The harmonic-analysis tag has no wiki summary.

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### How Does My Radio Work?

Bear with me for a moment while I invoke the real world; the main question at the end is purely mathematical.
I live in an area with $n$ AM radio stations and $m$ FM radio stations.
AM station ...

**20**

votes

**3**answers

1k views

### When are the eigenspaces of the Laplacian on a compact homogeneous space irreducible representations?

I was writing up some notes on harmonic analysis and I thought of a question that
I felt I should know the answer to but didn't, and I hope someone here can help me.
Suppose I have a compact ...

**18**

votes

**2**answers

1k views

### Is this statement which relates the Fourier transform of a function to its singularities correct?

I am working on a problem, which would possibly relate the Fourier transform/series with the jump singularities of the function where the function itself or one of its derivatives jump. ((some kind of ...

**17**

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**2**answers

587 views

### Image of L^1 under the Fourier Transform

The Fourier Transform $\mathcal{F}:L^1(\mathbb{R})\to C_0(\mathbb{R})$ is an injective, bounded linear map that isn't onto. It is known (if I remember correctly) that the range isn't closed, but is ...

**16**

votes

**6**answers

3k views

### Applications of Hardy's inequality

Every so often I would encounter Hardy's inequality:
Theorem 1 (Hardy's inequality). If $p>1$, $a_n \geq 0$, and $A_n=a_1+a_2+\cdots+a_n$, then
$$\sum_{n=1}^\infty ...

**16**

votes

**4**answers

4k views

### Learning roadmap for harmonic analysis

In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas ...

**15**

votes

**2**answers

769 views

### Borel set plus a closed set = Borel

Hi,
Let $R$ be equipped with the usual Borel structure. Let $F$ be a Borel subset and $E$ be a closed subset of $R$. Then $F+E=(f+e: f\in F, e \in E \)$ is Borel? If yes, is it true for any locally ...

**15**

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**0**answers

726 views

### The Fourier Transform of taking Eigenvalues

The purpose of this question is to ask about the Fourier transform of the map which associate to an $n$ by $n$ matrix its $n$ eigenvalues, or some function of the $n$ eigenvalues. The main motivation ...

**13**

votes

**1**answer

1k views

### Intuition for the Hardy space $H^1$ on $R^n$

the standard intuition for Lebesgue spaces $L^p(\mathbb R^n)$ for $p \in [1,\infty]$ are measurable functions with certain decay properties at infinity or at the singularities.
In particular, a ...

**13**

votes

**2**answers

245 views

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

I asked this in math.stackexchange, but it disappeared from the "main list" almost immediately, so I hope it will be appropriate as a separate question in MO.
For a given function $f\in C(G)$ on a ...

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votes

**1**answer

690 views

### Hearing the 17 planar symmetry groups

Though I'm sure it's really hard to work out for myself, does anyone know a reference for the spectra of the Laplacian on the 17 flat compact orbifolds that underlie the 17 planar symmetry groups.
...

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**3**answers

1k views

### Positive definite function zoo

I've asked the following question on math.stackexchange but there has been no response so I'll ask it again here:
A positive definite function $\varphi: G \rightarrow \mathbb{C}$ on a group $G$ is a ...

**11**

votes

**3**answers

1k views

### Fourier transforms of functions not in $L^2.$

This is probably something five-year-old physicists know, but here goes: Is there a standard methodology for computing Fourier transforms of things like $\log |x|$? Wolfram Alpha will happily give an ...

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**4**answers

1k views

### Where do the real analytic Eisenstein series live?

In obtaining the spectral decomposition of $L^2(\Gamma \backslash G)$ where $G=SL_2(\mathbb{R})$, and $\Gamma$ is an arithmetic subgroup (I am satisfied with $\Gamma = SL (2,\mathbb{Z})$) we have a ...

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**3**answers

1k views

### Harmonic analysis on semisimple groups - modern treatment

For my finals, I am digging through the book by Varadarajan An introduction to harmonic analysis on semisimple Lie groups. I find it a rather hard read and I feel it's a bit outdated now. Any ...

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**3**answers

1k views

### Is there a Plancherel Theorem for Gowers norms?

In the process of counting arithmetic sequences in sets, the Gowers norms
$$ ||f||\_{U^s[N]}^{2^s} = \frac{1}{N^s} \sum_{\vec{h} ,\\, n } \Delta_{h_1}\dots\Delta_{h_s}f(n) $$
where the sum is $ ...

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votes

**5**answers

493 views

### What are the best settings for the large scale geometry of locally compact groups?

My current research involves locally compact groups and from time to time I am tempted to check whether certain notions and statements of geometric group theory of finitely generated groups are still ...

**10**

votes

**3**answers

338 views

### Topology on the Unitary Dual

Suppose I have a locally compact topological group G. The unitary dual of G is the set of equivalence classes of irreducible unitary representations of G. Now, it seems to me that the sensible way of ...

**10**

votes

**1**answer

283 views

### Do circular pipes maximize flow rate?

Suppose that $U \subset \mathbb{R}^2$ is nonempty, open, connected and bounded. Consider a Poisseuille flow in the pipe $U \times \mathbb{R}$. That is: a time-independent incompressible flow of the ...

**10**

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**1**answer

318 views

### Locally Compact Quantum Groups application

Recently Mathematicians in harmonic analysis become more and more interested in Locally compact quantum groups and try to transfer concepts from abstract harmonic analysis to the setting of locally ...

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**3**answers

913 views

### Why is the dual of a torus the same as its fundamental group?

The set of continuous homomorphisms from a torus ${\mathbb T}^n = ({\mathbb R}/{\mathbb Z})^n \to {\mathbb R}/{\mathbb Z}$ can be identified with ${\mathbb Z}^n$ if we assign to each $k = (k_1, \ldots ...

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votes

**2**answers

275 views

### Entropy for Haar measure on $O(n)$

Let $G$ be a locally compact group. A measure $\mu$ is the right-Haar measure on $G$ if for every $g\in G$ and $E\subseteq G$ Borel set $\mu(Eg)=\mu(E)$. It is known that every locally compact group ...

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451 views

### Connections of results in Harmonic analysis in the theory of Transcendental Numbers

An entire function $f$ is said to be of exponential type if there exist constants $c$ and $k$ such that $|f(z)|\leq c e^{k |z|}$.
A famous result of Polya says if $f$ is an entire function of ...

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votes

**2**answers

664 views

### the convolution of integrable functions is continuous?

The question is simple but I still can't prove it or contradict it. Here it goes:
Suppose $f$ and $g$ are defined on the circle
(or, equivalently, $2\pi$ periodic functions) and Lebesgue ...

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votes

**2**answers

1k views

### Fourier transform of the unit sphere

The Fourier transform of the volume form of the (n-1)-sphere in $\mathbf R^n$ is given by the well-known formula
$$
\int_{S^{n-1}}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u) = (2\pi)^{\nu ...

**8**

votes

**1**answer

547 views

### How differentiable is the convolution of two continuous functions?

The question is really simple:
Given
$$
f, g\in C^\alpha_c(\mathcal{R}^d)
$$
is
$$
f*g\in C^d_c?
$$
I came up with a formal argument using the decay of the Fourier transform of continuous functions, ...

**8**

votes

**1**answer

522 views

### Removable sets for harmonic functions and hardy spaces of general domains

Hi,
Let $\Omega$ be a domain of the complex plane. The Hardy space $H^p(\Omega)$ is defined, for $1 \leq p<\infty$, as the class of functions $f$ that are holomorphic on $\Omega$ such that $|f|^p$ ...

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votes

**2**answers

745 views

### Is there a “right” proof of Riemann's Theta Relation?

Let $\theta$ denote the usual Jacobi Theta function (with auxiliary parameter $\tau = i$, for simplicity), i.e.
$$
\theta(z) = \sum_{n \in \mathbb{Z}} \exp(-\pi (a + n)^2 + 2 \pi i n z) \ .
$$
I'm ...

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174 views

### Laplace Transform in the context of Gelfand/Pontryagin

Question: Do quasi-characters properly generalize the Laplace transform or decompose functions in some setting in a way similar to how characters generalize the Fourier transform and decompose $L^1$ ...

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232 views

### Carleson's Theorem on Manifolds

Let $M$ be an oriented, compact, differentiable manifold with some Riemmanian metric $g$, so that $(M,g)$ has a nice volume form and one can define $L^2(M,g)$ as the completion of $C^\infty(M)$ under ...

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178 views

### when do norm-continuous unitary representations separate points of a group?

Recently I found in the web a discussion on the following question:
...

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**2**answers

1k views

### The dual group of $\mathbb Q$

What is the dual group of the additive group of rational numbers equipped with the standard topology inherited from $\mathbb R$? As a group, this dual group is isomorphic to $\mathbb R$ (see the ...

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votes

**3**answers

919 views

### A Question concerning the Fourier Transform of $\mathbb{R}$

Consider the classical Schwartz space $\mathcal{S}(\mathbb{R})$ together with the Fourier transform $\mathcal{F} : \mathcal{S}(\mathbb{R}) \rightarrow \mathcal{S}( \mathbb{R})$.
Consider the subspace ...

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votes

**1**answer

264 views

### Kakeya and Nikodym maximal functions

I've been working through part of Terry Tao's 1999 article "The Bochner-Riesz Conjecture Implies the Restriction Conjecture." (It appeared in the Duke Mathematical Journal.) A little more ...

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**1**answer

596 views

### approximate uncertainty principle for finite abelian groups

Edit: we cannot find such an example. It would imply a negative solution to the KS${}_2$ conjecture which has now been proven by Marcus, Spielman, and Srivastava in this paper.
In fact, their ...

**7**

votes

**2**answers

774 views

### Explaining Mukai-Fourier transforms physically

A core concept in mathematics, engineering, and physics is the Fourier Transform (FT) and its many variants (Generalized Fourier Series, Green's Function, Pontryagin duality).
The basic algorithm is ...

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votes

**2**answers

565 views

### Decomposing an arbitrary unitary representation of a connected nilpotent Lie group in terms of its irreps

For a locally compact (Hausdorff) abelian group $G$ we have following theorem (see e.g. Folland):
"For every (strongly continuous) unitary representation $(\pi,\mathcal{H_{\pi}})$ of $G$, there ...

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votes

**1**answer

671 views

### Norm concentration of trigonometric polynomials - Uncertainty principle

Hi all,
I am interested in the following question (which is quite similar to one I posed a long while ago): Let $P_{N}(t)=\underset{k=-N}{\overset{N}{\sum}}c_{k}e^{ikt}$ be a unit norm trigonometric ...

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votes

**1**answer

778 views

### First eigenvalue of the Laplacian on Berger spheres

Consider the Hopf fibrations $S^1\to S^{2n+1}\to CP^n$ and $S^3\to S^{4n+3}\to HP^n$. These are Riemannian submersions with totally geodesic fibers. Consider now their canonical variations (the ...

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votes

**1**answer

199 views

### C* algebras of Almost Periodic Functions

Suppose we take, for example, the $C^*$-algebra which is the sup norm closure of the exponentials $e^{2 \pi i ax}$ where $a \in \mathbb{Z} + \theta \mathbb{Z}$ for $\theta$ an irrational number. This ...

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### Why is it hard to obtain improved $L^6$ bound of eigenfunction of Laplacian on 2-dimensional compact Riemannian manifold?

Consider the $L^p$ estimate of the Laplacian on a compact boundaryless Riemannian manifold, suppose that $-\Delta_ge_{\lambda}=\lambda^2e_\lambda(x), x\in M$.
C.D. Sogge proved that we have the ...

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153 views

### Inferring asymptotic behaviour from the dominant pole of the Laplace transform

Hi,
I am reposting the following question with the hope that a more detailed description will lead to a more descriptive response:
dominant pole in the laplace transform
I have a vector function ...

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**0**answers

242 views

### Finding a dimension-free bound for a certain multiplier on Euclidean space

The following question is indirectly motivated by strong type maximal function estimates. Let $f\in L_{p}(\mathbb{R}^{n})$. For $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$ define $m(\xi)$ so ...

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559 views

### Hardy spaces: analysis <---> martingales

Let $H^p$ be the Hardy space of analytic functions on the open unit disk $D$: $f \in H^p$ if $f$ is analytic on $D$ and $\sup_{r < 1} \int_0^{2\pi} |f(re^{i\theta})|^p d\theta < \infty$.
...

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698 views

### Background for Hejhal's "The Selberg Trace Formula for $PSL(2, \mathbb{R})$

Reposted from math.stackexchange where my question received only five views and no answers...
I'm trying to learn the Selberg trace formula, but have very little background in harmonic analysis. I ...

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votes

**1**answer

570 views

### Does random matrix theory make any prediction for the eigenvalue distributions of compact Riemann surfaces?

Under RH, Montgomery has proven equidistribution results for the zeros of the Riemann Zeta function, which suggest a close connection of the distribution to certain results in Random matrix theory. ...

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**2**answers

219 views

### Structure of the unitary representation $L^2(N/M)$ when $N$ is a nilpotent Lie group

Hi All,
I am new to this (though I seem to be a latecomer); so forgive me if this is not your most favorite question:
I am trying to understand the structure (e.g., decomposition) of the unitary ...

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votes

**2**answers

584 views

### What is the simplest oscillatory integral for which sharp bounds are unknown?

I have either heard or read that sharp asymptotics and bounds for oscillatory integrals of the form
$ \int e^{i \lambda \Phi(x)} \psi(x) dx \quad \lambda \to \infty $
are unknown when the critical ...

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votes

**2**answers

119 views

### Spherical functions for sl(2,Q_p)

I kindly would like to ask you the following- I am refering to page
175 in the Book by Gelfand, Graev, Shapiro, etc, on "Automorphic forms
..."
My question to which I would kindly ask you to answer ...

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**3**answers

728 views

### Property/Relations using Fourier series/transform, which give complete information about all the jump singularities of a function.

Consider a function which has only jump singularities of the form of the function itself or one of its derivatives jumping. Now let $\hat{f}(k)$ be its Fourier transform/series. We know the decay of ...