The harmonic-analysis tag has no wiki summary.

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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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### 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 ...

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212 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 ...

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### 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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### 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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### 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 ...

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### 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 ...

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### 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 :
...

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### 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 ...

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### 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, ...

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### 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).

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### 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 ...

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### 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 ...

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### 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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### 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 ...

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### 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 ...

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### 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 ...

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### 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 :
...

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### 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 ...

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### 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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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\mathbb{Z}^n,$ ...

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### 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 ...

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### 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 ...

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### 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?

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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 ...

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### 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 ...

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### Fourier Transform on Locally compact quantum groups

I Have read some articles on locally compact quantum groups and the Fourier transform on them. I wonder yet why we define the fourier transform as an operator valued functions from $L^1(\mathbb{G})$ ...

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### Max of Fourier transform?

Let f be a real-valued function (or distribution) on $\mathbb{R}$. (You can assume it is nice in one way or another.) What would be some practical ways to bound
$\max_{\alpha \in \mathbb{R}} ...

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### 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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### Can one understand the Kelvin transform conceptually?

Let $U = \mathbf{R}^n - \{ 0 \}$, $n > 2$ and consider for a function $f \in C^2(U)$ the Kelvin transform
$$f^\star(x) = r^{2-n} f\left(\frac{x}{r^2}\right),$$
where $r = \lvert x \rvert$. One ...

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### Does there exists a necessary condition for Lp multiplier?

Let $1 \leq p \leq 2$. A measurable function $m(\xi)$ is called a $L^p(R^n)$ ($L^p$ for convenience) multiplier, if $$\|m(D)\varphi\|/\|\varphi\|_{L^p} \leq C , \varphi \in L^p
$$ for some constant ...

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### The Dunkl intertwining operator $V_k$ on $C(\mathbb{R}^d)$

The Dunkl intertwinig operator $V_k$ on $C(\mathbb{R}^d)$ is defined by:
$$V_k f(x)=\int_{\mathbb{R}^d}f(y)d\mu_x(y),$$
where $d\mu_x$ is a probability measure on $\mathbb{R}^d$ with support in the ...

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### Why groups that admit Folner Sequences are amenable

I've been looking at Folner's Condition recently, and I'm struggling to find a proof for why the existence of a Folner sequence on a locally compact group implies that it is amenable (and the converse ...

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### A question about the Beurling-Selberg majorant

Beurling's majorant is defined as the unique entire function $B(z)$ such that the Fourier transform of $B(x)$ is compactly supported in the interval $[-1;1]$, $B(x) \geq \text{sgn}(x)$ and $B(x)$ ...

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### Do the translates of integrable function approximate its radial part?

For an integrable function $f$ on $\mathbb R^n$ we consider its ``radial'' part
$$R(f)(x)=\int_{\mathrm{SO}(n)} f(kx)dk.$$ What is the minimal condition on $f$ so that the span of translates of $f$ ...

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### A question about the quotient measure on the ideles and the adeles

Let $F$ be a local field, then the additive Haar measure is easy to relate to the multiplicative Haar measure:
$$ \int_F f(x) d^+ x = \int_{F^\times} f(x) |x| d^+ x.$$
I know that the ideles have ...