Questions tagged [harmonic-analysis]

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-Paley theory, etc). Use the (wavelets) tag for questions on wavelets, and the (fourier-analysis) for more elementary topics in Fourier theory.

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10 views

What are some topics in representation theory of locally compact groups?

I am currently studying representations of locally compact groups and I find it a really interesting subject so I would like to know more. First I would like to ask if this is an active area of ...
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Probabilistic Hausdorff-Young Type Inequality

Let $1 \leq p <2$ and let $q$ be the Holder conjugate of $p$ so that $\frac{1}{p} + \frac{1}{q} = 1$. Show that for any $\epsilon >0$, there exists a Schwartz function $f \in S(\mathbb{R}^d)$, ...
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BMO space and representation theory

Does the BMO function have any applications in representation theory? Thanks!
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39 views

Problem of convergence in Schwartz class

Can someone help me to show the following: Let $\phi \in \mathcal{S}(\mathbb{R}^n)$ satisfying $$\partial^\alpha \phi(0) = 0,\;\;\; \forall |\alpha|<k,$$ for some integer $k>0$. Considere $f \...
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Questions while reading “$H^p$ spaces of several variables(Fefferman and Stein)”.

I have two questions while reading this paper: Hp spaces of several variables." Acta math 129 (1972): 167-193. Question 1. Line $7$ on page $147$ to line $13$ on page $148$ is a proof of "Theorem 3$\...
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What is the Schwartz-type space for Mellin transform?

It is well known that for $f\in S(\mathbb R)$, the Schwartz space, one can assert that $f^{(a)}$, $Ff^{(a)}$ (the Fourier transform of $f^{(a)}$) are also in $S(\mathbb R)$ for any $a=0,1,2,\ldots$. ...
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An equivalent formulation of a Fourier restriction inequality

This is a cursory remark in the text Oscillatory Integrals in Classical Analysis that I am unable to justify. Let $f \in L^p(\mathbb{R}^n)$. Then, the Fourier Restriction theorem states that $$\...
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Show that $f$ is $m$-times differentiable with $f^{(m)}\in L^2(\mathbb T)$ provided $k-m>\frac{1}{2}$.

Assume $f\in L^1(\mathbb T)$ and $\hat f(n)=O(|n|^{-k})$. Show that $f$ is $m$-times differentiable with $f^{(m)}\in L^2(\mathbb T)$ provided $k-m>\frac{1}{2}$. Since $\hat f(n)=O(|n|^{-k})$, we ...
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1answer
37 views

Defining left Haar measure on locally compact group

Let $G$ is a locally compact group that is homeomorphic to an open subset (say $U$) of $\Bbb R^d$ ,and let $\varphi$ be a homeomorphism of $G$ onto $U$. Show that if for each $a$ in $G$ the function $...
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Character of a non-compact topological group is not $L^2?$

I think this result is used in something I am reading though it's not completely clear to me. I am not very sure that if it's true, it should be. The statement is the following: A nontrivial ...
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24 views

Upper semicontinuous functions and the existence of continuous functions

So, I'm working on a proof of the lemma: If $u$ is subharmonic, then $\int_0^{2 \pi} u(a+r_1 e^{i \theta}) d \theta \leq \int_0^{2 \pi} u(a+r_2 e^{i \theta}) d \theta$, whenever $0<r_1 < r_2$. ...
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functional equation for adelic theta function

The following is taken from p. 53 of https://arxiv.org/pdf/1511.04265.pdf I understand how $\gamma_{\mathbb{A}}(x)$ is invariant under Fourier transform, but I don't see how we get $1/x$ in the ...
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28 views

Fourier coefficients of Eisenstein series

Several questions/reading reference requests for the following topics. I require some point-wise bounds on the absolute value of the Eisenstein series $E_{\mathfrak{a}}(\sigma_{\mathfrak{b}}z,1/2 + ...
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Abel transform for general $\nu$?

Everything I found on the internet about Abel transform is that it is defined by $$A:=F^{-1}H,$$ where $F$ is the Fourier transform and $H$ is the Hankel transform with Bessel function $J_0$ as its ...
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Isometrically embedding $L^1(H)$ into $L^1(G)$

Let $G$ be a locally compact group and $H$ a closed subgroup of $G$. Denote $\mu_G$ as Haar measure on $G$ and $\mu_H$ as Haar measure on $H$. Is it possible to embed $L^1(H)$ into $L^1(G)$? By Radon-...
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Stein's interpolation theorem (analytic interpolation)

Let $G^{\alpha}_k$ with $\alpha\in\Bbb C$ et $k\in\Bbb N^*$ be analytic family of operators such that: For $\tau\in\Bbb R$ $$\|G^{i\tau}_k f\|_{\infty}\leq C(1+|\tau|)^{\frac{1}{2}}\|f\|_1 $$ and ...
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Show that $\text{lip}_\alpha(\mathbb T)$ is the homogeneous subspace of $\text{Lip}_\alpha(\mathbb T)$ for $0<\alpha<1$.

Set $\mathbb T=\mathbb R/2\pi\mathbb Z$. And let $\text{Lip}_\alpha(\mathbb T)$, $0<\alpha<1$ denote the subspacce of $C(\mathbb T)$ consisting of the functions $f$ for which $$ \sup_{t\in\...
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Is a homogeneous Banach space on $\mathbb T$ always well defined?

A homogeneous Banach space $B$ on group $\mathbb T=\mathbb R/2\pi\mathbb Z$ is a linear subspace $B$ of $L^1(\mathbb T)$ having a norm $\|\ \|_B\ge\|\ \|_{L^1}$ under which it is a Banach space, and ...
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Is$ \prod_{n=1}^{\infty} (x-n\pi)$ a factor of $\sin x$?

As it holds for a polynomial it should be true for a power series too. Why not?
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Restricting the range of Riesz Kernel: Does the inequality still remains true?

I am reading something containing the following statement: We have seen that if $u$ is a smooth function defined on a ball $В \subset \mathbb{R}^n$ (possibly with infinite radius so that $В = \...
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Sup norm of Fourier transform of $ \frac{\sin |x|}{|x|^\lambda} \mathbb 1_{\{2^k\le |x| <2^{k+1}\}}, \ 0<\lambda<n $

It seems to me that in a paper of Charles Fefferman (open access), it is claimed in the introduction that (3rd page of the PDF file, 'page 11', $\lambda\in(0,n)$) $$\sup_{\xi\in\mathbb R^n}\left| \...
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A question from Stein's book, Harmonic Analysis, oscillatory integral of the second kind.

My question is about a claim in Stein's book"Harmonic Analysis:Real-Variable Methods, Orthogonality, and Oscillatory Integrals", page 379. It says that $$|K_{\lambda}(\xi,\eta)|\leq A_N(1+\lambda|\xi-...
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The Dirac measure as a weak limit of $L^2$ functions on a LCA group.

Let $K$ be a locally compact abelian group. In the proof of Proposition 2 (the proposition does not matter for my question) of this blog-post, Tao writes: $K$ comes with an invariant probability ...
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Prove that “$\sigma_n(S,t)\ge 0$ for infinitely many $n$'s” is enough for a series being a Fourier-Stieltjes series of a positive measure

Lemma 7.5(Katznelson's book on harmonic analysis) A series $S\sim\sum a_ne^{int}$ is the Fourier-Stieltjes series of a positive measure if, and only if, for all $n$ and $t\in\mathbb T$, $$ \sigma_n(S,...
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Prove that the linear operator $\Sigma_n$ on $B^\ast$ has norm $ \|\Sigma_n\|^{B^\ast}=1. $

A homogeneous Banach space $B$ on group $\mathbb T=\mathbb R/2\pi\mathbb Z$ is a linear subspace $B$ of $L^1(\mathbb T)$ having a norm $\|\ \|_B\ge\|\ \|_{L^1}$ under which it is a Banach space, and ...
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50 views

Trouble with Poisson integral

I'm continuing my studies about the space $\mathbb{T}$ and I reach the point in which are introced the Harmonic functions. Well up to now I have a little trouble with understanding the Poisson's ...
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35 views

Isometric isomorphism between Hardy Space $h^p(\mathbb{D})$ and $L^p(\mathbb{T})$

I know the question below is a known result but, I would need some help to prove it! Well, I know that in the Poisson integral induces an isometric isomorphism between $L^p(\mathbb{T})$ and the Hardy ...
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29 views

How Singular Integrals Theory is applied on Partial Differential Equations

Currently I'm interested in Singular Integrals Theory (I'm a beginner). I have read that this theory has deep relations with PDE's. For that reason I would like to know if there exists some web page, ...
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Calderon- Zygmund in torus

1) Is it true that for every $f \in L^p (\mathbb{T^n})$ (where $\mathbb{T^n}$ is the n-dimensional torus) there exists a unique $u \in W^{2,p} (\mathbb{T^n})$ : $$\Delta u = f$$ such that $u$ is ...
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1answer
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How exactly is the Fourier transform the same as the Gelfand transform?

This answer states the Fourier transform is the Gelfand transform on the Banach algebra $L^1(G)$ with convolution. I've read the resource linked in the answer, but I still have some confusion. My ...
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Special Schwartz function.

Prove that there exists $\varphi \in \mathscr{S}(\mathbb{R}^n)$(Schwartz space), satisfying $0\le \varphi \le 1$, $\text{supp}(\varphi)=\{x\in\mathbb{R}^n|\frac{1}{2}\le |x|\le 2\}$, such that $$ \...
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Fourier projections to polygons bounded on $L^p$ spaces?

The Hilbert transform $Hf(x) = \lim_{\epsilon\to 0} \frac1\pi\int_{|s|>\epsilon} \frac{f(x-s)}s \ ds$ is bounded on $L^p(\mathbb R)$ for every $1<p<\infty$, but not at the endpoints. Now, ...
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Proof of Bohr's inequality for Fourier transform

Suppose $g$ is differentiable such that $g, g'\in L^p(\mathbb{R})$ for all $p\in [1,\infty)$, and assume that the Fourier transform of $g$ satisfies $\widehat{g}(\xi)=0$ for all $\xi\in [-R,R]$ for ...
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estimate for exterior harmonic function

Problem Let $D\subset \mathbb{R}^3$ be a smooth, connected, bounded domain, consider Dirichlet boundary problem \begin{equation} \left\{ \begin{aligned} &\Delta u=0, &\mathrm{in}\ \ \mathbb{R}^...
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Computing the $L^p$ norm of an integral operator

Let $T : L^{p}(\mathbb{R}_{>0}) \to L^{p}(\mathbb{R}_{>0})$ be given by: $$ (Tf)(x) = \int_0^\infty \frac{f(y)}{x+y} \mathrm{d}y $$ I would like to show that this is a bounded operator for $ p \...
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Complete invariants and third-order polynomials

I am interested in understanding some general properties of complete invariant maps. An invariant is a map $$f: X \rightarrow Y$$ defined over objects $X$ and a transformation $\tau$ such that $f(...
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1answer
29 views

dimension of a vector space of harmonic functions among polynomials in two variables

I'm searching for harmonic functions inside a set of HOMOGENIOUS polynomials in two variables. Let's say that $$P_n = \{\sum_{i+j = n} a_{ij} x^i y^j \quad|\quad a_{ij} \in \mathbb{R}\}$$ Let's write $...
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1answer
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The Pontryagin dual of a discrete abelian group is compact

Suppose $G$ is a discrete abelian group. Show that $\hat{G}$ is compact. This exercise is the converse of this question: The Pontryagin dual of a compact abelian group is discrete An example of this ...
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Dyadic Decomposition For Oscillatory Integral

In Terence Tao's notes on oscillatory integrals, Tao mentions that if $a(x)$ is a smooth, compactly supported phase, with $a(x) = 1$ in a neighborhood of the origin, then for any $N$, $$ \int a(x) e^{...
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Control problem steady state

How can i solve this problem ? Find frequency $w_0$ such that $A_y <= 0.1$ for $w>w_0$ and $y_{ss}$ is the steady state solution. The function $G(s)$ is the transfer function and the input ...
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Inequality $|f'(0)|\leq 2|f(0)| \log{ \frac{1}{f(0)}}\leq 1 − |f(0)|^2.$

How to prove the following: If $f ∈ H(D)$ and $0 < |f(z)| < 1$ for all $z ∈ D$, then $$|f'(0)|\leq 2|f(0)| \log{ \frac{1}{|f(0)|}}\leq 1 − |f(0)|^2.$$ It seems to me that I should use Harnack's ...
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1answer
23 views

Dirac distribution and Sobolev spaces

I see the conclusion that $\delta_{x_{0}} \in H^{s}\left(\mathbb{R}^{n}\right)$ if and only if $s<-n/2$, where $ H^{s}\left(\mathbb{R}^{n}\right) $ is Sobolev space. from many places, and the ...
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Convergence of smoothing operators, Schwarz seminorms and mistakes in the book Modern Fourier Analysis

The question is about exercise 8.3.2 part (c) of Grafakos's book Moder Fourier Analysis. The hint seems to be wrong and as there has been a couple of situations like this with the book I want to make ...
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Need help understanding a proof on sets of divergence

I have few questions regarding the proof to the theorem : ( Katznelson "An Introduction to Harmonic Analysis" Chapter 2.3 What I am struggeling to understand is the last bit of the proof: Why does $ ...
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A question about generators of locally compact topological groups.

I have a question to a theorem(5.13) of Hewitt & Ross: Harmonic analysis volume 1: Consider a locally compact abelian group G. If there is a neighbourhood U of the neutral element e such that the ...
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1answer
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Why is $\int_{-\infty}^\infty |\text{sinc}(t)|\ dt=\infty$? [duplicate]

Can you help me with that: Why is $\int_{-\infty}^\infty |\text{sinc}(t)|\ dt=\infty$ ? I saw this in Signal Processing course and I can’t understand why this is true. Reference: https://dsp....
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28 views

Couterexample to Littlewood-Paley theorem

Let $d\geq 2$ and let $P_j$ be the Fourier multiplier defined on $L^2(\mathbb{R}^d)$ by $\widehat{P_kf}(\xi)=\mathbf{1}_{2^k<|\xi|\leq2^{k+1}} \hat f(\xi)$, for any $k \ge 0$. It has been proven by ...
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1answer
57 views

Reference on periodic Besov spaces

I am looking for a reference for the construction and the study of the main properties of Besov spaces on the torus (i.e. $\mathrm{B}_{p,q}^s(\mathbb{T}^d))$. Indeed, I know the classical $\mathrm{B}...
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1answer
34 views

Show that $T$ is a $(p,p)$-opeator.

Let $\varepsilon>0$, $f\in L^p(\mathbb{R})$ ($1\le p\le \infty$). Opeator $T$ is defined as $$ Tf(x)=\int_{|x-y|>1}\frac{f(y)}{|x-y|^{n+\varepsilon}}\,\mathrm{d}y. $$ Show that $T$ is a ...
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28 views

How close can $\mathrm{Ad}(g)x$ be to the identity?

Let $G = \mathrm{SL}(d, \mathbb{R})$ where $d \geqslant 3$. For $g \in G$, define the operator norm of $g$ as $$ \Vert g \Vert_\mathrm{op} := \max \big\{ \Vert gXg^{-1} \Vert_\rho : X \in \mathfrak{...

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