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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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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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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34 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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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 ...
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28 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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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 $\mathbb{R}$, ...
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1answer
63 views

where can I find the proof of this theorem of wavelet frames?

Let $\mathbb{K}$ be a local field. Definition of local field: Let $\mathbb{K}$ be a field and a topological space. Then $\mathbb{K}$ is called a locally compact field if both $\mathbb{K}^+$ ...
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15 views

Singular integral operator on a decay Hölder space

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

Neumann problem for Laplace equation on Balls by using Green function

It is well known that for Dirichlet problem for Laplace equation on balls or half-space, we could use the green function to construct a solution based on the boundary data. For instance, one could ...
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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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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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77 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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27 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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27 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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1answer
48 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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2answers
106 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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1answer
16 views

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 $r&...
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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((...
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66 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}\...
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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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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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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 ...
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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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2answers
104 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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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:...
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1answer
54 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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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 $...
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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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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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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/...
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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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63 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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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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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?
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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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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}\\ \...
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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" (...
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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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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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26 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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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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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 $\{e_\alpha\}\...
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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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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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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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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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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 ...