**56**

votes

**4**answers

3k views

### 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 ...

**34**

votes

**1**answer

2k views

### The Bourgain-Demeter-Guth breakthrough and the Riemann zeta function?

Yesterday Bourgain, Demeter and Guth released a preprint proving (up to endpoints) the so-called main conjecture of the Vinogradov's Mean Value Theorem for all degrees. This had previously been only ...

**26**

votes

**6**answers

8k 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 ...

**23**

votes

**3**answers

2k 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 ...

**20**

votes

**6**answers

4k 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 ...

**20**

votes

**1**answer

1k views

### A conjectured formula for Apéry numbers

A conjecture by the late Romanian mathematician Alexandru Lupas.
Posted in sci.math in 2005, but no proof was found.
Physicist Alan Sokal just reminded me of it, saying it was related to something he ...

**20**

votes

**1**answer

527 views

### Bounding Schur symmetric polynomials on the unit circle

Recall the Schur polynomial in $n$ variables, indexed by the partition $\lambda$, with $\ell(\lambda) \leq n$, is given by
\begin{equation}
s_\lambda(x_1,\ldots, x_n) = a_{\lambda + \delta}(x_1, ...

**18**

votes

**2**answers

3k 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 ...

**18**

votes

**0**answers

1k 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 ...

**17**

votes

**2**answers

893 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

**1**answer

2k 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 ...

**15**

votes

**2**answers

890 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**

votes

**4**answers

822 views

### Is the sequence of Apéry numbers a Stieltjes moment sequence?

Consider the sequence of Apéry numbers
$$
A_n = \sum_{k=0}^n \binom{n}{k}\binom{n+k}{k}\sum_{j=0}^k \binom{k}{j}^3
= \sum_{k=0}^n \binom{n}{k}^2\binom{n+k}{k}^2 .
$$
In an email, physicist Alan Sokal ...

**14**

votes

**3**answers

2k 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 ...

**14**

votes

**1**answer

3k 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 ...

**14**

votes

**2**answers

369 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 ...

**14**

votes

**0**answers

104 views

### A kaleidoscopic coloring of the plane

Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap ...

**13**

votes

**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 ...

**13**

votes

**3**answers

429 views

### Completeness of nonharmonic Fourier Series

I have the following question:
The Exponential System $(\exp(2\pi i n \cdot ))_{n\in \mathbb{Z}}$ constitutes an orthonormal basis of $L^2([-1/2,1/2])$.
Thus, certainly the oversampled system ...

**13**

votes

**1**answer

411 views

### Does there exist some $C$ independent of $n$ and $f$ such that $ \|f''\|_p \geq Cn^2 \| f \|_p$, where $1 \leq p\leq \infty$?

Let $f$ be a trigonometric polynomial on the circle $\mathbb{T}$ with $\hat{f}(j) = 0$ for all $j \in \mathbb{Z}$ with $\lvert j \rvert < n$. Does there exist some $C$ independent of $n$ and $f$ ...

**13**

votes

**1**answer

715 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.
...

**13**

votes

**0**answers

440 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 $\Gamma_2$ ...

**13**

votes

**0**answers

267 views

### A functional inequality about log-concave functions

Let $f,g$ be smooth even log-concave functions on $\mathbb{R}^{n}$, i.e.,$f=e^{-F(x)}, g=e^{-G(x)}$ for some even convex functions $F(x),G(x)$. Is it true that:
$$
\int_{\mathbb{R}^{n}} \langle ...

**12**

votes

**1**answer

804 views

### Where did the term “additive energy” originate?

A fundamental object in modern additive combinatorics and harmonic analysis is additive energy. Given a subset $A$ of (say) an abelian group $G$ the additive energy of $A$ is defined to be the ...

**12**

votes

**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 ...

**12**

votes

**2**answers

584 views

### Does the Legendre-Hadamard condition imply a generalized Gårding inequality?

For simplicity, we restrict to constant coefficients. Let $A^{ij}_{ab} \in \mathbb{R}$, $1 \le i, j \le n$ and $1 \le a, b\le m$, satisfy the Legendre-Hadamard condition:
$$
...

**12**

votes

**1**answer

286 views

### Definition of discrete spectrum and continuous and basic properties

I apologize if this is too basic for MO.
I have an embarrassing admission to make: I don't know the actual definition of the discrete/continuous spectrum of a reductive group $G/\mathbb{Q}$ (in the ...

**11**

votes

**3**answers

2k 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 ...

**11**

votes

**5**answers

613 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 ...

**11**

votes

**2**answers

636 views

### Motivation for BMO

At the moment, I don't have access to the early 1960's paper of John and Nirenberg that (from what I understand) introduced the space BMO (bounded mean oscillation). Why were John and Nirenberg ...

**11**

votes

**2**answers

272 views

### Heuristic interpretation of the 'third index' for Besov and Triebel-Lizorkin spaces

For $p,q \in (0,\infty)$ and $s \in \mathbb{R}$, one can define certain function spaces, $B_s^{p,q}(\mathbb{R}^n)$ and $F_s^{p,q}(\mathbb{R}^n)$, the Besov and Triebel-Lizorkin spaces respectively. ...

**11**

votes

**1**answer

486 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 ...

**11**

votes

**1**answer

412 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 ...

**11**

votes

**1**answer

356 views

### What's the role of $H^{p}(\mathbb{R}^{n})$ in modern (harmonic) analysis?

The classical theory of $H^p$,due to it's heavy dependence on the complex function theory(such as Blaschke products), seemed to have an insurmountable obstacle barrying its extension to several ...

**11**

votes

**0**answers

418 views

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

Question: Do quasi-characters or some other semi-group properly generalize the Laplace transform or decompose functions in some setting in a way similar to how characters generalize the classical ...

**10**

votes

**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 $ ...

**10**

votes

**3**answers

729 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

**2**answers

1k 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 ...

**10**

votes

**1**answer

888 views

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

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

**10**

votes

**0**answers

286 views

### Reciprocal polynomials with roots off the unit circle

A polynomial is called reciprocal if its coefficients read forwards are the same as those read backward - there is an obvious translation of reciprocal polynomials into cosine polynomials (since a ...

**10**

votes

**0**answers

501 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 ...

**9**

votes

**6**answers

1k views

### number theory which is close to analysis

I have basic training in Fourier and Harmonic analysis. And wanting to enter and work in area of number theory(and which is of some interest for current researcher) which is close to analysis.
...

**9**

votes

**5**answers

1k 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 ...

**9**

votes

**2**answers

2k 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 ...

**9**

votes

**3**answers

1k 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 ...

**9**

votes

**2**answers

427 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 ...

**9**

votes

**3**answers

561 views

### Historical developement of analysis and partial differential equations (especially in the 20th century)

Q: Is there a set of some comprehensive surveys or monographs describing (in
technical detail) the historical development of the various
subareas of analysis and partial differential equations?
...

**9**

votes

**1**answer

310 views

### Pontryagin dual of the surreal numbers?

Has any work been done on the Pontryagin dual of the surreal numbers (suitably topologized)? I have not been able to find anything and am not sure if this is still unknown.
Alternatively, has this ...

**9**

votes

**2**answers

549 views

### Fourier transform of the critical line of zeta?

This was asked on MSE and got a lot of upvotes but no answers, so I'm posting it here.
Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along ...

**9**

votes

**1**answer

142 views

### Nonconventional ergodic averages for commuting transformations

Let $S$ and $T$ be commuting measure-preserving transformations of a standard probability space $(X,\mu)$, so $S$ and $T$ define an action of $\mathbb{Z}^2$ on $(X,\mu)$. I am wondering about ...