All Questions
9 questions
20
votes
0
answers
333
views
Existence of orthonormal basis for $L^2(G)$ in $C_c(G)$
Remark: I cross-posted this question on MSE and added a bounty to it.
Suppose that $G$ is a locally compact (Hausdorff) group endowed with the Haar measure. It is well-known that the compactly ...
- 321
15
votes
1
answer
498
views
For what LCH groups is the Haar measure $\mu(U x U)$ bounded?
Let $G$ be a locally compact Hausdorff (LCH) topological group with left Haar measure $\mu$. Given a compact unit neighborhood $U$, consider the function
$$
\Phi: \quad G \to (0,\infty), \quad x \...
- 734
5
votes
0
answers
194
views
Haar mesure on $\mathrm{GL}_{d}(F)$
$\DeclareMathOperator\GL{GL}$Let $F$ be a $\mathfrak{p}$-adic field and $\mathscr{O}_{F}$ its valuation ring. For any measurable subset of $M_{d}(F)$ such as
$$
A=
\left( \begin{array}{ccc}
a_{11}+t^{\...
- 479
3
votes
1
answer
347
views
Subgroups of finite non-zero Haar measure of abelian locally compact groups
Is it true that every subgroup of finite non-zero Haar measure of an abelian locally compact group should be open and compact? This is obviously true for the case of discrete abelian groups. Thanks.
- 895
3
votes
0
answers
740
views
Justification of the convolution operation of $L^1(G)$ functions where $G$ is a LCA group (measurability)
Suppose $G$ is a locally compact abelian Hausdorff group (LCA), and $\lambda$ is the Haar measure on it. We all know the convolution of two $L^1(\lambda)$ functions $f$ and $g$ on $G$ is defined as
$$...
- 960
2
votes
1
answer
257
views
Haar measures of compact subgroups
Let $G$ be a locally compact group, $K$ a compact subgroup in $G$, and $\mu_K$ the normalized Haar measure on $K$:
$$
\mu_K(K)=1.
$$
Let us denote by $\widetilde{\mu_K}$ the measure on $G$ defined as ...
- 7,426
1
vote
0
answers
154
views
measure of Haar
Let $(G,K)$ be a Gelfand pair.
Why, for a function $f$ $K$-binvariant with respect to a compact subgroup $K$ of a group $G$, do we have the following equality:
$$ f(xy) = \int_K f(xky) \, dk$$
A ...
- 109
1
vote
1
answer
647
views
Haar measure coming from Pontryagin duality v/s Fourier inversion
Not research but advertising this question from mse in case someone wants to answer.
I'm struggling with some bookkeeping associated with the Pontryagin duality theorem. I'm thinking about the first ...
- 675
0
votes
0
answers
97
views
How large this subset is to say that it should equal the group?
Let $\alpha$ be a continuous automorphism on a compact group $G$ with normalized Haar measure $m$. We may say $\alpha$ is $n$-splitting, if the set
$$\text{Spl}_n(\alpha):=\left\{g\in G: \prod_{k=1}^...
- 2,118