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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Non uniform continuity of a function and almost periodicity

We say that a continuous function $f:\mathbb{R}\to\mathbb{C}$ is almost periodic in the sense of Bohr if: For every sequence $(t'_n)_{n\geq0}$, there's a sub-sequence $(t_n)_{n\geq0}$ such that ...
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46 views

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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41 views

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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14 views

$ 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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44 views

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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1answer
18 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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1answer
124 views

Prove that the Pontryagin dual of a locally compact abelian group is also a locally compact abelian group.

Let $ G $ be a locally compact abelian (LCA) group and $ \widehat{G} $ the Pontryagin dual of $ G $, i.e., the set of all continuous homomorphisms $ G \to \mathbb{R} / \mathbb{Z} $. Clearly, $ ...
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1answer
121 views

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 ...
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83 views

convolution of $f$ and $g$ is in $L^p$ where $f$ has compact support.

Suppose 1≤p≤∞ and $f∈L^1(G)$ and $g∈L^p(G)$. I know this is true $‖f*g‖_p ≤‖f‖_1 ‖g‖_q$ , Where ∗ is convolution of f and g. I want to prove that If G is not unimodular, we still have g∗f is in LP ...
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1answer
49 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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1answer
35 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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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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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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1answer
104 views

Show that $\log(x)$ is a Bounded Mean Oscillation (BMO)

As an extension of our class notes, we were asked to show that the function $w =\log(x)$ is a Bounded Mean Oscillation (BMO). First off, I believe our professor made a mistake, and really wanted us ...
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32 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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1answer
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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1answer
134 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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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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95 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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1answer
46 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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34 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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24 views

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 ...
3
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1answer
32 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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1answer
160 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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1answer
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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69 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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23 views

$|supp(v)|=0$ 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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23 views

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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60 views

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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36 views

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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1answer
47 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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1answer
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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1answer
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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1answer
19 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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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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1answer
51 views

How to show the completeness of the space of Fourier transforms $\mathcal{F}L^{1}$?

Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ...
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55 views

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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1answer
71 views

Hardy Space Cancellation Condition

I have been reading the chapter on Hardy spaces in Stein's Harmonic Analysis book, and I am having a lot of trouble figuring something out. The setting here is $\mathbb{R}^n.$ Let $f \in L^q$ be ...
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2answers
66 views

Involution on inverse semigroups

I'm trying to prove the following for inverse semigroups $\bf Def:$ an inverse semigroup $S$is a semigruop such that for each $x\in S$ the exists a unique $y\in S$ such that $xyx=x$ and $yxy=y$. An ...
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2answers
45 views

$f_{n}\to f$ 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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1answer
38 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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37 views

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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1answer
31 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 ...
3
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1answer
62 views

Biprojective $C^*$-algebra

Let $A$ be a Banach algebra. Define $\Delta:A\hat{\otimes}A\to A$ with $\Delta(\sum_{n=1}^\infty a_n\otimes b_n)=\sum_{n=1}^\infty a_nb_n$. Now $A$ is called biprojective if there exists a bounded ...
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18 views

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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52 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 ...
4
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1answer
86 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"
4
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1answer
80 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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49 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 $\mu$?
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1answer
92 views

Addition of two cosine waves with different periods

I was just wondering if anyone knows how to add two different cosine equations together with different periods to form one equation. Is there a way to do this and get a real answer or is it just all ...