# Tagged Questions

**2**

votes

**1**answer

162 views

### Fourier series of functions on compact groups

Let $G$ be a compact, second countable, Hausdorff topological group with the normalized Haar measure $\mu$. From Peter-Weyl's theorem we now that for any $f\in \mathrm{L}^2(G)$ the Fourier series of ...

**0**

votes

**1**answer

217 views

### When one can expect $\widehat{(fg)} = \hat{f} \ast \hat{g}$; $f, g\in L^{1} (G)$?

Let $f, g \in L^{1}(\mathbb T)= L^{1} ([-\pi, \pi))$. We define, the Fourier transform of $f$ as follows:
$$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt, \ (n\in \mathbb Z).$$
It is ...

**0**

votes

**0**answers

49 views

### How to get FourierāStieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)?

We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and
...

**14**

votes

**3**answers

460 views

### An algebraic approach to the thermodynamic limit $N\rightarrow\infty$?

In physics one studies quite often the thermodynamic limit or what we call the $N\rightarrow \infty$ behavior of a system of $N\rightarrow\infty$ particles. This is of particular relevance in the ...

**9**

votes

**2**answers

338 views

### Uncertainty principle on finite groups

For a finite group $G$ with normal subgroup $H$, the induced representation $\text{Ind}_H^G(1)$ decomposes as a sum of irreducibles with the multiplicities equal to the dimensions, because it is is ...

**3**

votes

**3**answers

597 views

### Peter-Weyl theorem as proven in Cartier's Primer

I'm reading Pierre Cartier's A primer of Hopf algebras to educate myself. In its subsection 3.3 (which doesn't need any Hopf algebra theory), he sketches a proof why compact Lie groups are algebraic. ...

**2**

votes

**0**answers

211 views

### Unitary Representations and Integral formulas

While reading the appendix to 4th chapter of Iwaniec and Kowalski's analytic number theory I came upon a remark relating unitary representations and some integral transforms involving J-Bessel ...

**11**

votes

**1**answer

356 views

### General Isoperimetric Inequality via Representation Theory of SO(n)

Is there a known proof of the $n$-dimensional isoperimetric inequality which generalizes Hurwitz's proof using Fourier analysis in the $2$-dimensional case?
Specifically, I imagine such a proof would ...

**4**

votes

**1**answer

518 views

### An approximate converse of discrete uncertainty principle

Let $f:\mathbb{Z}_n \rightarrow \{0, 1\}$ and let's normalize the Fourier transform $\hat{f}$ so that $\|\hat{f}\|_2 = \|f\|_2$, i.e.
$$\hat{f}(\xi) = \frac{1}{\sqrt{n}}\sum_{x \in ...

**2**

votes

**1**answer

288 views

### Plancherel measure on Homogeneous spaces.

Does any one know what the correct formulation of the plancherel theorem should be for Homogeneous spaces. More specific I am looking for a statement like: there is a unique measure in $\mu$ in $\hat ...

**52**

votes

**10**answers

6k views

### Is Fourier analysis a special case of representation theory or an analogue?

I'm asking this question because I've been told by some people that Fourier analysis is "just representation theory of $S^1$."
I've been introduced to the idea that Fourier analysis is related to ...

**4**

votes

**1**answer

280 views

### closed irreducible subspaces in L²(R)

Let R be the set of real numbers, then R acts on LĀ²(R) by translation. This is a unitary representation, which is far away from being irreducible. So what are the closed irreducible subspaces?
This ...

**5**

votes

**3**answers

780 views

### Reading for finite Fourier Analysis

Can anyone recommend some good reading for Fourier Analysis (and the Fourier transform) over finite abelian groups? I've found it given brief descriptions in both books on representation theory and on ...