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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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.
2
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1answer
36 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 $L^2(G)$ ...
5
votes
1answer
130 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 ...
5
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1answer
142 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 ...
3
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1answer
91 views

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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69 views

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 ...
3
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1answer
440 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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1answer
63 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 ...
2
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1answer
50 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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1answer
37 views

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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45 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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1answer
32 views

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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0answers
73 views

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 ...
6
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1answer
134 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 ...
2
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48 views

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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55 views

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 ...
5
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1answer
70 views

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 $\|f-g\|_p<\epsilon$ ...
3
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29 views

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.
2
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1answer
31 views

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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1answer
29 views

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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28 views

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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2answers
68 views

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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2answers
329 views

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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votes
2answers
62 views

Group with topology which is not topological group

What will example of a group $G$ with topology $\tau$ such that $f: G \to G$ such that $f(x) =x^{-1}$ and $g: G \times G \to G$ such that $g((x,y)) = x * y$ (where $*$ is binary operation on $G$) both ...
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0answers
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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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33 views

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 ...
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0answers
685 views

How to find the Total Harmonic Distortion of a Periodic Signal through MATLAB?

How to find the Total Harmonic Distortion of a Periodic Signal through MATLAB? I just need help in confirming if my way of approach to finding the THD seems valid, I'm new to MATLAB so I'm not quite ...
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1answer
90 views

Big theta notation of harmonic series

I want to prove that big theta notation of the harmonic series is $\Theta(\log n)$. I want to work with integral to show that. I attempted this: $$\ln(n)=\int^n_1 \frac{dx}x \le \sum _{k=1} ^n ...
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Connections of results in Harmonic analysis in the theory of Transcendental Numbers

Note :This question is proposed 2 years ago in MO , I see it appropriate for stackexhange math, i posted it here as it's unsolved problem and has a connection with Transcendental Numbers , mayeb we ...
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Fourier transform of $\frac{1}{x_1^2+x_2^2+x_3^2}$ [duplicate]

How can I find Fourier transform of $$\frac{1}{x_1^2+x_2^2+x_3^2}?$$
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37 views

example of maximal operator that is integrable

We know that there are no nonzero functions $f \in L^1(\mathbb R^n)$ such that $Mf \in L^1(\mathbb R^n)$, where $Mf$ is the Hardy Littlewood maximal function. Can we find a maximal operator that is ...
4
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1answer
90 views

Technical question about Strichartz estimate's proof.

I was studying the proof of Strichartz estimates from the book "Semilinear Schrödinger equation" of T. Cazenave. The proof is divided in several steps. Here we can assume ...
1
vote
1answer
73 views

Improvement of weak type inequality for Hardy-Littlewood Maximal inequality

Let $B(x,R)$ denotes the ball in centered at $x\in \mathbb{R}^n$ with radius $R$. The centered Hardy-Littlewood maximal operator $M$ is defined by \begin{equation} Mf(x)=\sup_{B(x,R)} ...
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1answer
79 views

Harmonic analysis on discrete groups

I am currently confused about the notion of generalized Fourier transforms on discrete groups (I'm only concerned about discrete groups here, not necessarily abelian). On finite discrete groups, I ...
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1answer
22 views

Hilber transform on [0,1)

Let $\mathbb{T}=[0,1)$ and $H$ be a Hilbert transform on $L^p(\mathbb{T})$ when $2\leq p< \infty$. If $f$ is $L^p$ and $f_n$ is trignometric polynomial such that $f_n\rightarrow f$ in $L^p$ sense. ...
9
votes
1answer
86 views

What are the “right” spaces for the Laplace transform

There are for example several canonical spaces to define the Fourier transform (i.e. Schwartz's space). Is there also a particularly suitable space to define the Laplace transform, so that the Laplace ...
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vote
1answer
42 views

Hilbert transform on $L^p(\mathbb{T})$

Let $\infty >p\geq 2$, then for $f\in L^p(\mathbb{T})$ (here $\mathbb{T}=[0,1)$), show that for any real-valued trigonometric polynomial $f$, we have $H(f^2-(Hf)^2)=2fHf$. The hint is to use the ...
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26 views

Looking for a proof involving the Harmonic number [duplicate]

Prove that: $\displaystyle \sum_{k=1}^{\infty} \frac{H_k}{k^q} = (1 + \frac{q}{2})\zeta(q + 1) - \frac{1}{2}\cdot \sum_{n=1}^{q-2}\zeta(k+1)\zeta(q-k)$ It looks tough just to start off with. Any ...
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38 views

question about property of $L^p$ Lipschitz space

$f\in L^p$ is said to satisfy $L^p$ Lipschitz condition of order $\alpha$ if there exists $C>0$ such that $\displaystyle|h|^{-\alpha}\Big(\int_{\mathbb{R}^d}|f(x-h)-f(x)|^p ...
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An interesting proof using Green's representation formula?

Let $B_R(0)$ be a ball in $\mathbb R^3 $ and define $$u(x)=\int_{B_R(0)}\frac{1}{|y-x|}dy$$ Prove that $$ u(x) = \begin{cases} \frac{2}{3}\pi(3R^2-|x|^2) & \quad \text{for $0 \le|x|\le R$ ...
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How to construct $\operatorname{End}(V_{\pi})$ using a representation $\pi$

Let $(\pi, V)$ be a representation of the group $G$. To make the setting as general as possible, I will not put any restrictions on $\pi, V$, and $G$ from the beginning. By the very definition, for ...
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1answer
64 views

Fourier transform of power function

Assume that $$\hat f(x)= (2\pi)^{-n/2} \int_{\mathbb{R}^n} f(y) e^{-i\left<x,y\right>} dy$$ is the Fourier transform of a function $f$. What is $\hat f$ if $f(x)=|x|^{2-n}$?
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Could characters in harmonic analysis be generalized into $S^2$?

Consider a locally compact abelian (LCA) group $G$. The start of commutative harmonic analysis is the fact that the collection of characters $\chi : G \to S^1$ (thought of as $S^1 = \mathbb{T} ...
2
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1answer
108 views

Questions about Fourier Series

I have recently started looking at the topoic of Fourier series. Consider the space of square integrable functions $L_{2}[0,2\pi]$. Where we define the inner product as $(f,g):= \int_{0}^{2\pi}fg dx$ ...
2
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1answer
48 views

Equivalence of Schwartz Space Definition

I've come across two definitions for what it means for a function to be in $\mathfrak S$, the Schwartz space. A function $f \in \mathfrak S$ if $f \in C^\infty$ and for all $j, k \geq 0$ integers, ...
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35 views

Pattern emerging from expansion of sphere in spherical harmonics

Seeing as one can express any function on the sphere in terms of the spherical harmonics, I am interested in what the sphere itself looks like when expanded. To get a function for the sphere, I use ...
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1answer
62 views

Definition of Zygmund class

I need help with showing that a function f belongs to the Zygmund class. Only helpful suggestions please, no full solution. I am here to learn for myself. This is work for school (we are allowed to ...
3
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1answer
60 views

Integrals of compactly supported functions of positive type

Consider a continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$, supported on $[-1,1]$, of positive type. Assume $f(0) = 1$; what is the "largest" area $\int f\,dx$ that can be achieved? To be ...
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35 views

Poisson integral with discontinuous $U$

Let $U$ be a piecewise continuous function and bounded for all real numbers. Then define the Poisson Integral for the UHP to be (It can be deduce from the one for the unit circle). ...
2
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1answer
67 views

About the Fourier transform of the surface measure of the unit sphere

Let $d\sigma$ denote the surface measure on $\mathbb{S}^{n-1}$. To compute its Fourier transform $$ \hat{d\sigma}(\xi)=\int e^{-i x\cdot \xi}\, d\sigma(x), $$ a standard technique (cfr. Folland's ...