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2
votes
0answers
243 views

Continuity of multiplicative character

Let $G$ be a discrete group and $\beta (G)$ denote the Stone-Cech compactification of $G$, a right topological semigroup. By a multiplicative character, I mean a mapping that preserves multiplication ...
6
votes
2answers
261 views

Orthonormal basis for $L^2(G/H)$.

Let $G$ be a locally compact group and $H$ be a closed subgroup of $G$. Is there any way to define a reasonable orthonormal basis for $L^2(G/H)$? By "reasonable" I mean elements of the orthonormal ...
4
votes
1answer
249 views

Support of functions on compact groups, and their Fourier transforms

I make my question more precisley : Let $f$ be a conjugation invariant function on a compact semi-simple Lie group $G$. So $f$ can be regarded as a function on the maximal tours ( or Lie algebra of ...
2
votes
1answer
263 views

Characters separating points on Maximal Torus modulo Weyl group?

Let G be a compact Lie group, for example, SU(n). Let T be its maximal torus. Let W be its Weyl group. Every finite-dimensional representation of G has a character, which is a function on G, T and ...
1
vote
0answers
114 views

L^p norms of Littlewood-Paley blocks

Is it possible to show that all the $L^p(\mathbb{R}^n)$ norms of a given Littlewood-Paley block are equivalent or else find a counterexample? One inequality follows from the Bernstein inequalities if ...
2
votes
2answers
254 views

Factorization of antisymmetric bounded holomorphic functions

A basic principle in complex function theory is that one can split off zeros of holomorphic functions in a similar way as for polynomials: If $f$ is holomorphic near $0$ and $f(0) = 0$, then $f(z) = ...
2
votes
2answers
348 views

Lebesgue differentiation theorem beyond Euclidean spaces

To formulate Lebesgue differentiation theorem one needs a metric and a measure. Apart from the Euclidean spaces i.e. $\mathbb R^d$, the theorem holds true for homogeneous groups (e.g. Heisenberg ...
4
votes
1answer
152 views

variation of the obstacle in the obstacle problem

Suppose $D \subset \Bbb C$ with smooth boundary. Let $f \in C^{1,1}(D)$. Let $\varphi$ be the supremum of all members in the set $$\lbrace g \in C^{\infty}(\overline{D})| g \ is \ subharmonic \ and ...
3
votes
2answers
410 views

Quotient of a compact Lie group by maximal Torus

I start with a noncompact connected semisimple Lie group with finite center $G$ and fix a maximal compact subgroup $K$ of $G$. I am considering these compact groups $K$. If $\mathbb T$ is the ...
15
votes
0answers
772 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 ...
1
vote
2answers
301 views

Harmonic polynomials on complex 2-space

Consider a complex-valued harmonic polynomial $f$ on ${\Bbb C}^2$ and assume that $f(0)=0$. Suppose also that $f$ does not vanish on the unit sphere $S^3\subset{\Bbb C}^2\simeq{\Bbb R}^4$. Does it ...
0
votes
1answer
258 views

Fixed norm problem for analytic functions

Hi there, I have the following problems on my hand: Given arbitrary positive sub-harmonic function $l(x,y)$ on plane, for all analytic functions: $f=a(x,y)+ib(x,y)$ on complex plane, consider the ...
2
votes
0answers
182 views

Using the Mellin transform to invert ill-posed problems of harmonic transforms

I will try to explain the problem using just words. Let's see how far I get! I will use the harmonic transform of the Radon transform for 2 or more dimensions as an example, but there is a large ...
3
votes
2answers
463 views

Corona Theorem in several variables

Hallo, I have read about the Corona Theorem (see link:http://en.wikipedia.org/wiki/Corona_theorem). From this one ca deduce that: Let $f_{1}, ..., f_{n}$ be holomorphic bounded functions on the unit ...
1
vote
2answers
200 views

Alternate definitions of $C^{1,\alpha}$ and $C^{1,\alpha}(\bar{D})$ maps

My question is about the precise definition regarding the following: Let $f$ be an orientation-preserving $C^1$ diffeomorphism of the unit circle $S^1$. So $f'(b)$ exists and can be thought as a ...
6
votes
3answers
786 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 ...
4
votes
3answers
429 views

Meaning of a quote of Doubilet,Rota and Stanley on harmonic analysis and combinatorics

The beginning of the paper On the foundations of combinatorial theory. VI. The idea of generating function (1972) link text says that Since Laplace discovered the remarkable correspondence ...
3
votes
1answer
407 views

Uncertainty principle (really for Mellin, but never mind that!)

Is there a smooth funtion $f:\mathbb{R}\to \mathbb{C}$ such that (a) $f(x)$ decreases faster than $e^{-e^x}$ when $x\to \infty$, (b) $\widehat{f}(t)$ decreases faster than $e^{-|t|}$ when $t\to ...
3
votes
1answer
148 views

Subharmonic envelope

I came across a more complicated version of the following problem. It is so elementary, I think that there had to be some research done on this in the past. If someone has any ideas please let me ...
6
votes
5answers
764 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 ...
5
votes
1answer
224 views

What is the Plancherel Measure for $\textrm{SL}_3(\mathbb{Q}_p)$?

I am looking for a description of the Plancherel Measure of $\textrm{SL}_3(\mathbb{Q}_p)$. Has this been calculated yet? I've search many places for it, but I've only found results on real/complex ...
1
vote
2answers
233 views

Approximating a function by fractional powers

Say I have a continuous function $f$ defined on a compact interval $I$ on the real line. As is well-known, I could approximate $f$ arbitrarily well by polynomials. Given $R>0$, how well can we ...
10
votes
1answer
336 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 ...
7
votes
1answer
704 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 ...
5
votes
1answer
272 views

What properties should a transform have to deserve the descriptor Fourier?

Two MO questions, "Heuristic behind the Fourier-Mukai transform" and "Explaining Mukai-Fourier transforms physically," compel me to ask these two related questions: 1) What properties do you feel are ...
7
votes
1answer
604 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 ...
0
votes
1answer
141 views

On a limit at the boundary of $\mathbb{D}$ related to complex and harmonic analysis

Let $p(z,t)=\frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2}$ be the Poisson kernel on the open unit disk $\mathbb{D}$, fix $0<\alpha<1$ . Let $a\in \partial\mathbb{D}=S^1$ be fixed. Then my question is : ...
0
votes
1answer
129 views

What is the limit of the derivative of the harmonic extension/Dirichlet solution in $C^{1,\alpha}$ cases ?

Let $f:\mathbb{S}^1 \to \mathbb{S}^1$ be an oreintation-preserving homeomorphism. Denote by $H(f)$ the complex harmonic extension/solution in $\mathbb{D}$ to the Dirichlet problem with boundary data ...
1
vote
1answer
440 views

Is $\int_{t\in S^1} |t-\zeta|^{\alpha}p(z,t) |dt| \leq K|z-\zeta|^{\alpha}, 0< \alpha < 1$ for uniform $K$?

I asked the question before, but didn't get any reply, so I took the liberty to ask again. Let $\zeta\in S^1$(unit circle in the complex plane) and $z\in \mathbb{D}$. Fix $0< \alpha < 1$. Then, ...
1
vote
1answer
225 views

References on hyperbolic harmonics

I am looking for good and elementary references on hyperbolic harmonics (which form an orthonormal basis spanning the space of functions on the unit pseudo-sphere).
1
vote
0answers
144 views

Bounds on norm of harmonic function on degenerating hyperbolic surface

Suppose I have a Riemann surface with a shrinking geodesic which is degenerating towards a surface with a cusp, and I consider a neighborhood of the shrinking geodesic, $C_[a,b] = [a,b]\times ...
1
vote
0answers
131 views

Stein inequality

Dear all, we have by Stein that for any sequence $0 \le r_k \le 1$, and any functions $f_1, \cdots, f_n$ which are holomorphic on a neighbourhood of the unit disk, $$\| (\sum_{k =1}^n |f_k(r_k e^{i ...
6
votes
2answers
228 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 ...
0
votes
1answer
135 views

Convolution operators defined by compactly supported distribtion

Let $\mu_{n}$ be the unit measure over $S^{n-1}$,and consider the convolution operator$$Tf=\mu_{n}\ast f,\quad f\in \mathcal{S}$$ then,it's well-known that T can be extend to a bounded operator on ...
0
votes
0answers
122 views

The dense subspace of Hardy Space $H^p$

A famous result is that $H^p\cap L^1$ is a dense subspace of $H^p$. We need the fact that $\displaystyle\sup_{0<s\leq M}|(P_s*P_t*f-P_s*f)(x)|$, where $P_t$ is the Poisson kernel on $R^n$ and $f\in ...
6
votes
0answers
178 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 ...
4
votes
1answer
217 views

When is Prim(A) of an infinite discrete group hausdorff ?

Does anyone know, if the following result has been proved ? Let G be an infinite discrete group. A = L1(G) it's algebra and Prim(A) the set of prime ideals with spectral topology. The result is : ...
0
votes
1answer
494 views

Wave equation v.s.Schrödinger equation

The motivation of comparison of this two kind of operators is that,$$\partial_{tt}-\Delta=(\partial_{t}-i\sqrt{-\Delta})(\partial_{t}+i\sqrt{-\Delta})$$ From the above that a wave operator can be ...
6
votes
2answers
602 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 ...
1
vote
1answer
211 views

Harmonic/conformal map composition between manifolds in either order?

Suppose $\mathcal{M}$, $\mathcal{N}$, and $\mathcal{P}$ are Riemannian manifolds (compact and of dimension 2, if it matters). It seems well-known that if $\phi:\mathcal{M}\rightarrow\mathcal{N}$ is ...
0
votes
2answers
247 views

Convergence of Fourier series for $C^p$ functions

Let $f \in C^p[0,2\pi]$ and periodic. Denote $\omega_p$ as the moduli of continuous of $f^{(p)}$. Then $ |f - S_Nf| \le K \frac{\log{N}}{N^p}\omega_p(2\pi/N), $ where $S_N$ is the Fourier partial sum ...
2
votes
0answers
353 views

L^1-convergence of convolution exponential

Consider a differential equation \begin{eqnarray*} \frac{d}{d\tau}q^{\tau}\left(x\right)=Aq^{\tau}\left(x\right)+h\star q^{\tau}=Aq^{\tau}\left(x\right)+\int ...
0
votes
1answer
366 views

Pointwise limit at Lebesgue's point

Dear MOs, I am sorry if this problem is too elementary for someone. I just want to get confirmation. Suppose $f\in L^1(R^d)$. Since almost all points are Lebesgue points by the Lebesgue ...
3
votes
1answer
355 views

Ask for theory about the weighted L^2(R^d) space.

Dear MOs, I am now considering the following norm: $$ ||f||_{H}^2 := \iint f(x) H(x,y) f(y) d x d y\:. $$ where the integral is over the whole space $R^{2d}$ and $H(x,y)$ is some non-negative ...
4
votes
0answers
160 views

Fourier analysis on crystallographic groups

It seems pretty clear a priori that Fourier analysis on a discrete cocompact group of motions of $\mathbb{E}^n$ should be quite tractable (since, by Bieberbach, such groups are almost $\mathbb{Z}^n,$ ...
3
votes
1answer
395 views

modules over group algebras

Let $G$ be a locally compact group. Then we can define a modular action of $L^1(G)$ on $L^\infty(G)$ by $$ (f.u)(t)=\int f(s)u(st) ds $$ and $$ (u.f)(t)=\int f(s)u(ts) ds $$ for $f\in L^1(G)$ and ...
4
votes
0answers
214 views

Smoothness of the convolution of a singular measure with itself

Let $\gamma:I\subset\mathbb{R}\rightarrow\mathbb{R}^2, s\mapsto \gamma(s)$, denote the arclength parametrization of a smooth, convex curve $\Gamma:=\gamma(I)\subset\mathbb{R}^2$. Equip $\Gamma$ with ...
1
vote
0answers
158 views

Injective modules over Fourier algebra

Is there any article on injective modules over Fourier Algebras? Do we have anything about injectivity of $A(G)$ as a $A(G)$-bimodule?
51
votes
4answers
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 ...
5
votes
2answers
415 views

Is there any way to generalize the Laplacian to finite groups?

The group theoretic interpretation of harmonic analysis was born out of the observation that the discrete Fourier transform on a signal of length $n$ was precisely the Fourier transform of the finite ...