3
votes
0answers
48 views

Building up the Fourier inversion theorem on locally compact abelian groups

I'm reading through Folland's Abstract Harmonic Analysis and I've come to a bit of a road block with some machinery developed for the Fourier inversion theorem. We know that the Fourier transform of a ...
1
vote
1answer
13 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 ...
0
votes
0answers
8 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
1answer
38 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
0answers
26 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) ...
2
votes
0answers
21 views

Induced measure on dual group and kernel of fourier transform

Let $G$ be a locally compact Hausdorff abelian group (LCA). Then the Fourier transform gives a map from functions on G to functions on $\hat G$ $$f\mapsto \hat f(\xi) = \int_G f(x)\Psi(x,\xi) ...
0
votes
1answer
64 views

The dual of L^1(G) for a locally compact group G

I might be missing something, but most literature on topological groups and harmonic analysis that I've encountered mention that $L^\infty(G)$ can be naturally identified with the dual of $L^1(G)$ by ...
1
vote
1answer
54 views

When does a left Haar measure on a locally compact group restrict to a left Haar measure on a locally compact subgroup?

Given a locally compact group which is not $\sigma$-compact, there exists a $\sigma$-compact subgroup $H$ of $G$ which is open and closed. A Remark in Folland's, A Course in Abstract Harmonic ...
3
votes
1answer
85 views

Is there a formula for the Haar measure on a product of groups?

Let $ (G_{n})_{n \in \mathbb{N}} $ be a sequence of locally compact topological groups with a corresponding sequence $ (\mu_{n})_{n \in \mathbb{N}} $ of Haar measures. Is there a way to construct a ...
1
vote
1answer
56 views

Convolution of $L^1(G)$ functions with elements of $M(G)$.

Let $G$ be a non-discrete locally compact group with left Haar measure $\mu$. There is an isometric embedding of $L^{1}(G)\to M(G), f\mapsto fd\mu$. Since $G$ is not discrete, the point-mass measure ...
3
votes
0answers
76 views

Mapping $G$ into its group algebra as left multiplication. Continuous?

I am reading an appendix on Group algebras which contains the following Proposition which I am trying to prove: Proposition: Let $G$ be a locally compact group, with $\zeta\in L^{p}(G)$ fixed. ...
1
vote
1answer
36 views

Characterisation of the spectrum of certain unitary representations on $L^2(G)$

I have been informed of the existence of theorems in harmonic analysis that will allow me to calculate the spectrum of given unitary operators $L^2(G)$ where $G$ is a locally compact group. So far I ...
2
votes
1answer
207 views

Are nilpotent Lie groups unimodular?

The continuous homomorphism $\Delta:G \rightarrow \mathbb{R}^{\times}_+$ is defined by \begin{equation*} \int_G f(xy)dx = \Delta(y)\int_Gf(x)dx \end{equation*} where $dx$ is a left Haar measure on ...
3
votes
2answers
191 views

Shifting a function is continuous

I'm slightly puzzled by the following: if $g(t)$ is a function in $L^q(X)$ then we can show that $g(t-x)$ is continuous function of $t$, i.e. for $\varepsilon > 0$ we can find $\delta$ such that ...
7
votes
1answer
501 views

A net version of dominated convergence?

Let $G$ be a locally compact Hausdorff Abelian topological group. Let $\mu$ be a Haar measure on $G$, i.e. a regular translation invariant measure. Let $f$ be fixed in $\mathcal{L}^1(G, \mu)$. ...
12
votes
1answer
553 views

Properties of Haar measure

Let $G$ be a locally compact group (but not discrete) and let $m$ be its left Haar measure. Is it true that $\forall \epsilon$ $\exists$ $C$ such that $C$ is a compact neighborhood of the identity and ...
3
votes
1answer
261 views

Left regular representation of $L^1(G)$ for a locally compact group $G$

Let $G$ be a locally compact group (not discrete) and let $L$ be the left regular representation of $A = L^1(G)$ on itself i.e. $L: A \to \mathcal{B}(A)$ where $L(f): A \to A$, $L(f)(g) = f*g$. I want ...
2
votes
1answer
645 views

Young's inequality for discrete convolution

Young's inequality for convolution of functions states that for $f\in L^p(\mathbb{R}^d)$ and $g\in L^q(\mathbb{R}^d)$ we have $$\|f\star g\|_r\le\|f\|_p\|g\|_q$$ for $p$, $q$, $r$ satisfying ...
11
votes
1answer
340 views

Stone-Weierstrass implies Fourier expansion

To prove the existence of Fourier expansion, I have to solve the following exercise, which supposedly follows from the Stone-Weierstrass theorem: Let $G$ be a compact abelian topological group ...
2
votes
1answer
186 views

Surjective endomorphism preserves Haar measure

How to prove the following statement: Let $G$ be a compact topological group and let $m$ be the Haar measure on it. Let $\varphi$ be a continuous endomorphism of $G$ onto $G$, i.e., the map $\varphi$ ...
3
votes
0answers
145 views

Steinhaus theorem in topological groups

Let $(G,\cdot)$ be a locally compact Abelian topological group with Haar measure. It is known that if $B$ is a measurable subset of $G$ of finite and positive Haar measure then $int(B \cdot B)\neq ...
7
votes
1answer
814 views

Theorem of Steinhaus

The Steinhaus theorem says that if a set $A \subset \mathbb R^n$ is of positive inner Lebesgue measure then $\operatorname{int}{(A+A)} \neq \emptyset$. Is it true that also ...
6
votes
0answers
420 views

Positive definite function zoo

A positive definite function $\varphi: G \rightarrow \mathbb{C}$ on a group $G$ is a function that arises as a coefficient of a unitary representation of $G$. For a definition and discussion of ...
0
votes
1answer
255 views

Transitive group actions and homogeneous spaces

Given a topological group $G$ and a space $X$ with a transitive $G$ action, let $G_x$ be the isotropy group of a point. In Folland "A course in harmonic analysis", there is a statement that $X$ is ...
4
votes
1answer
558 views

Reference request: Fourier and Fourier-Stieltjes algebras

I would like to learn the basic theory of Fourier algebras and Fourier-Stieltjes algebras. In particular, I want to know how these two objects are defined in the case of not necessarily abelian ...