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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Harmonic functions using complex analysis

Suppose we want a harmonic function $f$ in the first quadrant with the boundary condition that for $\arg(z) = \theta$ where $\theta$ is fixed and along $\Im(z) = 0$ we have $f = 1$. Since ...
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I “proved” the first Laplacian eigenvalue has many eigenfunctions. Where's my mistake?

(For simplicity, say our domain $D \subset \Bbb R^2$, $\partial D$ is nice and smooth, and $\overline D$ is compact.) Let $\lambda_1$ be the first eigenvalue of the problem $\Delta u + \lambda u = 0$ ...
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42 views

A sequence of neighbourhoods with decreasing Haar measure in a non-discrete group

Let $G$ be a non-discrete (LCH) group. How can one find a sequence $(V_n)$ of compact, symmetric neighbourhoods of the identity element $1$ such that $\mu(V_n)\to 0$, where $\mu$ denotes the Haar ...
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62 views

Every unitary representation is a direct sum of cyclic representations.

Every unitary representation is a direct sum of cyclic representations. it can be proved without the Zorn's Lemma ?
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Non-unitary representation

How to prove $\pi :\mathbb R\to \mathbb C^2$, defined by $t\mapsto \begin{pmatrix} 1 & t\\ 0 & 1\end{pmatrix}$ is a non-unitary representation? Is the following correct? $\pi$ is a ...
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Want to prove certain sum representation of $\cot(x)$

So here is my problem, I would like to prove an identity I found in a book which was given without a proof. Namely $$-i\sum_{n\in\mathbb Z} \operatorname{sign}(n)\cdot e^{i2\pi nx}=\cot(\pi x)$$ I ...
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How to make sense of the Fourier transform of this distribution

I want to compute the Fourier transform of this distribution: $$D(f)=\int_{\mathbb{R}} f(t,t^2) \frac{dt}{t}$$ ($f$ a Schwartz function on $\mathbb{R}^2$, the integral interpreted with a Cauchy ...
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$ S_\lambda f = \sum_{j=1}^\infty \chi_{(-\infty,\lambda_j]}E_j f =(2\pi)^{-1}\int_{\mathbb{R}}\widehat{\chi}_{(-\infty,\lambda]} (t)e^{itP}f \,dt$

Let $P$ be a self-adjoint elliptic pseudo-differential operator on $M$ (compact manifold). I can not demonstrate that $$ S_\lambda f = \sum_{j=1}^\infty \chi_{(-\infty,\lambda_j]}E_j f ...
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20 views

Definition of Uniform Continuity on an LCA group

I was working on an exercise from Tao's An Epsilon of Room to show the existence of Haar measure on an LCA group. For one of part of the problem we have ($G$ an LCA group, $C_c(G)^+$ denoting the ...
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53 views

Mean Value Property and Harmonic functions: a simple exercise

Let $f:[a,b]\to\mathbb{R}$ such that for every $h>0$ such that $$(x-h,x+h)\subset[a,b]$$ we have $$f(x)=\frac{1}{2}(f(x-h)+f(x+h)).$$ How can I conclude that $f$ is harmonic in $[a,b]$? My idea ...
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304 views

Fourier Transform on compact groups

I'm trying to get my head around the concept of Fourier Transform on a compact group. The standard definition is $$\widehat{f}(\pi)=\int_Gdg\,f(g)\pi(g)$$ where $\pi\in\widehat G$, the Pontryagin ...
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36 views

pointwise limit of uniformly bounded sequence in $A(\mathbb T)$ is again in $A(\mathbb T)$?

Let $\mathbb T$ be a circle group, and $\hat{f}(n)= \frac{1}{2\pi}\int_{0}^{2\pi} f(t) e^{-int} dt;$ $(n\in \mathbb Z, f\in L^{1} (\mathbb T)).$ Put $A(\mathbb T)= \{f\in C(\mathbb T): \hat{f}\in ...
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22 views

$f\mapsto \sum_{n\in \mathbb Z} |\widehat{F(f)}(n)|$ lower semi continuous?

Let $T$ be a circle group, and $\hat{f}(n)= \frac{1}{2\pi}\int_{0}^{2\pi} f(t) e^{-int} dt;$ $(n\in \mathbb Z, f\in L^{1} (\mathbb T)).$ Put $A(\mathbb T)= \{f\in C(\mathbb T): \hat{f}\in ...
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35 views

Poisson Integral Formula for Complex Function

We are given the domain $\Omega = \{ |z| \le 1, \text{Im}z \ge 0\}$ and that for some analytic function $F$, $|F(z)| \le a$ on $|z| = 1$ in the upper half plane and $|F(z)| \le b$ on $\text{Im}z = 0$. ...
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28 views

Approximating a mean on $L^{\infty}(G)$ by a norm $1$ non-negative $f\in L^{1}(G)$

Let $G$ be a locally compact group. A mean $M$ on $L^{\infty}(G)$ is a continuous linear functional on $L^{\infty}(G)$ such that $1 = \langle 1 , M\rangle = \|M\|$. My Exercise: Show that the set ...
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137 views

Is every integrable function on the real line with compact support also square integrable?

I wonder that whether every integrable function on the real line with compact support is also square integrable ? In other words, is $L^1_c(\mathbb R)\subseteq L^2(\mathbb R)$ holds true? Thanks in ...
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48 views

Extension of character in Banach algebras

Let $A$ be a Banach algebra. The continuous linear functional $\phi:A\to\Bbb{C}$ is called character if it is non-zero multiplicative function i.e., for every $a,b\in A$ we have ...
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10 views

The convergence of Maximal function

Suppose $\Omega$ is open bounded and I have $u\in BV(\Omega)\cap L^\infty(\Omega)$. Take molifier of $u$, say $u_\epsilon$, then I have that $u_\epsilon\rightarrow u$ in $L^1$ and also $||D ...
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96 views

What is the Fourier series of $e^{\mu\cos\theta}$?

Motivation: I want to solve this convolution problem on the circle: find $f$, given $g$ and $$ g(\theta) = \int_{S^1} e^{\mu\cos(\theta-\phi)}g(\phi)\ d\phi. $$ To do this, I want to find the Fourier ...
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35 views

Unimodular groups

Let F be a non-archimedean local fields of characteristic p (for p any given prime number): the field of formal Laurent series Fq((T)) over a finite field Fq (where q is a power of p). Is GL(n,F) ...
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33 views

Complex exponential is not Fourier multiplier on $L^p$

I am having difficulty to show that the function $m(\xi):= e^{i|\xi|^2}$ is not a Fourier multiplier on $L^p$ when $p\neq 2$. Note that $m:\mathbb{R}^n\to \mathbb{C}$ is called an $L^p$ Fourier ...
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basics on spectral synthesis…

I am reading basic material on spectral synthesis (such as Rudin's book, and some papers of Herz), and have found various definitions for the spectrum of $f\in L^\infty$: 1.- the one I am used to is ...
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34 views

$L^{\infty}(G)$ 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 $x\in A, f\in A^{*}$, $x.f:y\mapsto f(yx)$ ...
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71 views

Proof that $\frac{dx\ dy}{x^2+y^2}$ is a Haar measure on the multiplicative group $\mathbb C\setminus\{0\}$

How can it be proven that for every Borel subset of $\mathbb{C}\setminus\{0\}$ as A we have $\mu(cA)=\mu(A)$? $$ ∬_{cA} \frac{dx\ dy}{x^2+y^2}=\iint_{A} \frac{dx\ dy}{x^2+y^2} $$ I'm confused...
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192 views

Is it true that the Fourier coefficient of convolution is the product of the coefficients?

what I mean by the title is the following: if we define the convolution between two $2\pi$-periodic, $C^1$ functions as $f*g(x) = (2\pi)^{-1}\int_{-\pi}^\pi f(x-y)g(y)dy$, is it true that the Fourier ...
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35 views

$f, \hat {f} \in L^{p} \cap L^{\infty} \implies f\in A(\mathbb R)$?

$1\leq p \leq \infty $. We put, $$X_{p}= \{f\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R) :\hat{f}\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R)\};$$ and we consider the algebra of Fourier ...
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$|supp(v)|=0$ implies the existence of $\lim_{\epsilon \to 0}\int_{|\theta -\phi|>\epsilon}\cot{(\pi(\theta-\phi))}dv(\phi)$

Let $v$ be a complex Borel measure on $[0,1]$ and $m$ be the Lebesgue measure. We define the support of measure by $$supp(v) = [0,1]-\cup\{I \subset [0,1]: v(I)=0\}$$ where $I$ is an interval. ...
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Real part of a holomorphic function is bounded by polynomial then the holomorphic function is a polynomial [duplicate]

Let $u$ be a harmonic function on $\mathbb{R}^2\cong \mathbb{C}$ such that $Ref= u$ where $f$ is an entire function. If $|u(z)|\leq |z|^n$ for any $z\in\mathbb{C}$, then $f$ is a polynomial of degree ...
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Density of a set of functions in Schwartz space

I have a difficulty doing the following problem: Let $S(\mathbb{R}^n)$ be the Schwartz space. I need to determine whether the following set of functions $A$: $$A= \{f\in S(\mathbb{R}^n): ...
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Jucys-Murphy elements confusion

I am taking a class called "Harmonic Analysis on Finite Groups" and am studying for an exam. We have recently been talking about the representation theory of the symmetric group (over $\mathbb C$). ...
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49 views

Characteristic Function as Fourier Multiplier

In lecture notes I have, it is mentioned that the characteristic function $\chi_I$ of an interval in $\mathbb{R}$ is an $L^p$ Fourier multiplier for $1 < p < \infty$. I thought this would be ...
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37 views

$f, \hat{f} \in L^{1}(\mathbb R) \implies \widehat {\text{Re}(f)}, \widehat{\text{Im}(f)} \in L^{1}(\mathbb R)$?

Let $f:\mathbb R \to \mathbb C$ such that $f(x)= (f_{1}(x), f_{2}(x))$; where $f_{1}(x)=\text{Re}(f(x))=\text{the real part of} \ f $ and $f_{2}(x)=\text{Im}(f(x))= \text{ the imaginary part of} \ ...
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55 views

Why are function spaces typically defined on open sets?

I never really bothered to ask this question and now it seems silly...but why do we always seem to define function spaces $X(U)$ (e.g. $L^2(U), BV(U)$ for open sets $U\subset\Bbb{R}^n$? What breaks if ...
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20 views

how to prove translation identity: $(\mathcal{F}^{-1}m\mathcal{F}f)(x)= e^{ikx}[\mathcal{F}^{-1}m(\cdot+k)\mathcal{F}(e^{-iky}f(y))](x)$?

Let $m, f\in S(\mathbb R^{n})$=The Schwartz space. My question: How to prove: $$(\mathcal{F}^{-1}m\mathcal{F}f)(x)= e^{ikx}[\mathcal{F}^{-1}m(\cdot+k)\mathcal{F}(e^{-iky}f(y))](x);$$ where ...
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47 views

Essential hypothesis of Fourier Inversion formula

Let $f\in L^{1}(\mathbb R)$ and we define its Fourier transform as follows: $\hat{f}(\xi)=\int_{\mathbb R} f(x)e^{-2\pi i \xi\cdot x} dx, (\xi \in \mathbb R);$ and we define $f^{\vee}(x):=\hat{f}(-x)= ...
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62 views

Solving the equation $\int G(t) dt =\frac{\sin x}{x}$

I have to solve the equation $$\int_{\mathbb R} \frac{f(t)}{1+(x-t)^2} dt =\frac{\sin x}{x}.$$ I tried change of variables to make the $\frac{1}{1+(x-t)^2}$ part resemble $e^{h(x)}$ so I can use the ...
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Real version of the Jensen's formula.

Prove the Jensen's formula $$\int_{T}f(z+re^{2\pi i\theta})d\theta-f(z)=\iint_{D(z,r)}\log{\frac{r}{|w-z|}}\Delta f(w)dm(w)$$ where $w$ is in $D(z,r)$ and $f$ is a two-dimensional $C^2$ ...
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33 views

On convergence rate of kernel approximation

Let $\{K_\epsilon\}$ be a sequence of mollifiers, (or often take heat kernels). We have known from classical analysis that if $f$ is uniformly continuous then the error ...
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$f_{n}\to f$ in $L^{1}\implies \hat{f_{n}}\to \hat{f}$ in $L^{1}$?

Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R .$ Suppose there exists $f_{n}, \in L^{1}(\mathbb R)$ such ...
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39 views

Estimative with kernels of the Riesz transform

Let $x=(x_1,x_2)$ and $k(x)=\frac{x_1}{|x|^3}$, $|x|>0$. Condider $x,y\in\mathbb{R^2},\xi=|x-y|, \tilde{x}=\frac{x+y}{2}$ then $|k(x-t)-k(y-t)|\leqslant C\frac{|x-y|}{|\tilde{x}-t|^3}$ when ...
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Differentiabilty of certain convolution

Let $\Gamma$ be a smooth closed curve. Suppose $f\in L^2(\Gamma,ds)$ and $g$ is defined everywhere on $\mathbb{C}$ with compact support. Moreover $\frac{dg}{dz}$ exists everywhere but point $z=0$. ...
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When can every unitary representation of a discrete abelian group be written as a direct sum of irreducible representations?

Suppose $\Gamma$ is a discrete abelian group. A unitary representation of $\Gamma$ is group homomorphism $\pi: \Gamma\to U(H)$ where $H$ is a complex Hilbert space and $U(H)$ is the group of unitary ...
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$\Bbb A_K'$ is a one dimensional $\Bbb A_K$ module

Let $\Bbb A_K'$ be the dual to the group of adeles $\Bbb A_K$ of some field $K$. Then $\Bbb A_K'$ is an $\Bbb A_K$ module by the prescription $$a\cdot \Psi(x) \mapsto \Psi(ax)$$ But why is $\Bbb ...
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75 views

Uniqueness for second order PDE $-\Delta u = c(u)$

I'm trying to solve the following problem: Let $\Omega$ be an open set in $\mathbb R^n$ and consider the equation \begin{cases} -\Delta u = c(u) & \text{ on } \Omega \\ u = 0 & \text{ on } ...
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82 views

A function $f:\mathbb{R} \to \mathbb{R}$ with infinite norm but finite weak seminorm

Let $0<q\leq p<\infty$. For $f:\mathbb{R}\to \mathbb{R}$, we define the norm \begin{equation} \|f\|_{p,q}=\sup_{a\in \mathbb{R},r>0} r^{\frac{1}{p}} \left(\frac{1}{r} \int_{a-r}^{a+r} ...
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66 views

An $L^1$ function whose Fourier series converges but not to itself

Do we have an $L^1$ function whose Fourier series converges almost everywhere but not to itself?
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88 views

Integrability of function and its Fourier transform implies differentiabilty

Is the the following true: "Assume $f\in L^1[0,1]$ and $\hat{f}\in L^1(\mathbb{R})$, then $f$ is differentiable a.e"
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59 views

Invariant functions under integral transforms

We all know Fourier transform has invariants such as $e^{-x^2}$, and another MSE post has shown the non-existence of invariant function under Hilbert transform using Fourier transform. I am wondering ...
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54 views

Does a complex polynomial can be approximated by a Euler-type exponential function?

Can one uniformly approximate a complex polynomial with a Euler-type exponential function? I mean: let $E\subset\Bbb C$ be a compact set with the connected complement $\Omega:= \bar {\Bbb C}\setminus ...
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51 views

Zero convolution of a function with a measure

Suppose $0\not=f\in L^1_{loc}(\mathbb{R}^2)$ and $\mu$ is a positive Borel measure with compact support. Given $f\ast\mu=0, $ what can be said about $\mu$?