with $f(x) = (1+\|x\|)^{-\frac{1}{2}}$

Let $f(x) = (1+\|x\|)^{-\frac{1}{2}}$ for $x \in \mathbb{R}^n$. This is clear that $f\star g \notin \mathcal S$ where $\mathcal S$ is algebra of Schwartz functions on $\mathbb{R}^n$ and $ g \in \...
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0answers
41 views

How to calculate this integral tends to zero?

I've posted this before, but I was unable to solve this... Setting : U is a bounded Lipschitz domain in the complex plane Consider the following classical Dirichlet problem for the Laplace operator:...
2
votes
1answer
48 views

How much does the $L^p$ norms say about a function?

Let's say we have two positive, decreasing function $u$ and $v$ on $[0,+\infty)$, and we know that $\|u\|_{L^p}=\|v\|_{L^p}$ for all $p\ge1$, can we say something about $u$ and $v$? Do they have to be ...
0
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0answers
67 views

Representation of a real function through a Fourier Transformation

I 'm trying to do some calculations regarding some differential equations and I came across an interesting way to express a real function through a double integral of the form: $f(x)=\frac{1}{\pi}\...
0
votes
1answer
29 views

A continuous action of a compact group on a uniform space is equicontinuous?

I am stuck on how to prove the statement "A continuous action of a compact group $G$ on a uniform space $X$ is equicontinuous." So, essentially we want to show that for every entourage $\alpha$ of $...
0
votes
0answers
28 views

Fourier Inversion and Convolution

For $f, g$ in Schwartz function, I have $\widehat{D^\alpha f}=|\xi|^\alpha{}\hat{f}(\xi).$ My question is that if $$(|\xi|^\alpha\widehat{fg}(\xi))\check{}=(|\xi|^\alpha{})\check{}*(\widehat{fg}(\xi)...
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0answers
25 views

Calderon's commutator and differential operator[Edited]

First, $$C_1f(x)=p.v.\int_\mathbb{R}\frac{A(x)-A(y)}{(x-y)^2}f(y)dy$$ Here, $C_1$ is the Calderon's first commutator. The author says, the $L^p$ boundedness of $C_1$ is coincides with $[|D|,A]$ where $...
1
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0answers
40 views

Fourier transform without using Lebesgue measure

Let $\mathbb{L}^p(\mu)$ be a space such that $$ \mathbb{L}^p(\mu) = \left\{f:\mathbb{R}\to \mathbb{R} \mbox{ measurable}: \|f\|_{L^p(\mu)} = \left(\int_0^{+\infty} \big|f(x)\big|^pd\mu(x)\right)^{1/...
1
vote
1answer
39 views

Harmonic forms and functions on compact manifolds

I hope my question is not stupid, I'm studying chapter 5.1 of Claire Voisin's book "Hodge theory and complex geometry". Let $X$ be a compact manifold and $A^k(X)$ be the space of $C^\infty$ forms on $...
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0answers
19 views

Differential operator with absolute sign?

From Classical and Multilinear Harmonic Analysis - Schlag, Muscalu There is a operator denoted by $|D|$, where $D$ is a convenient notation for derivative operator. What does $|D|$ mean? Is it $|Df|$?...
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0answers
27 views

What does the explicit formula means in this sentence?

I'm reading Classic and Multilinear Harmonic analysis vol.2 - Muscalu, Schlag In page 134, it says, $$\int_{\partial B(x,\epsilon)}-F(y-x)\frac{\partial u}{\partial\nu}(y)d\sigma(y)=0$$ can be ...
1
vote
1answer
41 views

Completeness of $(BMO(\Bbb R^n),||\cdot||_{\ast})$

Recall that $$ BMO(\Bbb R^n):=\left\{f\in L_{loc}^1(\Bbb R^n)\;\mbox{modulo constant functions, such that}\\ \forall B\subseteq\Bbb R^n\;\mbox{ball}, \exists\alpha(B)\in\Bbb R\;\;\mbox{such that}\\ \...
7
votes
1answer
157 views

Prove that $\operatorname{p.v.}(k\;*f)$ does not exist if $k(x)=|x|^{-n+i\gamma}$ and $f\in\mathcal{C}_c^1$

I put a bounty only because I need quickly a solution, NOT because I know it's difficult - maybe it is, maybe not. I'm trying to do it, but without results. If I get some "intermediate result" (...
0
votes
1answer
41 views

Math-english for non-natives: What does “supported in” mean?

As a non-native English speaker, I am struggling with the following sentence: "Fix a function $f:\mathbb{R}\to\mathbb{C}$ such that $f$ is supported in the unit Ball." Does this mean $\...
6
votes
1answer
79 views

A version of Hörmander multiplier theorem

Let $m>n/2$ be an integer. Let $h\in H^m_{loc}(\mathbb{R}^n)$ satisfy that $\displaystyle \exists M>0,\forall R>0,\sum_{|\alpha|\le m}\int_{\frac R2\le|w|\le2R}R^{2|\alpha|}|\partial^\alpha ...
0
votes
1answer
27 views

If $f\in S(\mathbb R^n)$ (schwarz space), why $f\in L^p(\mathbb R^n)$?

Let $$\mathcal S(\mathbb R^n)=\left\{f\in \mathcal C^\infty (\mathbb R^n)\mid \forall N\in\mathbb N,\forall \alpha \in\mathbb N^n, \sup_{x\in\mathbb R^n}|(1+|x|^N)\partial _x^\alpha f(x)|<\infty \...
1
vote
1answer
21 views

Space of Riesz transforms is closed

Let $B=\bigoplus_{j=0}^nL^1(\mathbb R^n)$ a Banach space with norm $\|(f_0,\ldots,f_n)\|=\|f_0\|_{L^1}+\cdots+\|f_n\|_{L^1}$. Define $$S=\{(f_0,f_1,\ldots,f_n):f_j=R_jf_0,\quad j=1,2,\ldots,n\}\subset ...
0
votes
0answers
23 views

Controlling $\dot W_{k,1}$ norm of a schwarz function

If $\phi \in \mathscr{S}(\mathbb{R})$ then does it follow that there is $c$ s.t. $||\nabla^k \phi||_{L_1} \leq c^k$? From the definition I have only been able to get that this is true for all $k$ in ...
3
votes
1answer
53 views

If Fourier sequence $\{H_n\}$ of a function $f$ converges almost everywhere to a function $g$, then $f=g$ almost everywhere?

if $\{H_n\}$ fourier sequence of an $L^2$ function $f$ converges almost everywhere to a function $g$ then $f=g$ almost everywhere Where $H_n = n$th fourier series of $f$. True or False?
0
votes
1answer
28 views

Solve for bound of $\sigma(n)$ from harmonic series.

I am given the harmonic series, $H_n$. How can I show that for $n\geq 1$, $$\sigma(n)\leq H_n+e^{H_n}\ln{H_n}$$ By the way, if you're not familiar with it, $\sigma(n)$ is the sum of all positive ...
0
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0answers
27 views

Solving traveling wave usin the shooting method

The spatially-dependent Hodgkin-Huxley equation for a cylindrical dendrite or unmyelinated axon: where $\frac{a}{2\rho}\frac{\partial^2V}{\partial x^2}$ is a diffusion term $a$ is the fiber radius, ...
2
votes
0answers
28 views

Planar sets in R^{2} with bounded Fourier transforms

I have a question which I'd be greatly happy to hear an answer for (because it sounds really cool!). Say I'm given some "nice" region $\Omega$ in $[0,1]^{2}$, how can one determine all the $p>0$ ...
2
votes
1answer
40 views

Proving $\|[b,T](f)\|_{p}\le C\|b\|_{BMO(\mathbb{R}^{n})}\|f\|_{p}$ using the Fefferman-Stein inequality

Let $1<p<\infty$ and $1<r<\infty$ and let $\mathcal{M}_{r}(g)$ denote $\mathcal{M}(|g|^{r})^{\frac{1}{r}}$, where $\mathcal{M}$ is the Hardy-Littlewood maximal function. Also let $T\in ...
8
votes
3answers
394 views

Triangle inequality fails in $L^{1,\infty}$

It can be proved that $\forall\varepsilon>0$ there exists $C(\epsilon)>0$ such that for all $f,g\in L^{1,\infty}(\Bbb R^n)$ we have that $$ ||f+g||_{1,\infty}\le(1+\varepsilon)||f||_{1,\infty}+...
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0answers
117 views

Several questions about the proof of this lemma

I am reading this paper. I have Several questions about the proof of the lemma 3.3. How to Prove 1,2,3,4?
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0answers
22 views

Equivalent conditions for weak $L^p$ spaces for $p\leq 1$

I have difficulty doing the following exercise from Tao's real analysis book: Let $X$ be $\sigma$-finite measure space and $0<p\leq 1$. Then show that the following are equivalent: $f$ is in ...
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0answers
36 views

Fundamental solution for the p-harmonic and p-biharmonic equation

I am working on $p$-Laplace equation. that is $$\tag{1} -\text{div}(|\nabla u|^{p-2}\nabla u)=\delta_0 \;\; in\; \mathcal{D}'(\mathbb{R}^2), $$ and the $p$-bilaplace equation, that is $$\tag{2} \...
0
votes
1answer
46 views

Sequences $(U_n)$ of neighborhoods of $0$ in a LCA group with $m(U_n)\to 0$

Let $G$ denote an infinite compact abelian group with Haar measure $m$ (so $m(G)=1$). Given a neighborhood $U_1$ of the unit $0$ in $G$ we can find a symmetric neighborhood $U_2$ of $0$ such that $...
3
votes
1answer
33 views

What is duality argument for the operator on $L^p-$ spaces?

Suppose that the operator $T: L^{p}(\mathbb R) \to L^{p}(\mathbb R)$ (say for instance, some nice convolution operator) is bounded for $1\leq p \leq 2.$ At various, places we see that (for ...
0
votes
1answer
26 views

Riesz basis of Paley-Wiener space.

Let us consider the Paley-Wiener space: $$PW_\pi:=\{f\in L^2(\mathbb R)\cap C(\mathbb R),\ \operatorname{supp}\hat f\subset [-\pi,\pi] \}.$$ Let $\{\lambda_n\}_{n\in\mathbb Z}$ be a sequence of real ...
1
vote
0answers
34 views

Does Fourier Algebra of locally compact group separate compact sets of the group?

Let $G$ be a locally compact group. Consider the left regular representation $\lambda$ over $L^2(G)$. Then according to Eymard, Fourier algebra of $G$, $A(G)$ is the set of all coefficients of $\...