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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Calculate $\mathscr{F}((1+t)^{-3})$

Let $$f(t)=\cases{\frac{1}{(1+t)^3}&t>0\\0&t<0}$$ Does: a.$\hat{f}$ is differentiable? b.$\hat{f}\in L^1(\mathbb{R})$? c.$\hat{f}\in L^2(\mathbb{R})$? Seems like we need to calculate ...
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harmonic function on manifold

Let M be a 2 dimensional manifold. $h:M\rightarrow R$ be a harmonic function from manifold to real line. G is group that act by isometry. $g*h(x)=h(g(x))$. Let $W=\{x|h(x)=t\}$ that is the level set ...
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Proving a sum is finite using Equidistribution

Let $\phi:\mathbb{R\to R}$, be an integrable function with finite integral on $[0,1]$($\int_{[0,1]}\phi(x)dm<\infty$) and $\phi(x)=\phi(x+1)\forall x\in \mathbb{R}$. Prove that ...
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Is each multiplicative linear functional on $L1(SL(2,R):SO(2,R))$ triviall?

We know that if $G=SL(2,R)$ and $H=SO(2,R)$ as a compact subgroup of $G$, then $\{ξ∈\hat{G};ξ|_H=1\}$ is triviall ($\hat{G}$ is the characters group). Can we conclude that each multiplicative linear ...
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Harmonic analogue of the Weierstrass approximation theorem

The Weierstrass approximation theorem says that, given any continuous function $f(x)$ on a closed interval, there is a polynomial which approximates it arbitrarily closely. I'm looking for a theorem ...
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39 views

Differential operator that must be a constant multiple of the Laplacian

Can you help me show that if $L$ is a partial differential operator given by $$L=a{{\partial^2}\over{\partial{x^2}}}+b{{\partial^2}\over{\partial{y^2}}}+c{{\partial^2}\over{\partial{x}\partial{y}}}$$ ...
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What are single layer and double layer potentials?

I have never encountered the terms "single layer potential" and "double layer potential" in my (under)graduate studies, although there was a firm development of partial differential equations and some ...
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49 views

Gradient curve of a harmonic function

I am reading the paper "Energy of Harmonic function and Gromov proof of Stalling theorem" https://www.math.ucdavis.edu/~kapovich/EPR/energy.pdf I have no clue about the lemma 8.4(i). What is gradient ...
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Basic Fourier analysis questions [closed]

Could someone point me in the right direction to prove the following: Let $f \in L^{1}(\mathbb{T})$ and suppose $S_{N}f \to g$ in $L^{p}(\mathbb{T})$ then $\|f-g\|_{L^{1}(\mathbb{T})}=0$, where $1 ...
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32 views

Group $C^*$-algebra elements as limit of self-adjoint integrable functions

Assume $G$ is a locally compact abelian group and let $C^*(G)$ denote its group $C^*$-algebra. I am reading a proof that uses the 'fact' that some $f\in C^*(G)$ is a limit of self-adjoint functions ...
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How can I prove that $\max_{\partial K}\varphi$ does exist?

Let $\varphi:\Omega\to[-\infty,+\infty[$, where $\Omega\in\Bbb C$ is a domain and $\varphi$ is upper semicontinous, i.e. $\varphi(z_0)\ge\limsup_{z\to z_0}\varphi(z)\;\;\;\forall z_0\in\Omega$. How ...
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42 views

Is $\|x\|^6 \sin^6 \|x\|^6$ harmonic?

Suppose the function $$ u(x)=\|x\|^6 \sin^6 \|x\|^6$$ for $x \in \mathbb{R}^d$, where $$\|x\| = \sqrt{x_1^2 + \ldots +x_d^2}.$$ How can I decide if the function $u$ is harmonic in the unit ball ...
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37 views

Does there exist a non-negative valued compactly supported function such that its Fourier transform only vanishes at a given point?

My question is as follows: Given $t_0\in\mathbb{R}$. Does there exist a non-negative valued compactly supported function $f\in L^1(\mathbb{R})$ such that its Fourier transform, $\widehat f\left( t ...
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31 views

estimating a convolution type maximal function

Let $\phi : \mathbb{R}^n \rightarrow \mathbb{R}_{+}$ be a $C^1$ function with $supp(\phi) \subset B(0,1)$ and $\int \phi = 1$. Define $$\phi_t(x) := t^{-n} \phi({x/t})$$ and set $$ M_{\phi} f(x) := ...
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Have a question about $L^p$ multiplier.

I'm studying $L^p$ multiplier. While reading the book, it says "The characteristic function of the unit disk is not an $L^p$ multiplier on $\mathbb{R}^n$ when $n\ge2$ unless $p=2$." How can I verify ...
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Trigonometric polynomials on non-compact and non-abelian groups

Hewitt and Ross define trigonometric polynomial on a locally compact group $G$ as a linear combination of matrix elements of continuous unitary irreducible representations of $G$: $$ f(t)=\sum_{i=1}^n ...
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an oscillatory integral with two parameters

Consider $$I(a,b)=\int_{\mathbb{R}}e^{i(ax^2+bx)}\psi(x)\,dx$$ where $\psi$ is smooth and supported in $\{x:|x|\in[1/2,2]\}$. How to control $I(a,b)$ in terms of $a$ and $b$? Moreover, is there an ...
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28 views

construction of a partition function

Let $q$ be a large positive integer. How to construct a smooth function $\phi$ with the following properties? i) $\sum_{a\in\mathbb{Z}}\phi(q(x-a/q))=1$ for any $x\in\mathbb{R}$ ii) For any ...
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Integration over ideles over $\Bbb{Q}$ , Tate`s thesis special case

Let $f\in S(A_\Bbb{Q})$ that is $f$ is adelic Schwartz-Bruhat function over $\Bbb{Q}$, such that all its components in the finite places are characteristic functions of the corresponding ring of ...
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Tate thesis : Global functional equation [closed]

It will be very helpful if someone tells me how to do EXERCISE 1 here. I have done part $2$. I cannot do part $1$ and part $3$. I tried part $1$ by decomposing $Z(f,s)$ as the product of its local ...
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inserting absolute value in Hilbert transform and a discrete version of Hilbert transform

It is well known that the Hilbert transform $H(f)(x)=p.v. \int\frac{f(x-y)}{y}dy$ is bounded on $L^p(\mathbb{R})$ for $p\in(1,\infty)$. I want to consider some variants of $H$. 1) What happens if we ...
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126 views

Does band-limited imply continuous?

I was hoping someone could point me to an article or text which explores the connection between the continuity of a signal in the time domain and it being band-limited in frequency domain. (Update) ...
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Calculating derivative of a radial function: am I doing it right?

Let $f: \mathbb R^n \setminus \{0\} \to \mathbb R^n \setminus \{0\}$ be such that $f$ only depends on the distance from the origin, that is, $f=f(r)$ where $r = \sqrt{\sum_{i=1}^n x_i^2}$. I am ...
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“Tube domain over symmetric cone” thesis references?

My professor has told me to read "tube domain over symmetric cone". While saying so he said something related to complexificatin of real Lie algebra and representation theory. What is connection ...
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38 views

In the real spherical harmonics, where does the sqrt(2) factor come from?

The real spherical harmonics can be written in terms of the complex spherical harmonics: $$ Y_{\ell m} = \begin{cases} \displaystyle \sqrt{2} \, (-1)^m \, \operatorname{Im}[{Y_\ell^{|m|}}] & ...
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If $-\Delta u=f$ and $f$ is analytic, then $u$ is analytic as well.

I know the theorem that if $-\Delta u=f$ and $f$ is analytic, then $u$ is analytic as well. The book I know that contains the proof of this theorem all use the approach with respect to complex ...
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1answer
52 views

A problem for laplace operator in Sobolev space

Suppose $u\in L^2(\Omega)$, then for any $\phi\in C_c^\infty(\Omega)$ we have $$ \int_\Omega v\,\phi\,dx=\int_\Omega u\Delta \phi\,dx $$ Then can I conclude that $u\in H_0^1\cap H^2(\Omega)$ and ...
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Limit of bounded harmonic functions is harmonic

I am trying to solve this old qual problem: Suppose $\{u_n\}_{n=1}^\infty$ is a sequence of functions harmonic on an open set $U \subset \mathbb{C}$ and uniformly bounded by 1. Suppose there is a ...
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The regularity of elliptic equation

Define $$ Lu:=-\partial_j(a_{ij}\partial_iu)+b_i\partial_iu+cu $$ where $a_{ij}$, $b_i$, and $c$ are all of $C^\infty(\bar \Omega)$ and we assume $\Omega$ is open bounded in $R^N$, smooth boundary, ...
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Using taylor formula to compute laplace operator

Suppose $u\in C^2(\Omega)$ and $x\in \Omega\subset \mathbb R^N$. I am trying to prove that $$ \Delta u(x)=\lim_{r\to 0} \frac{2N}{r^2} \left[\frac{1}{\alpha(N)} \int_{\partial ...
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Group C*-algebra of an abelian discrete group

Let $A = \mathbb C[G]$ be the group ring of all finitely supported functions $f\colon G \to \mathbb C$ of a discrete abelian group $G$ with the usual convolution product, and involution defined by ...
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Harmonic Analysis on the Affine Group

In my previous question, I asked about harmonic analysis on the group $\operatorname{SL}(3, \mathbb{R})$. The representation theory of this group appears to be quite complicated, so I am now looking ...
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the (2,2,1) boundedness of a “product” operator

This question is from MO: http://mathoverflow.net/questions/191551/the-2-2-1-boundedness-of-a-product-operator It seems easy but turns out very difficult. Let $\{E_j\}_{j\in\mathbb{Z}}$ and ...
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Decomposition and harmonic analysis of $L^2(S^n)$

Deitmar and Echterhoff write in their book Principles of Harmonic Analysis that `It follows from the Peter-Weyl theorem that the $SU(2)$ representation on $L^2(S^3)$ is isomorphic to the orthogonal ...
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Finding an example

I am looking for an example to have two distinct decreasing functions $g_1$ and $g_2$ between [0,1], such that their ratio ($\phi=\frac{g_1}{g_2}$) is a periodic function with period one, moreover for ...
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Sublinearity of Hardy-Littlewood Maximal Function on Sobolev Spaces

Define the (centered) Hardy-Littlewood maximal function by $$\mathcal{M}f(x)=\sup_{r>0}\dfrac{1}{m(B(x,r))}\int_{B(x,r)}\left|f(y)\right|dy,\ f\in L_{\mathrm{loc}}^{1}(\mathbb{R}^{d})$$ We say ...
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Questions about definitions of crossed product $C^\ast$-algebras and group $C^\ast$-algebras

In Dana P. Williams's book Crossed Products of $C^\ast$-Algebras, the author defined the crossed product of a $C^\ast$-algebra $A$ by a local compact group $G$ as the completion of $C_c(G,A)$ with ...
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68 views

The Cauchy problem for Laplace equation in unit cube

Here is the question: We are given a laplace equation $\Delta u=0$ in $Q:=(0,1)\times(0,1)$. Q1: What are some conditions you can put on this to get uniqueness/existence? Q2:What if you wanted to ...
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relationship between BMO norm and $L_p$ norm

Are there any relationship between the BMO norm of a function and its $L_p$ norms? For example, one norm is controlled by the other for functions of some special class.
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Decomposition of regular representation

Let $G$ be a compact group. Then there is an isomorphism $L^2(G)\simeq \bigoplus_{\tau\in \hat{G}} V_{\tau}\otimes V_{\tau^*}$ which intertwines the conjugation action of $G\times G$ on $L^2(G)$ ...
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Harmonic Analysis on the real special linear group

I would like to understand the representation theory and generalized Fourier transform of $SL(3, \mathbb{R})$ in as concrete a manner as possible. My ultimate goal is to develop an algorithm that can ...
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130 views

The well-posedness of Laplace equation on half-space

In $2$ dimension, take $-\Delta u=0$ on $\{(x,y\},y\geq 0\}$ with $u(x,y=0)=f(x)$, $u_y(x,y=0)=g(x)$ where $f$ and $g$ are smooth function. I want to justify whether this problem is well posed. My ...
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The Neumann Problem on a Half-space when dimension is $2$

Take $\Omega:=\{x=(x_1,x_2):\,-\infty<x_1<\infty,\,x_2>0\}$, i.e., the half-space, and I am interested in the Neumann problem \begin{cases} \Delta u=0&x\in \Omega\\ ...
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Functions over a finite domain that cannot be represented by Fourier series

Given a double Fourier series for some $f:[0,L]\times [0,R]\to \mathbb{R}$ of the following form $$\sum_{k,l=0}^{\infty}a_{kl}\cos\left(\frac{2k\pi ...
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1answer
304 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 green function to construct a solution based on the boundary data. For instance, one could find a ...
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52 views

Jump discontinuity of a function and its analytic phase

Let $f:\mathbb{R}\to \mathbb{R}$ and $f \in BV(0,1)$ with a jump discontinuity at $x = a\in(0,1)$. Let $f_h$ be its Hilbert transform and let $$f_A(x) = f(x) + i f_h(x)$$ Is it true that the function ...
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36 views

A nonlinear Poisson equation problem. (Related to Laplace equation)

Let $\Omega\subset R^N$ be open bounded. Define \begin{cases} \Delta u = f(u) &x\in\Omega\\ u=1 & x\in\partial \Omega \end{cases} Q1: Suppose $f(u)=u^m$ where $m$ is odd. Prove that if there ...
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The tangent hyperplane to the graph of harmonic function

This is an interesting question I found online about Laplace equation. We define a function $u$ :$R^N\to R$'s graph to be the set $\{x,u(x):\,x\in R^N\}\subset R^{N+1}$. Then I want to prove that ...
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36 views

Left and Right Discrete Maximal Functions

Define the uncentered maximal function $$\widetilde{M}f(n)=\sup_{s,r\in\mathbb{Z}^{+}}\dfrac{1}{s+r+1}\sum_{k=-r}^{k=s}\left|f(n+k)\right|,$$ where $\mathbb{Z}^{+}=\left\{0,1,2,\ldots\right\}$. Define ...
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Restriction estimates

What is the defining property of what someone in the harmonic analysis community would call a "restriction estimate?" I see sobolev norms, Fourier transforms, and inequalities relating these. The ...