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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6 views

Do conformal maps preserve subsolutions of elliptic PDE?

The fact is well-known for the Laplace equation for regions in $\mathbb R^n$ but I'm wondering if it extends to general elliptic PDE.
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40 views

Intuition regarding $\lim \lVert u_r - u \rVert_{p}=0$

I have some trouble intrepreting the following statment If $u$ is harmonic in $D$ and has bouned means for order p on circles of radius $< 1$ then $\lVert u \rVert_{p}=\lVert u \rVert_{L^{p}}$ ...
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30 views

Do we have correspondance of all of $L^{p}(T)$ and $h^{p}(D)$ $p\in (1,\infty)$?

Let $h^{p}(D)=\{u ;u \ $harmonic in D and bounded means of order p$ \}$ Its well known that the Hardy space "sits nice" in $L^{2}$ by considering Fourier series. Do we have analagous result for $h^{...
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1answer
27 views

Can a schwartz class function be dominated by an exponential?

Given a function $f$ from Schwartz class, does there exist a constant $C$ such that $|f(x)|<Ce^{-|x|}$. For me its seems true, if it is not true, any counterexample would be very illustrative to me
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34 views

On an inequality for harmonic functions

I'm trying to understand the following; $| u(z)|^{p}= |\int P_{z}u \ d\lambda |^{p} \le (\int P_{z} |u |^{p} \ d\lambda) (\int P_{z})^{p-1} $ where $P_z$ is the Poisson kernel and $u$ is harmonic in ...
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94 views

Why this abuse of notation correctly solves the heat equation

Here's a stupid method I observed to solve the heat equation in $\mathbb R^d$, \begin{align*} \partial_tu=\Delta u,\quad u|_{t=0}=u_0. \end{align*} Pretend that $\Delta$ is a constant so this just ...
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37 views

The eigenfunction of modified 1-laplace equation

Let $\Omega\subset \mathbb R^2$ be a bounded, smooth boundary domain. I am interested in the following operator $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{|\nabla u|^2+\epsilon}}\right)=0. $$ ...
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41 views

Getting the DFT of irregularly spaced points

I am trying to estimate the discrete Fourier transform of a discrete surface, $x:\{1,\dots,N\}\times \{1,\dots,N\} \to\mathbf{R}$, given a sparse set of samples on the grid. If we had all the ...
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393 views

If $f$ is a smooth real valued function on real line such that $f'(0)=1$ and $|f^{(n)} (x)|$ is uniformly bounded by $1$ , then $f(x)=\sin x$?

Let $f : \mathbb R \to \mathbb R$ be a smooth ( infinitely differentiable everywhere ) function such that $f '(0)=1$ and $|f^{(n)} (x)| \le 1 , \forall x \in \mathbb R , \forall n \ge 0$ ( as usual ...
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22 views

How theorem 3.1 is a consequence of lemmas 3.2-3.4?

In the article "Tight wavelet frames on local fields", the author states that "Theorem 3.1 is an easy consequence of lemmas 3.2-3.4". How?
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17 views

Complex interpolation between $H^1$ and $L^1$

We have the complex interpolation $[H^1,L^p]_\theta=L^q$ for $1<p<\infty$ where $H^1$ is the Hardy space and $\frac{1}{q}=1-\theta+\frac{\theta}{p}$. This raises the obvious question as to ...
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46 views

Elementary proof of $C^\infty_c$ is dense in $L^p (L^q)$ mixed space

As it well known, $C^\infty_0 (\mathbb{R}^n)$(the space of infinitely differentiable functions with compact support) is dense in $L^p (\mathbb{R)^n}$ Here I want to consider the same result with ...
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1answer
24 views

Fourier coefficents of harmoinc $L^{1}$ functions in the disk

I just did an exercise in a some lecture notes, my result implied that the Fourier coefficients of an harmonic $L^{1}$ function $u$ was the integer values of the borel measure in the Poisson integral ...
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1answer
19 views

Image under Hilbert Transform in $L^1$

I have a question concerning the proof of following proposition: Proposition: Let $\phi\in S(\mathbb{R})$ be given. Then $H\phi \in L^1(\mathbb{R})$ if and only if $\int_{\mathbb{R}}\phi(x)dx=0$. ...
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26 views

Why is this true about MRA-wavelets?

In the article Characterization of wavelets and MRA wavelets on local fields of positive characteristic, in page 11, As seen in the picture, The author refers to 16(Multiresolution analysis on local ...
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1answer
22 views

$G$ is dense in $X^*$ in weak* sense if and only if $G$ is total set

I have some question on functional analysis. Recently, I'm reading an article of Coifman and Weiss, "Extensions of hardy spaces and their use in analysis". They proved some important theorem to me by ...
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1answer
43 views

Null space of the Laplacian operator?

(I guess the answer to my question is well-known in harmonic analysis, but I consider it in the framework of Schwartz distributions and in any dimension, and could not find a satisfactory answer.) ...
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32 views

Determining a function is harmonic from mean value property for just three(?) radii.

This theorem is well-known (maybe it can be called Morera's theorem): A continuous function satisfying the mean value property on balls is harmonic. I was recently surprised to hear in a talk ...
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22 views

BMO and truncation

I'm trying to solve Exercise 7.1.4 of Grafakos' book $\textit{Modern Fourier Analysis}$: consider two real numbers $K<L$ and a function $f \in BMO(\mathbb{R^n})$. Define the truncation of $f$ $$f_{...
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27 views

convolution of two decaying polynomial

Show that if the functions $f,g$ defined on $\mathbb{R}^n$ satisfying $|f(x)| ≤ A(1+|x|)^{−M} $and $|g(x)| ≤ B(1+|x|)^{−N}$ for some $M,N > n$, then $|(f ∗g)(x)| ≤ ABC(1+|x|)^{−L} $, where $L = ...
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1answer
16 views

Why $|K(x-y)-K(x)|\leq B|y|\left(\frac{|x|}{2}\right)^{-n-1}$?

Let $K:\mathbb R^n\backslash \{0\}\longrightarrow \mathbb C$ differentiable s.t. $|K(x)|\leq B|x|^{-n}$, and $|\nabla K(x)|\leq B|x|^{-n-1}$. Suppose $|x|>2|y|$. By IVT, $$K(x-y)-K(x)=-\int_0^1 \...
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44 views

What does $f$ continue on $\mathbb S^1$ mean?

Let $\mathbb S^1:=\mathbb R/\mathbb Z$. What does $f$ continuous on $\mathbb S^1$ mean ? That it's continuous over $[0,1)$ or $[0,1]$ ? I would say $[0,1)$ but I have doubt since we sometimes take the ...
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32 views

Pointwise convergence of Fourier series in two dimensions

By Carleson's Theorem, we know that for every $f\in L^2(\mathbb{T})$ $$ f(x)=\lim_{N\rightarrow\infty}\sum_{k=-N}^N\hat{f}(k)e^{2\pi ikx}\;\text{ a.e.} $$ Suppose now that $f\in L^2(\mathbb{T}^2)$. ...
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25 views

Are holder space dense in $L^p$ ? or in Schwarz space?

I recall that $f\in \mathcal C^\alpha ([0,1[)$ where $\alpha \in (0,1)$ if $$[f]_\alpha :=\sup_{x,y\in\mathbb [0,1[}\frac{|f(x)-f(y)|}{|x-y|^\alpha }<\infty .$$ Does those space are dense in $L^p$ ...
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26 views

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 ...
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26 views

Why $(2\pi x)^\alpha f(x)(2\pi i\xi)^\beta e^{-2i\pi x\cdot \xi}=(\partial _x)^\beta [(2\pi x)^\alpha f(x)]e^{-2i\pi x\cdot \xi}$?

Why $$(2\pi x)^\alpha f(x)(2\pi i\xi)^\beta e^{-2i\pi x\cdot \xi}=(\partial _x)^\beta [(2\pi x)^\alpha f(x)]e^{-2i\pi x\cdot \xi} ?$$ Indeed, to prove that the fourier transform is in the Schwarz ...
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35 views

Question on Schwarz space why is it $\sup_{x\in \mathbb R^n}|(1+|x|^N)\partial _x^\alpha f(x)|<\infty $

Why Schwarz space is given by the set of $f\in \mathcal C^\infty (\mathbb R^n)$ s.t. $$\sup_{x\in \mathbb R^n}|(1+|x|^N)\partial _x^\alpha f(x)|<\infty ,$$ where $\alpha \in \mathbb N^n$ and $N\in\...
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59 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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13 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 ...
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68 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 $\lambda>0$, $(\...
5
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1answer
75 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 !
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28 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 ...
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28 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 ...
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38 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. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...
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1answer
31 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 ...
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18 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 $m\in \mathbb{Z}^+$, ...
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21 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 ...
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17 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 ...
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2answers
62 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)^{...
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30 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. ...
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33 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 $(\...
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44 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
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1answer
54 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
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109 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

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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33 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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42 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
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1answer
49 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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68 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
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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 $...