The harmonic-analysis tag has no wiki summary.

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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 ...

**7**

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**1**answer

616 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 ...

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votes

**1**answer

144 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 :
...

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**1**answer

142 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 orientation-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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**1**answer

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, ...

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**1**answer

261 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).

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148 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 ...

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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**

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**2**answers

233 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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**1**answer

138 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 ...

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129 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 ...

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182 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 ...

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**1**answer

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 :
...

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**1**answer

509 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 ...

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621 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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vote

**1**answer

216 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 ...

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250 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 ...

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364 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 ...

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**1**answer

390 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 ...

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**1**answer

368 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 ...

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

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**1**answer

396 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 ...

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216 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 ...

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160 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?

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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 ...

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429 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 ...

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460 views

### 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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583 views

### 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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809 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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**1**answer

460 views

### 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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378 views

### 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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202 views

### 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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420 views

### 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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**1**answer

382 views

### 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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89 views

### 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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307 views

### 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 ...

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**1**answer

175 views

### spectrum of Banach algebras

Let $G$ is a locally compact group (non-Abelian)
Why $sp(L^1(G))$ , i.e. the set of all nonzero bounded multiplicative functionals on $L^1(G)$ is a locally compact group.
Even for any noncommutative ...

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### A nice overview (and maybe derivation) of the Poincaré transformations of the Vector Spherical Harmonics

With $Y_{lm}(\vartheta,\varphi)$ being the Spherical Harmonics and $z_l^{(j)}(r)$ being the Spherical Bessel functions ($j=1$), Neumann functions ($j=2$) or Hankel functions ($j=3,4$) defining ...

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### 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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376 views

### Measure Theoretic view of Hardy Littlewood Circle Method

Is it possible to view the Hardy-Littlewood Circle method as the Fourier transform with respect to the Lebesgue measure on [0,1) for an appropriate generating function defined in terms of additive ...

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326 views

### Generalized widths and reverse Urysohn inequalities

This question is inspired by the discussion in MO questions "Local minimum from directional derivatives in the space of convex bodies" and "Bodies of constant width?" about generalized notions of ...

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**1**answer

359 views

### Estimating oscillatory integral

Say we are given an oscillatory integral of the form
$\Psi(x)=\int_{-\infty}^\infty e^{i\psi(x,t)} a(t)dt$.
where $a(t)$ is a sufficiently nice function. When, for instance, $|\psi(x,t)_t| \gg ...

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873 views

### Automorphic Forms on product of groups $G\times H$

Dear all, I have some difficulty in understanding the notion of automorphic forms on product of groups.
Let $G$, $H$ be two reductive groups defined over a number field
$F$. Let $\mathcal{A}(G)$ be ...

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211 views

### is there a relation between the complex Hardy spaces and the Hardy spaces of harmonic analysis?

Maybe my question is just a matter of knowing the right equivalent definition.
The question is whether there is some relation between
$ H^p(D^2)$, defined as the space made of the analytic ...

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**1**answer

430 views

### Hardy-Littlewood maximal function

We know that Hardy-Littlewood maximal function is $(p,p)$ for any $p>1$. But one proves first that it is weak type $(1,1)$ and then use interpolation. I am just curious to know: is there a way of ...

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285 views

### Wiener Tauberian Theorem for nonunimodular group

Is there a nonunimodular group for which Wiener's Tauberian theorem is true?
Is a locally compact topological group whose volume grows polynomially with radius always unimodular?

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416 views

### Calderón's Complex Interpolation: what is the corresponding classical theorem?

This question is closely related to my answer to Dan's question, which contains the definitions of some terms I use here. In addition, the notion of exact interpolation functor of exponent $\theta$ is ...

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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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386 views

### Conformal transformations and harmonic analysis on the sphere

Consider the $n$-dimensional sphere $S^n$. I'm especially interested in the $n=4$ case. The Hilbert space $L^2(S^n)$ can be decomposed into a direct sum of eigenspaces of the Laplacian, which are ...

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**1**answer

168 views

### Oscillatory integral decay & sublevel set growth

I am trying to understand how estimates on sublevel integrals imply estimates on oscillatory integrals. Specifically in this article by M. Greenblatt it says on page 7:
By well-known methods ...