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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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Surface measure as a finite Borel measure in $\Bbb R^n$

Given the result in http://www.math.jyu.fi/research/pspdf/236.pdf (theorem 1.2) I want to prove that for $ f \in L^2 (S^{n-1}) $ we have $$ \| \hat{f} \|_{L^q( \mathbb{R^n} )} \lesssim_{n,q} \| f \...
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38 views

Additive-Group-Homomorphisms from a Local Field to the Non-zero complex numbers

I'm quite familiar with the $p$-adic numbers and $p$-adic analysis, so I already know that any continuous group homomorphism $\varpi_{p}:\left(\mathbb{Q}_{p},+\right)\rightarrow\left(\mathbb{C}\...
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Sublinear operators and sublinear functions

Sublinear functions are defined as functions which are positive homogeneous and sublinear, i.e. $f( \alpha x) = \alpha f(x) \, \, \, \, \forall \alpha \geq 0 $ $f(x+y) \leq f(x)+f(y)$ In harmonic ...
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1answer
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In Banach *-algebra, every positive linear functional is bounded.

While proving this in class, Our Prof. did it into two cases: One case was if Banach *-algebra has the unit $u$. then he used the following result in his proof: "if $A$ is Banach *-algebra with unit $...
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Bounded linear operator in subset implies bounded in the whole space?

Let $T$ be an operator from $L^p$ to $L^p$, $S$ a dense subset of $L^p$, such that for any $f \in S$, $||T(f)|| < C||f||$. Then, can we affirm this property extends to all $f$ in $L^p$. If not, ...
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20 views

The density of $C_0^{\infty}$ in Morrey spaces

Recall that $f \in L^{p,\lambda}=L^{p,\lambda}(\mathbb{R}^n)$ if $$ \|f\|_{L^{p,\lambda}}=sup_{r>0,x_0 \in \mathbb{R}^n}\left[r^{\lambda-n}\int_{B(x_0,r)}|f(y)|^pdy\right]^{1/p} < \infty $$ I ...
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Inequality on Schwartz functions

Let $w:\mathbb R^n\to \mathbb R$ be a fixed strictly positive Schwartz funcion with the following property: $w(x)\geq 1$ on the unit ball $B(0,1)$ in the physical space. $\widehat {w^{1/2}}$ is ...
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1answer
28 views

Fourier transform of $\exp(-z^k)$: How can one quatify its decay?

Consider the Fourier transform of $\exp(-z^k)$ where $k$ is a positive integer. As the function is analytic, I expect it to have exponential decay at infinity. Is there some known theorem giving a ...
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1answer
52 views

Number System with Torsion

Introduction: A number system, long known but seldom seen, is (re)introduced for which some elements are torsion and some are torsion-free. A topology question and an analytic number theory question ...
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Calderon-Zygmund Theorem for Hardy space

Let $f\in\mathcal{H}^{1}(\mathbb{R}^{3})$ be a function in the calssical real Hardy space. $u$ solves the equation \begin{equation} -\Delta u=f\ \text{in}\ \Omega \end{equation} \begin{equation} u=0\...
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Is there a way to concretely see cyclic sub-representations of the “Euclidean group” on $\mathbb Z$?

Let $\mu$ be a finite Borel measure on $S^1$. We have an action of $\mathbb Z$ on $L^2(S^1, \mu)$ defined by $n\cdot \varphi = e^{2\pi i n}\varphi$. The following is a standard theorem in functional ...
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26 views

$L^2$ norm of Riesz Potential of Disk

I am interested in evaluating, or more precisely finding asymptotics in terms of $s$, of the following integral: $$2^{-s}\pi^{-1}\frac{\Gamma(\frac{2-s}{2})}{\Gamma(\frac{s}{2})}\int_{|x|\leq (2\pi)^{-...
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18 views

Class of function BV and finite oscillation

There is relation a function with Bounded Variation and a function with finite oscillation. A function with bounded variation has finite oscillation. But is there any function which has finite ...
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Evaluating this limit in Fourier analysis

In the study of Fourier transform of the standard Cantor measure, I came across the following problem: For $k\geq 1$, let $$ S_k=\sum_{m=3^{k-1}}^{3^k}\,\,\,\prod_{j=1}^\infty \cos^2\left(\frac {2m}{...
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How to use Van der Corput's lemma to get the following estimates?

Recently, I'am reading Tao's article Spherically Averaged Endpoint Strichartz Estimates for The Two-dimensional Schrodinger Equation, and the link of the article is there Spherically Averaged ...
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1answer
19 views

$\ell^1$ Space of a Group $G$

If $G$ is a topological group, how is $\ell^1(G)$ defined (is it necessary to require that $G$ be a topological group?)? From my understanding, it consists of all functions (continuous?) $f : G \to \...
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convolution of the multiplier and integrable function

Suppose $m(x)$ is a $L^p(R^n)$ multiplier, $f\in L^1(R^n)$,prove that the convolution $m\star f$ is also a $L^p(R^n)$ multiplier, and satisfy $$\|T_{m\star f\quad}\|_{L^p\rightarrow L^p}\leq\|f\|_1\|...
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1answer
24 views

Fourier exponential series of odd functions does not have even coefficients?

Consider the exponential fourier series $$ f(t)= \sum\limits_{n = - \infty }^\infty {c_n e^{i n\omega _0 t} } $$ if $f$ is odd $f(-t)=-f(t)$, then how is this visible in the series? Are there only ...
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Harmonic analysis needed for the Langlands program

I am sorry if this questions turns out to be "too broad" or "unclear", since I really know very little about the Langlands program. What I am wondering is that what kind of harmonic analysis is used ...
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Obtaining scaled Fourier restriction estimate

I have a restriction estimate: \begin{equation} \| \widehat{fd\sigma}\|_{L^2(V)} \le C\|V\|_{\mathcal{L}^{2,p}}^{1/2} \| f \|_{L^2(\mathbb{S}^{n-1})}. \end{equation} where $V$ is a positive function ...
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1answer
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weighted inequality of Fourier transform

For $1<p\leq2$, prove that $$\|\hat{f}\|_{L^p(R^n,|x|^{n(p-2)}\quad dx)}\leq C\|f\|_{L^p(R^n,dx)}$$ $\hat{f}$is the Fourier transform of $f$. It is trivial if $p=2$, I try to use holder ...
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2answers
18 views

If $u$ subharmonic, why $U_s=\{x\mid u(x)=s\}$ is open?

Let $U$ a domain and $u\in \mathcal C^2(U)\cap \mathcal C^1(\bar U)$ and let $s=\sup_{U}u$. Let $U_s=\{x\mid u(x)=s\}$. I want to prove that $U$ is clopen. I proved that it's closed (because $U=u^{-1}\...
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30 views

How do I obtain a solution of this Helmholtz equation?

I want to get a solution of the following Helmholtz equation. $$ \Delta u+ k^2 u = f, \qquad x \in \mathbb{R}^{n}, \; k>0$$ Using the Fourier transform, I have \begin{align*}u&=\int_{\mathbb{...
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14 views

Convolution of trig polynomials over a Group

I want to prove that $T(G)=T(G)*T(G)$ where G is an infinite compact abelian Hausforff Topological group. I'm trying to start this but really im confused with the convolution. Say $f,g \in T(G)$ I ...
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10 views

harmonic function degenerating at infinity

If $\Omega\subset R^n$ is a bounded domain with smooth boundary. Prove that there is a positive harmonic function $u$ on the complement of $\overline{\Omega}$, with $u=1 $on $\partial\Omega$ and ...
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2answers
42 views

Measurability of maximal functions

I have been reading Fourier Analysis by J. Duoandikoetxea. I got stuck on proving the measurability of maximal functions. The general maximal function/operator in this book is from the following ...
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Some explanatios in Wollf's Harmonic Analysis notes

I read these notes in harmonic analysis: http://www.math.ubc.ca/~ilaba/wolff/notes_march2002.pdf I started to study the proof of proposition 8.2 (page 54) The author constructs a measure using ...
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Is characteristic function radial function

I am starting to study the radial function. I am confusing whether the characteristic function $$ \mathbf{1}_{x_1\geq 0}(x),\quad x=(x_1,x_2,x_3)\in \mathbb R^3 $$ is radial function?.
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Siegel Transforms on Homogeneous Spaces?

For any integer $n \geq 2,$ we may identify the space of unimodular lattices in $\mathbb{R}^n$ with the homogeneous space $X_n := \mathrm{SL}_n(\mathbb{R})/\mathrm{SL}_n(\mathbb{Z})$ via the ...
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Reference Request: Martingales - Convex Analysis (-Harmonic Analysis) relationship

During my undergraduate studies I've encountered martingales, convexity (and also partial differential equations). Fast forward a few years, as a PhD student in applied math, I often find myself ...
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How can I prove that the Hilbert transform on the 1-torus doesn't map $L^1(\mathbb{T})$ into itself?

Let $\mathbb{T}$ be the 1-torus. Then, it is well defined the Hilbert transform: $$\mathcal{H}:L^1(\mathbb{T})\to L^0(\mathbb{T}), \vartheta\mapsto\int_{-\pi}^\pi f(\vartheta-t)\cot\left(\frac{t}{2}\...
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Boundness of Fourier thansform

Given $1\leq p\leq\infty$, prove that if it exists a constant C s.t. $$\|\hat{f}\|_{p'}\leq C \|f\|_p$$here $1/p+1/p'=1;$ then $1\leq p\leq 2$ I know that if $1\leq p\leq 2$, then it exists a ...
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2answers
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Why are there two different recurrences for Gegenbauer polynomials?

As I mentioned previously, I've been reading up on Gegenbauer polynomials in preparation for a blog post on the kissing number problem—specifically, the Delsarte method. To make a long story short, ...
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1answer
164 views

How to show a tempered distribution is a Schwarz function?

Given a tempered distribution $T$, in order to show that it is a Schwartz function, does it suffice to prove for any $f$ Schwartz, $T(f) = \int g f$ for some $g$ Schwartz? Now if $T$ is a tempered ...
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1answer
34 views

How to Calculate the Fourier transform of $\frac{1}{x-c}$?

How to Calculate the Fourier transform of $\frac{1}{x-c}$? $c$ here is a complex number which does not lie on the real line. First I should figure out how to interpret this function as a tempered ...
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An attempt to gain insight into a fascinating result on harmonic functions

I'm fascinated (probably by my lack of understanding of the topic) by the following result discussed in this paper Let $u,v$ be two harmonic functions on a compact domain $K$ such that their set of ...
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12 views

the infinity norm of triangular polynomial

Given $P(x)$ a triangular polynomial with order $n$, prove that $\|p'\|_{\infty}\leq4\pi n\|p\|_\infty$. I suppose $p(x)=\sum a_k\exp(2\pi ikt)$ and fail to solve it. I have trouble in the relation ...
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Question on a step of the proof of Theorem 1.25 of Introduction to Fourier Analysis on Euclidean Spaces

Theorem 1.25: Suppose $ \phi \in L^1(\mathbb{R^n}) $ and $ \int_{\mathbb{R^n}} \phi =1 $ . Also, let $\phi_{\epsilon}(x)=\frac{\phi\left(\frac{x}{\epsilon}\right)}{\epsilon^n}$.Moreover , suppose ...
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Kernel on Grassmannian

The Binet-Cauchy kernel on Grassmannians given in terms of principal angles $\theta_i$ is $K_{bc}^2 := \prod_i \cos^2(\theta_i)$, and is known to be positive definite. However, $L_{bf} := \prod_i \cos(...
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A question about Fourier transform of measures

Let $G$ be a locally compact abelian group and $M(G)$ the Banach space of all complex Radon measures on $G$. The convolution on $M(G)$ is defined by specifying a linear functional on $C_0(G)$ (and ...
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2answers
51 views

Relation between Holder norm and fractional laplacian

Let $f$ a function smooth sufficiently, e.g $f\in\mathcal{S}(\mathbb{R})$ ($\mathcal{S}(\mathbb{R})$ denote the Schwartz space), and consider $0<\beta,\alpha<1$ such that $1-\beta\leqslant \...
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1answer
42 views

How to calculate $c_a$ where $\left(f\mapsto\int_{\mathbb{R}}\frac{f(t)-f(0)}{|t|^{a}}dt\right)=c_a\mathcal{F}_x(|x|^{-1+a})$

From this question I know that for every $a\in\mathbb{R}$ there exists a unique radial positive homogeneous tempered distribution of degree $a$, up to a multiplicative constant. Also, it is easily ...
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20 views

Understanding Gelfand–Raikov theorem

The context is unitary representations of locally compact topological groups. Theorem (Gelfand–Raikov). Let $G$ be a locally compact topological group. Then $G$ is separated by its irreducible ...
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Dyadic decomposition in $\mathbb{R}$.

Consider the classical dyadic decomposition of $\mathbb{R}$ $$\{\Phi_j\}_{j\in\mathbb{Z}}\in \mathcal{S}(\mathbb{R}) \quad\text{ where }\quad \operatorname{supp} \widehat{\Phi}_j\subset A_j:=\{\xi\in\...
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A problem from Folland: constructing Haar measure from Lebesgue measure

The following is a result from A Course in Abstract Harmonic Analysis by G.B. Folland. It also appears as an exercise in the real analysis text by the same author. The context is Haar measure on a ...
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Interchanging Integration Order involving Fourier Transform

$$f(\omega,u):=\frac1{\omega+iu}$$ where $i$ is the imaginary unit number. We see that the integral of a Fourier transform $$\int_1^\infty du\int_{-\infty}^\infty d\omega\,f(\omega,u)\,e^{-i\omega x}=...
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45 views

Can we extend the Riesz potential convolution operator for the Laplacian to a continuous operator from $L^p$ to $\mathcal{S}'$ if $p\ge\frac{n}{2}$?

If $n\ge3$ and $\omega_n$ is the $n-1$-dimensional Hausdorff measure of the unit sphere in $\mathbb{R}^n$, define: $$K_n(x):=\frac{1}{(n-2)\omega_n} \frac{1}{|x|^{n-2}}.$$ Then $K_n$ is locally ...
2
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1answer
102 views

Shattering with sinusoids

Let $d \geq 2$ and $K$ some positive integer. Consider distinct points $\theta_1, \ldots, \theta_K\in \mathbb{T}^d$ and (not necessarily distinct) $z_1, \ldots, z_K \in \{-1,1\}$, does there exist an ...
2
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1answer
54 views

Can we say anything about the first distributional derivatives of $g$, where $g$ is the solution to $-\Delta g =f\in L^p$ given by Riesz potential?

If $n\ge3$ and $\omega_n$ is the $n-1$-dimensional Hausdorff measure of the unit sphere in $\mathbb{R}^n$, define: $$K_n(x):=\frac{1}{(n-2)\omega_n} \frac{1}{|x|^{n-2}}.$$ Then $K_n$ is locally ...
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0answers
15 views

How to show that $g^**g$ is a function of positive type, $g \in L^1(G) \cap L^2(G)$?

Let $G$ be a locally compact abelian Hausdorff group (with Haar measure $d\mu$). Call a function $h \in L^\infty(G)$ a function of positive type if $$ \int_G (f^* *f)h \, d\mu \geq 0 \ \ \ \forall f \...