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, ...

learn more… | top users | synonyms

1
vote
1answer
45 views

Open subgroup and group algebra

Let $G$ be a locally compact group and $H$ be an open subgroup of $G$. Consider the group algebras $L^1(G)$ and $L^1(H)$ with convolution product and consider $L^1(H)$ as a subalgebra of $L^1(G)$ ...
1
vote
1answer
17 views

Discontinuous characters on locally compact abelian groups

I'm doing some reading out of Rudin's Fourier Analysis on Groups and Folland's Abstract Harmonic Analysis for research. Particularly I am on characters and dual groups and I've noticed that neither ...
4
votes
1answer
49 views

Weak* convergence in $(L^q)^*$ and convergence in $L^p$

Let $\{\phi_n\}$ be a sequence in $L^p(X)$. Assume that $\phi_n\to \phi$ weak* under the natural identification $L^p\cong (L^q)^*$. Of course, it is not true in general that $\phi_n\to \phi$ in $L^p$ ...
1
vote
0answers
31 views

I am having a hard time understanding just how quotients such as $\mathbb{T}$ are constructed.

I encountered the quotient $\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}$ while beginning to study harmonic analysis. Despite having taken elementary abstract algebra and number theory, I am still having ...
4
votes
1answer
48 views

Fourier transform of a potential

I need help computing the distributional inverse Fourier transform of the function $1/|x|^2$ in dimension two. The integral makes sense written as \begin{align} 1/2\pi \int_{\mathbb{R}^2} e^{ix\xi} ...
5
votes
1answer
70 views

Using Plancherel's Theorem to Prove the Gauss Sum

I'm interested in proving the following: Where $p$ is an odd prime and $z$ is a primitive $p$th root of unity, we let $Q(p)=\sum^{p−1}_{k=0}z^{k^2}$. Prove: $|Q(p)|=\sqrt{p}$. Specifically, I want ...
0
votes
0answers
24 views

non degenerate representation

Let π:G →L(H) be a unitary representation Then the map π1: $\mathbb L^1(G)$→L(H) is nondegenerate. where ’nondegenerate’ is meant in the sense that For every non-zero $\xi \in \mathcal{H}$ ...
3
votes
1answer
53 views

A questions about the schur's lemma

Schur's lemma is this: If (ρ1,V1) and (ρ2,V2) are irreducible representations of a group G, then any nonzero homomorphism ϕ:V1↦V2 is an isomorphism. or Schur's Lemma. a. A unitary ...
2
votes
1answer
42 views

If the Fourier transform of a probability measure goes to zero at infinity, can the measure have a point mass?

Let $\mu$ be a probability measure on $\mathbb{R}$. Is the following implication true? $$ \widehat{\mu}(y) \rightarrow 0 \text{ as } |y| \rightarrow \infty \quad \Rightarrow \quad \mu(\{x\})=0 \quad ...
2
votes
0answers
58 views

Is every unitary representation a direct sum of irreducible subprepresentations?

I've read that any unitary representation of a compact group decomposes as a Hilbert space direct sum of irreducible representations. In the book I'm reading this is stated as a prong of the ...
2
votes
1answer
47 views

Module algebras

Spectrum: For Banach algebra $A$ spectrum is denoted by $\sigma(A)$ and defined as the set of all non-zero bounded linear multiplicative function from $A$ to $\Bbb C$.(Function $\psi:A\to\Bbb C$ is ...
0
votes
0answers
9 views

Kernel of a representation and central projections

Let $G$ be a locally compact group, and $B(G)$ be the Fourier-Stieltjes algebra of $G$. I'm trying to see why the following is true: let $\pi$ be a representation of $G$, the the kernel of $\pi$ in ...
1
vote
2answers
47 views

Why are all Laplacian eigenfunctions on the square obtained by separation of variables?

Let our domain $\Omega = (0,2\pi)\times(0,2\pi)$ be a square with sides parallel to the axes. Consider the Dirichlet eigenvalue problem $\Delta u+\lambda u=0$ with Dirichlet boundary condition $u=0$ ...
2
votes
0answers
66 views

Relation between fractional integral operator and solution of poisson equation

For $0<\alpha<d$, fractional integral operator $I_{\alpha}$ is defined by $$I_{\alpha}f(x)=\int_{\mathbb{R}^d} \frac{|f(y)|}{|x-y|^{d-\alpha}} dy$$ for any suitable function on $\mathbb{R}^d$. ...
3
votes
1answer
35 views

Application of weak $L^p$ estimate besides for proving boundedness of some linear operator

For all $1\leq p< \infty$, weak-$L^p(\mathbb{R}^d)$ space is defined as a set of all functions $f$ such that $$\gamma^p|\{x\in \mathbb{R}^d: |f(x)|>\gamma\}|<\infty$$ for every ...
0
votes
0answers
31 views

functions of positive type

We say that $∅∈\mathbb L^∞(G)$ is of positive type, iff $$ \iint f(x)\overline{f(y)} \phi(y^{-1} x)\;dy\;dx≥0 $$ for all $f∈ \mathbb L^1(G)$ The set of all continuous functions of positive type ...
0
votes
0answers
24 views

Computing the modularity function

Let B be the group of real matrices of the form \begin{pmatrix} 1 & x\\ 0 & y\end{pmatrix} with y ≠ 0. Then the modular function Δ is given by $$Δ\begin{pmatrix}1 & x \\ 0 & ...
2
votes
0answers
40 views

*-homomorphism and *-representation

I understand the concept of the unitary representation of G . A unitary representation of G is group homomorphism π:G→U(H) where H is a complex Hilbert space and U(H) is the group of unitary operators ...
2
votes
1answer
83 views

Continuity under the integral sign?

I've been reading Folland's Harmonic analysis book, in which he claims the following on page 56: Suppose $G$ is a locally compact (and of course Hausdorff) topological group G, $H$ a (closed) ...
2
votes
0answers
41 views

Can a continuous function on a compact group $G$ be interpreted as the sum in $C(G)$ of its Fourier series?

For a given function $f\in C(G)$ on a compact group $G$ its Fourier transform is defined as the family of operators $$ \widehat{f}_\sigma=\int_Gf(t)\cdot\sigma(t^{-1}) \ \text{d}\ t,\quad ...
4
votes
1answer
31 views

restriction of unitary operator is unitary?

Let $\mathcal{U}: \mathcal{H} \rightarrow \mathcal{H}$ be a unitary operator on a Hilbert space $\mathcal{H}$. If $\mathcal{K}\subset \mathcal{H}$ is a closed subspace such that ...
1
vote
0answers
45 views

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 ...
12
votes
1answer
201 views

A property of exponential of operators

Let $X$ be a Banach space. $A\in B(X)$ is a bounded operator. we can define $e^{tA}$ by $$e^{tA}=\sum_{k=0}^{+\infty}\frac{t^kA^k}{k!}$$ I am interested in this property: If $x\in X$, such that the ...
0
votes
1answer
50 views

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 $\Delta u + \lambda u = 0$ ...
1
vote
1answer
24 views

Covolution (space) over compact Lie groups

Let $G$ be a compact Lie group. Is there any way one can characterize the functions $\phi$ of the form $\phi=\psi\ast \psi^\ast$ in $C^\infty(G)$ where $\psi\in C^\infty(G)$? Here as usual ...
1
vote
1answer
40 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 ...
1
vote
1answer
43 views

proof that$ L^1 (G)$ is a subspace of $M(G)$

Let G be a locally compact group, and let $M(G)$ be the space of complex Radon measures on G. I identify the function f with the measure $f(x) \rm dx$ . but How do I prove this inclusion?؟ . .
3
votes
1answer
67 views

The function $f(t)=2+\sin(t)+\sin(t\sqrt2)$

The function $f$ defined on $\mathbb{R}$ by $$f(t)=2+\sin(t)+\sin(t\sqrt2)$$ can never reach $0$. Can we find some sequence $(t_n)_{n\geq0}$ such that $$\lim_{n \to \infty}f(t_n)=0 \ \ \ ?$$ Or in ...
0
votes
1answer
48 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 ?
2
votes
2answers
82 views

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 ...
1
vote
2answers
44 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 ...
1
vote
0answers
39 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 ...
0
votes
0answers
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 ...
0
votes
0answers
43 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 ...
0
votes
1answer
15 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 ...
0
votes
0answers
80 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, $ ...
1
vote
1answer
117 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 ...
2
votes
0answers
73 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 ...
0
votes
1answer
37 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 ...
1
vote
1answer
32 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 ...
1
vote
0answers
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 ...
1
vote
0answers
20 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 ...
1
vote
1answer
85 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 ...
0
votes
0answers
22 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 $\text{Im}z = 0$. ...
2
votes
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 ...
7
votes
1answer
127 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 ...
0
votes
0answers
9 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 ...
1
vote
0answers
82 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 ...
6
votes
1answer
41 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 ...
1
vote
0answers
28 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) ...