Questions tagged [pontrjagin-duality]
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16
questions
4
votes
2answers
138 views
Measure algebra on the Bohr compactification vs the bidual algebras
The following question probably reduces to some standard abstract harmonic analysis Twister play, but I'd still welcome some comments on it.
Let $G$ be a locally compact Abelian group and let $bG$ ...
7
votes
0answers
169 views
Duality of Hopf algebras and duality of spectra
Let $S$ be the sphere spectrum, and for $X$ a topological space, let $S(X)$ be the mapping spectrum from the free loop spectrum on $X$ to the sphere spectrum. This is an $E_\infty$ ring spectrum (also ...
4
votes
0answers
409 views
Is Serre duality related to Pontryagin duality?
I am wondering if there is some relationship between Serre duality and Pontryagin duality for compact complex manifolds. In this case Serre duality reduces to the commutativity of Hodge-star operator ...
6
votes
1answer
361 views
Pontriagin reflexivity of the character group
For an Abelian topological group $G$ by $G^{\wedge}$ we denote the Pontryagin dual of $G$, i.e. the group of continuous homomorphisms $G\to\mathbb T:=\{z\in\mathbb C:|z|=1\}$. The group $G^{\wedge}$ ...
16
votes
1answer
905 views
A possible mistake in Walter Rudin, “Fourier analysis on groups”
I have the following lemma 4.2.4 on page 80 in the book (we have locally compact abelian topological groups $G_1, G_2$ and their duals $\Gamma_1, \Gamma_2$):
Suppose $E$ is a coset in $\Gamma_2$ ...
2
votes
0answers
167 views
Conceptual explanation for Poisson summation formula
The Poisson summation formula says that for a Schwartz function $f : \mathbf R^d \to \mathbf R$ and its Fourier transform $\widehat f$, we have
$$\sum_{n \in \mathbf Z^d} f(x) = \sum_{n \in \mathbf Z^...
1
vote
1answer
203 views
Creating Duals in A Category
Before stating my question I would like to provide afew motivating examples:
Examples:
In the category of Finitely-generated-projective $R$-modules, we have that:
$M^{\vee}:=Hom_R(M,R)$ satisfies: $...
2
votes
1answer
221 views
Bohr compactification and “discretization”
Let $G$ is a compact group. We can form the Pontriagin dual $\widehat{G}$ of $G$: it is then discrete space. One can consider the Bohr compactification $b\widehat{G}$ of $\widehat{G}$ which is compact ...
2
votes
1answer
338 views
What is the dual of a pre-injective map?
In [M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math.
Soc. (JEMS) 1 (1999), 109–197], Gromov introduces the notion of pre-injective map. Recasting this notion in the setting of ...
2
votes
1answer
2k views
Proof that the Pontryagin dual of a topological group is a topological group
I'm looking for a proof that the Pontryagin dual $G^*$ of a topological group $G$ is a topological group.
It's very easy to prove that $G^*$ is a group, my troubles are in proving that the map $G^* \...
5
votes
4answers
3k views
Quick computation of the Pontryagin dual group of torus
I'm looking for a quick way to compute the Pontryagin dual group of the n-dimensional torus $\mathbb{T}^n$ (with $\mathbb{T} := \mathbb{R} / \mathbb{Z}$). The only way I know is from "Dikran Dikranjan ...
17
votes
6answers
2k views
Discrete-compact duality for nonabelian groups
A standard property of Pontrjagin duality is that a locally compact Hausdorff abelian group is discrete iff its dual is compact (and vice versa). In what senses, if any, is this still true for ...
1
vote
1answer
996 views
Fourier Transform of measure on Banach Space (a question about Pontryagin Duality)
The following definition is given as the Fourier transform of a Borel probability measure $\mu$ on $E$, a Banach Space (Real):
$\hat{\mu}: E^*\rightarrow \mathbb{C}$ defined by
$\hat{\mu}(x^*):=\...
11
votes
6answers
6k views
Two reference requests: Pinsker's inequality and Pontryagin duality
Sorry for such a newbie post and for asking two unrelated references in one shot.
First, I am interested in any proof of Pinsker's inequality.
Second, I wonder what is the best place to read about ...
11
votes
1answer
8k views
Fourier transforms of compactly supported functions
One manifestation of the uncertainty principle is the fact that a compactly supported function $f$ cannot have a Fourier transform which vanishes on an open set. As stated, this phenomenon applies ...
4
votes
2answers
1k views
Injective modules and Pontrjagin duals
Forgive me for this naive question.
We consider the following lemma and its proof in Lang's algebra, Third Ed., published 1999, Chap. 20, section 4, page 784.
Every module is a submodule of an ...
