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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.

Harmonic analysis can be considered a generalisation of Fourier analysis that has interactions with fields as diverse as isoperimetric inequalities, manifolds and number theory.

Abstract harmonic analysis is concerned with the study of harmonic analysis on topological groups, whereas Euclidean harmonic analysis deals with the study of the properties of the Fourier transform on $\mathbb{R}^n$.

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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 zeroes are exactly the same. Let $f=\frac{u}{v}$.…
user67803
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Minimize $\int_{-\pi}^{\pi}|x-a-be^{3ix}-c^{5ix}|^2dx$ for $a,b,c \in \mathbb{R}$

Minimize the term $\int_{-\pi}^{\pi}|x-a-be^{3ix}-c^{5ix}|^2dx$ for $a,b,c \in \mathbb{R}$ Thoughts- So it's a rather common problem, i know that if $ \{1, e^{3ix}, e^{5ix} \} $ span a vector space, then the minimum of the term is the minimum of $…
Yariv Levy
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Background for Hejhal's "The Selberg Trace Formula for $PSL(2, \mathbb{R})$

I'm trying to learn the Selberg trace formula, but have very little background. I was referred to Dennis Hejhal's The Selberg Trace Formula for $PSL(2, \mathbb{R})$ but just got the book and was dismayed to learn that that the author assumes…
Jonah Sinick
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Maximum principle of harmonic function without mean value formula

Are there any way to prove maximum principle of harmonic functions without the mean value formula? In other words I would like to show $$ \max_{\overline{\Omega}}(f)=\max_{\partial \Omega}(f) $$ for a harmonic function $f$ on a bounded domain…
Pooya
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"Reverse" Triangle Inequality in weak $L^p$

Consider the $L^p$-weak quasi-norm, for $p>1$, that is $$\|f\|_{L^p_w(U)}=\sup_{t>0} t \mu\big(\left\{x \in U : |f(x)| > t \right\}\big)^{1/p}$$ where $\mu$ is the Lebesgue measure on $\mathbb{R}^n$. Is it true that there exist two positive…
user361486
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distribution is a limit of polynomials

I have a question about distributions. If we consider locally integrable function, then it is a limit of polynomials in the sense of $L^1$ convergence in any fixed compact set. Is every distribution also a limit of polynomials in the distribution…
Ale
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Are holder space dense in $L^p$ ? or in Schwarz space?

I recall that $f\in \mathcal C^\alpha ([0,1[)$ where $\alpha \in (0,1)$ if $$[f]_\alpha :=\sup_{x,y\in\mathbb [0,1[}\frac{|f(x)-f(y)|}{|x-y|^\alpha }<\infty .$$ Does those space are dense in $L^p$ space ? Or in Schwarz space ? I'm asking this…
MSE
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$L^p$ bounds for linear operators

A typical way to show $L^{p}$ boundedness of a linear or sub-linear operator is to show a weak type $(1,1)$ bound and an $L^{p}$ bound for some $p$, and then combine these two using interpolation to prove $L^{p}$ bounds for all $p$ between $1$ and…
user8621
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Decay of heat kernel on $\mathbb{T}$

I'm studying Muscalu and Schlag's Classical and Multilinear Harmonic Analysis, v. 1. One problem asks to study the heat equation on $\mathbb{T}$, i.e. $$u_{t} = u_{\theta \theta} \quad \text{on} \, \mathbb{T}, \quad u(0,\cdot) = u_{0}$$ for some…
user81375
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For Dirichlet's problem on the unit disk, show that the solution is harmonic.

I am trying to prove that $u(re^{i\theta}) = \sum\limits_{n=-\infty}^{\infty}\hat{f}(n)r^{|n|}e^{in\theta}$ is the solution to the Dirichlet problem on the unit disk if on the boundary of the unit disk, we have the Fourier series expansion of the…
Justin
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Fourier transform of power function

Assume that $$\hat f(x)= (2\pi)^{-n/2} \int_{\mathbb{R}^n} f(y) e^{-i\left} dy$$ is the Fourier transform of a function $f$. What is $\hat f$ if $f(x)=|x|^{2-n}$?
user190146
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Haar measure on $ \mathbb{R} × \mathbb{T}$ and on dual $\mathbb{R} × \mathbb{T}$

I've solved this exercise somewhat.To complete it please help me Haar measure on G $=$ translation invariant on G $$ μ(A)=μ(A+t)$$ if $ G=\mathbb{R}$ then Haar measure on G is lebesgue measure. and if $G=\mathbb{T}$ then Haar measure on G is…
user147972
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Suppose G is unimodular. If f is in LP(G) and g is in Lq(G) where 1 < p,q < ∞ and 1/p+1/q=1, then f * g ∈Co(G) and ||f * g||sup ≤ ||f||p ||g||q

Suppose G is unimodular. If f is in LP(G) and g is in Lq(G) where 1 < p,q < ∞ and 1/p+1/q=1, then f * g ∈Co(G) and ||f * g||sup ≤ Ilfllp llgllq. * is convolution f and g. I read the proof of this theorem.This is proof of the folland's…
user147972
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Absolute value operation in frequency domain

Let $f\in L^2(\mathbb{R}^d)$ be a real, positive function and $h\in L^2(\mathbb{R}^d)$ a complex function with compact support in frequency domain and $0\notin \text{supp }\hat{h}$. I am looking for an expression of $\mathcal{F}(|f\star h|)$ in…
Armin
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$L^{\infty}(G)$ as a Banach $L^{1}(G)$-bimodule. (Check my logic please!)

Background: Given a Banach algebra $A$, we can turn $A^{*}$, the Banach space dual of $A$, into a Banach $A$-bimodule via the following module actions: For $x\in A, f\in A^{*}$, $x.f:y\mapsto f(yx)$ and $f.x:y\mapsto f(xy)$. Example: Let $G$ be a…
roo
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