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, ...

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Estimation with Hölder condition

Do you have any hints about how to prove (or find a counterexample) that, given $f \in \mathcal{C}^1 ( \mathbb{R}^n \smallsetminus \{ 0 \}) $ such that $$\int_{|x|=r} f(x) \, dS(x) = 0$$ for all ...
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11 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: ...
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55 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 = ...
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8 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 ...
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51 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: ...
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6 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 ...
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19 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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45 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 ...
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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 ...
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98 views

Reference for Harmonic Analysis?

I'm looking primarily for references for Harmonic Analysis. I'm mostly considering Doran&Fell or Deitmar, but I have access to lectures using Stein as well. The important thing is covering ...
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34 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 ...
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1answer
52 views

Prove that the Pontryagin dual of $\mathbb{R}$ is $\mathbb{R}$.

From Wikipedia: the group of real numbers $\mathbb{R}$, is isomorphic to its own dual; the characters on $\mathbb{R}$ are of the form $r \to e^{i\theta r}$. How can I prove this assertion?
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1answer
24 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 ...
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20 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 ...
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20 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 ...
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27 views

Prove: For a harmonic function $u$, $|z|^{1-\epsilon}|u(x,y,z)| \leq (x^2+y^2)^{0.5-\epsilon}$ implies $u = 0$ for $0 < \epsilon < 0.5$

Prove: For a harmonic function $u: \mathbb{R}^3 \rightarrow \mathbb{R}$, if for all $(x,y,z) \in \mathbb{R}^3$ $|z|^{1-\epsilon}|u(x,y,z)| \leq (x^2+y^2)^{0.5-\epsilon}$ then $u = 0$ for $0 < ...
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29 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} ...
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1answer
36 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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60 views

Fourier transform of a Gaussian

I am trying to solve the following exercize: Show that Fourier transform of a Gaussian (a function of the form $Ae^{-\frac{x^2}{\sigma^2}}$) is also a Gaussian. So I did the required calculation (I ...
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17 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 ...
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50 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?
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22 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 ...
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38 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}\\ ...
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156 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" ...
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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 ...
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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 ...
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10 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 ...
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24 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 ...
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18 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 ...
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55 views

Compactum of Banach algebra

I need an example of Banach algebra $A$ and a left non-trivial closed ideal $I$ with all of following properties: There exists a bounded approximate identity in $I$ for $I$ i.e., a net ...
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25 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, ...
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15 views

Invariant subspaces of $L^{\infty}(G)$

I am reading a book named "Lectures on Amenability". The author Dr. Volker Runde defines the following : Let $G$ be a locally compact group, and let $E$ be a subspace of $L^{\infty}(G)$ containing ...
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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 ...
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1answer
60 views

When does the Fourier algebra coincide with the Fourier-Stieltjes algebra?

For a given locally compact group $G$ the Fourier-Stieltjes algebra $B(G)$ is defined as the algebra of matrix coefficients of unitary representations $\pi:G\to B(H)$. Similarly, the Fourier algebra ...
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390 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 $$ ...
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1answer
38 views

Is the action on $L^2$ arising from a measure preserving action continuous?

Let $G$ be a locally compact topological group, $X, \mu$ a probability space, and $G\times X \rightarrow X$ a measurable group action which preserves $\mu$ (i.e. $\mu (gA)=\mu(A)$) . Does it follow ...
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83 views

Do we have the pointwise bound $\left|\tilde{f}\right| \lesssim_d Mf$?

Edit. I know I am missing a mean value on all integrals, but unfortunately I do not know if it is possible to make an integral sign with a horizontal slash through it with MathJax. For any locally ...
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0answers
26 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 $p>0$ ...
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1answer
35 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 ...
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103 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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32 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} ...
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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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42 views

Is this a right calculation of Fourier transform?

I am trying to solve: Calculate Fourier transform (on $-\infty < x < \infty $) of: $$ f(x) = \begin{cases} 1, & -L<x<L \\ 0, & |x|\geq L \end{cases} $$ My ...
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1answer
53 views

Questions about Haar integral for the group $GL_2(\mathbb{R})$.

I have some questions about Haar integral for the group $GL_2(\mathbb{R})$. How to show that a Haar integral for the group $GL_2 (\mathbb{R})$ is given by \begin{align} I(f ) & = \int_{\mathbb{R}} ...
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1answer
76 views

How the second and third equalities can be achieved?

I am reading this paper. In the Proof of Lemma 3.3, How the second(*) equality can be achieved? How can i use Parseval's identity in third(**) equality?
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1answer
30 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 ...
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1answer
39 views

Does the Hilbert transform of Schwarz function decay far away

The Hilbert transform $H$ of Schwarz functions can be defined as \begin{equation} Hf(x)=\int_{|y|<1}\frac{f(x-y)-f(x)}{y}dy + \int_{|y|>1} \frac{f(x-y)}{y}dy. \end{equation} I would like to ...
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12 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 ...
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32 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 ...
2
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1answer
62 views

Fourier series of dirac delta

Let $f \in S(\mathbb{R}^n)$ is it true that $$\frac{1}{(2\pi)^n} \lim_{\epsilon \rightarrow 0} \sum_{z\in \mathbb{}{Z^n}} \int_\mathbb{R^n} f\left( \frac{x}{\epsilon} \right) e^{iz (x-a\epsilon)} dx = ...