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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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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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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The existence of sub-harmonic function

Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Given function $g\in C(\partial\Omega)$, I am trying to find a sub-harmonic function $u$ such that $u=g$ on $\partial\Omega$. ...
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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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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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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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52 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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23 views

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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103 views

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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107 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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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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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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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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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 ...
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Hahn-Banach proof of existence of Haar measure

I'm reading these notes of Terry Tao on the Haar measure (and related topics) on a locally compact Hausdorff group $G$. When he goes through the construction of the Haar measure, he does so by way of ...
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80 views

Is there a “unifying framework” for harmonic analysis?

Recently, I was exposed to a basic harmonic analysis course. Although the course is almost over, I still can't put my finger on what harmonic analysis is about. I have a vague idea that it is ...
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use of Strichartz estimates

I have been learning(understanding) the Strichartz estimates. My naive Questions: (1) Can you give some ideas, how Strichartz estimates plays role in solving nonlinear dispersive equations(e.g. ...
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Estimating the (double) Riesz transform.

I'm trying to verify the following estimate, which appears in a paper I'm reading. It seems I'm missing something easy, I just can't figure this out. $\textbf{Background}:$ For a function $f \in ...
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Approximating $L^p$ functions using Schwartz functions with compact support on the Fourier side

For $1\leq p<\infty$, how would you show for any $f\in L^p(\mathbb{R})$ and given $\epsilon>0$, there exists $L<\infty$ and $g\in \mathcal{S}(\mathbb{R})$ such that $\|f-g\|_p<\epsilon$ ...
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An $f\in H^{1/2}$ with self-convolution, showing it is an $C^1$ function.

If $f\in H^{\frac{1}{2}}(\mathbb{R})$ is a Sobelev 1/2 function that $f=f*f$, then how do you show that $f\in C^1$ with a bounded derivative.
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Showing $f$ is an $L^p$ function if $f$ is "self-convoluted.

If $f$ is $L^2(\mathbb{R})$ and $f=f*f$, show that $f$ is $L^p$ for $2\leq p\leq \infty$.
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Time series closed form similar to harmonic series

I want to find the closed form for the following: $$ S(k, \alpha) = \sum_{t=T-k}^T \frac{\alpha^{-t}}{t} $$ when $\alpha \in (0,1)$. For harmonic series there is an easy way to upperbound: $$ H(k) ...
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Plancherel's theorem variants

How would you prove a variant form of Plancherel theorem: If $(c_n)_{n\in\mathbb{Z}}$ are coefficients and $\sum_{n\in\mathbb{Z}}|c_n|^2<\infty$, then there exists a unique function $g\in L^2(0,1)$ ...
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Is “being harmonic conjugate” a symmetric relation?

The question is: Prove or disprove the following: If $u,v:\mathbb{R}^2 \to \mathbb{R}$ are functions and $v$ is a harmonic conjugate of $u$, then $u$ is a harmonic conjugate of $v$ (in other words, ...
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Hardy–Littlewood-Sobolev inequality without Marcinkiewicz interpolation?

Here is the statement of the Hardy–Littlewood–Sobolev theorem. Let $0< \alpha< n$, $1 < p < q < \infty$ and $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$. Then: $$ \left \| ...
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Orthonormal system

Let $\varphi\in L^2(\mathbb{R})$, prove that $\{e^{2\pi i m x}\varphi(x)\}$ is an orthonormal system iff $$\sum_{n\in\mathbb{Z}}|\varphi(x-n)|^2=1 \ \ a.e \ x$$ How do you prove this. The hint is ...
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Can a “tangent vector of a discrete group” be extended to a tangent vector of its $C^*$-algebra?

This is related to my recent question in MO. I am sure this is trivial, but I have no intuition here, so my apologies from the very beginning. Let $G$ be a discrete group, $A$ a $C^*$-algebra, and ...