The harmonic-analysis tag has no wiki summary.

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249 views

### High dimensional beta integral (question following the previous post)

Hello,
This post is a question following the previous post. In one dimensional case, we have
$$
\int_0^x |y|^{1-\alpha} |x-y|^{1-\beta} d y = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} ...

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430 views

### High dimensional beta integral (a typo in Stein's book “singular integrals”)

Hello,
When I read Stein's book of Singular Integrals, at p. 118, there is an obvious mistake:
$$
\int_{R^n} |x-y|^{-n+\alpha} ...

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**1**answer

443 views

### Measures and structure on conjugacy classes

Given a locally compact group $G$, does there exist a measure $\nu$ on the conjugacy classes $conj(G)$ such that for $f \in C_c(G)$
$$ \int_G f(g) d \mu_G(g) = \int_{conj(H)} \int_{G / G_\gamma} ...

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146 views

### Regularity properties of the derivatives of a particular function on $D \times D\to \bar{D} $ ?

This question might sound a little less rigorously formulated, but I hope the question still makes sense.
Let $h: S^1 \to S^1$ be an oriention-preserving homeomorphism and let $p(z,t) = ...

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**1**answer

156 views

### Local nonarchimedean Sobolev inequality

Let $B$ be the unit ball in $\mathbb{R}^n$. A local version of the Sobolev inequality on $\mathbb{R}^n$ says that for any $p\in[1,\infty]$ there exist constants $C >0$ and $k \in \mathbb{N}$ such ...

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255 views

### When does a LCA group not contain a (closed) infinite cyclic subgroup?

If $G$ is an LCA (locally compact abelian) group, is there any 'nice' sufficient (or preferably necessary and sufficient) criteria for when $G$ does not contain a closed (and hence discrete in the ...

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**1**answer

533 views

### Multiplicity of eigenvalues of the Laplacian on quaternionic projective space

Using the classic spherical harmonics theory, one obtains the $k$-th eigenvalue of the $n$-dimensional round sphere $S^n$ to be $k(k+n-1)$, and its multiplicity is $\binom{n+k}{k}-\binom{n+k-1}{k-1}$, ...

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**1**answer

595 views

### Endpoint Strichartz Estimates for the Schrödinger Equation

The non-endpoint Strichartz estimates for the (linear) Schrödinger equation:
$$
\|e^{i t \Delta/2} u_0 \|_{L^q_t L^r_x(\mathbb{R}\times \mathbb{R}^d)} \lesssim \|u_0\|_{L^2_x(\mathbb{R}^d)}
$$
$$
2 ...

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2k views

### How to do DFT for irregular sampling period ?

I have two vectors: $\vec{a}$ and $\vec{t}$: $a_k$ is the sampling value taken at $t_k$. I need to do DFT, but the sampling period is irregular. I've learned about Frames but unsure how to use the ...

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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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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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993 views

### Carleson's Theorem (on the Adeles and other exotic groups)

I have redone this question:
On $\mathbb R^n$ the Carleson Operator if defined by
$$Cf(x) = \sup_{R>0} \left \vert \int_{B_R(0)} e^{2\pi i x\cdot \xi} \widehat{f}(\xi) d \xi \right \vert. $$ ...

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**1**answer

229 views

### Inducing from cocompact subgroups

Consider a locally compact group $G$ and a cocompact subgroup $H$, is it known that the induction of an irreducible representation $\pi$ of $H$ to $G$ decomposes discretely into a direct sum of ...

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259 views

### Traceable representation of reductive group over a p-Adic field.

I have a question on how to define a trace of a unitary representation. If $G$ is a reductive group over a p-adic field it is known ( I do not know who prove it) that is of type I. Knowing this we ...

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341 views

### Zero sets of harmonic fucntions

Can a two variable Harmonic function f(x,y) be zero on a curve with a cusp?

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779 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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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 ...

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**1**answer

646 views

### Schwartz space inequality

Let $g$ be a function in the Schwartz space $\mathscr S (\mathbb R)$. Show that for any $l \ge 0$, we have $\sup_x |x|^l |g(x-y)|\le A_l (1+|y|)^l$ by considering separately the cases $|x|\le 2|y|$ ...

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### Exponential sums for beginner.

What are the good books, online lecture notes or starting material on exponentials sums with applications in number theory for a beginner, apart from N. M. Korobov's book? The book or notes should ...

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323 views

### Maximal function related to the Ornstein-Uhlenbeck operator.

On $\mathbf R^d$ the Ornstein-Uhlenbeck operator is defined as ($\partial_i = \frac{\partial}{\partial x_i}$).
$$L = \frac12 \sum_i \partial_i^* \partial_i$$
where $\partial_i^* = -\partial_i + 2 ...

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968 views

### Suitable references for the the Stone-von Neumann Theorem

Hi all,
I am working on a mathematical physics project now and I need to understand the Stone-von Neumann Theorem properly. Wikipedia says that it is any one of a number of different formulations of ...

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470 views

### Are there explicit formulas for spherical functions on oriented real grassmannians?

Let $p$ and $q$ be integers. The group $K=SO(p) \times SO(q)$ can be naturally seen as a subgroup of $G=SO(p+q)$. The quotient space $G/K$ is identified with the space of oriented $p$-dimensional ...

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### 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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**1**answer

606 views

### A counterexample in Littlewood-Paley theory.

Are there any (at least mildly) explicit counterexamples to the statement
$$
\sum_{m \in \mathbb{Z}} \|P_m f\|_p \lesssim \|f\|_p?
$$
(Or some good reason to expect this to be false?).
Here $P_m$ is ...

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**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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**1**answer

425 views

### Is there any result about the uniform convergence rate of multi-dimensional Fourier series

For example in the 1-dimensional case, it is known that if f satisfies the α-Hölder condition, then
$|f(x)-(S_Nf)(x)|\le K \frac{\ln N}{N^\alpha}$
where $S_N f$ is the n-term partial sum of the ...

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394 views

### Capacity of Balls in Hyperbolic Space

Given $M$ a Riemannian manifold and $\Omega\subset M$ the capacity of $\Omega$ is defined as
$$
\mathrm{cap}(\Omega)=\inf \int_{M\setminus\Omega}{|\mathrm{grad} \varphi|^2 dV}
$$
where $\varphi$ ...

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**1**answer

234 views

### Spherical transforms and rotations

Consider a function $f$ on $S^2,$ and its spherical transform $\hat{f}.$ Let $r$ be a rotation by some $\rho \in SO(3).$ Is there some nice formula for $\widehat{f \circ r}?$ I have found some ...

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**1**answer

617 views

### Rate of convergence of smooth mollifiers

How does one figure out/prove the rate of convergence (in some norm) of mollifiers given a function bounded in some other norm (say Sobolev space, Besov space)? Also, is there a dimensional analysis ...

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889 views

### L1 distance from a trigonometric susbspace

How to check, whether the $L^{1}$ distance between a finite exponential sum $S_{F}(x)=\sum\limits_{n\in F} \exp(inx)$ and the $L^{1}$-closure of subspace $\mathrm{span}\left(\exp(inx): n\in ...

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### 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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569 views

### Uniform boundedness of an $L^2[0,1]$-ONB in $C[0,1]$

Assume that we have an orthonormal basis of smooth functions in $L^2[0,1]$. Are there useful practical criteria to determine whether the sup-norm of the basis functions has a uniform bound? I am sure ...

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318 views

### Does Hölder continuity imply smoothness for the CMC equation: $u:D^2\rightarrow\mathbb{R}^n$, $\Delta u = 2H\partial_xu\times\partial_yu$, $H$ constant?

Context: I am currently reading through the freely available lecture notes from Tristan Riviere (here) on the applicability of integration by compensation in the analysis of various geometrically ...

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843 views

### Reference for complex analysis jargon

I am not a (complex) analyst but it seems that some of the questions I am working on are related to the following concepts:
logarithmic capacity
transfinite diameter
Green's function of a compact ...

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121 views

### Asymptotic rearrangement

I had some trouble coming up with a good title for this question. Here is the setup. Suppose you have two infinite sets of (positive real, say) numbers $\{a_k\}$ and $\{b_k\}$ such that the ...

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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 ...

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**1**answer

1k views

### Relation between complex analysis and harmonic function theory [closed]

There are some theorems in harmonic function theory that resemble results in complex analysis, like:
Holomorphic functions and complex functions are analytic;
Cauchy's integral formula in complex ...

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165 views

### Fredholmness and invertibility in a C* algebra generated convolution-type operators

Let $PC$ be the algebra of complex-valued, piecewise-continuous functions from $[-\infty,+\infty]$, $SO$ be the algebra of bounded, continuous, complex-valued functions on $\mathbb R$ which are slowly ...

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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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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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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 ...

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703 views

### distribution of {na} when a is irrational number

(by $\{x\}$ I mean the fraction part of the real number $x$)
If $a$ is an irrational number and $n$ is a integral number, what is the distribution of $\{na\}$? I'm asking for some continuous function ...

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**1**answer

330 views

### absolute convergence of the Fourier coeff for Hölder continuous function

I saw the following theorem in the wiki page:
http://en.wikipedia.org/wiki/H%C3%B6lder_condition
if $f$ satisfies the $\alpha$-Hölder condition
$| f(x) - f(y) | \leq C \, |x - y|^{\alpha}$ for ...

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**1**answer

538 views

### The Dirichlet heat semigroup, $L^1_\delta$, and the dimension shift phenomenon

In relation to the question on the Hardy inequality and the answer by Terry Tao, I've always been curious about the following:
Let $U \subset \mathbb{R}^n$ be a bounded domain of class $C^2$, $(e^{-t ...

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**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 ...

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**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 ...

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920 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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**3**answers

599 views

### Plancherel formula for special linear group

I am looking for a comprehensible material covering Plancherel formula for $SL(n,\mathbb{R})$ and $SL(n,\mathbb{C})$. Of course, I wouldn't mind reading an explanation for general semisimple Lie ...

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236 views

### faithful representation of locally compact group

I have been thinking about existence of faithful representation of locally compact groups. This representation exists for example for compact lie groups. But I am curious to know if one can say some ...

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1k 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 ...