The harmonic-analysis tag has no wiki summary.

**0**

votes

**0**answers

112 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

**0**answers

172 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

**1**answer

213 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

**1**answer

479 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

**2**answers

584 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

**1**answer

208 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

**2**answers

245 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

**0**answers

342 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

**1**answer

353 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

**1**answer

349 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

**0**answers

155 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

**1**answer

392 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

**0**answers

212 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

**0**answers

155 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

**4**answers

2k 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

**2**answers

400 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 ...

**2**

votes

**2**answers

416 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 $L^1(\mathbb{G})$ ...

**4**

votes

**1**answer

485 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}} ...

**7**

votes

**2**answers

775 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 ...

**4**

votes

**1**answer

378 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 ...

**4**

votes

**2**answers

352 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 ...

**3**

votes

**0**answers

195 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 ...

**1**

vote

**0**answers

368 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 ...

**2**

votes

**1**answer

308 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 $B(x)$ ...

**6**

votes

**0**answers

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 $f$ ...

**1**

vote

**1**answer

297 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 ...

**0**

votes

**1**answer

172 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 ...

**1**

vote

**1**answer

192 views

### 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 ...

**9**

votes

**0**answers

451 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 ...

**4**

votes

**0**answers

365 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 ...

**4**

votes

**1**answer

290 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 ...

**3**

votes

**1**answer

338 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 ...

**2**

votes

**3**answers

862 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 ...

**0**

votes

**0**answers

202 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 ...

**4**

votes

**1**answer

417 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 ...

**3**

votes

**2**answers

279 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?

**3**

votes

**0**answers

365 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 ...

**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 $ ...

**1**

vote

**2**answers

355 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 ...

**1**

vote

**1**answer

164 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 ...

**11**

votes

**3**answers

1k 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 ...

**0**

votes

**0**answers

174 views

### Optimal pointwise estimate of the gradient of harmonic functions in the unit ball

Hello.
How to prove that
$$\sup_{\gamma\ge 0}\int_0^1 (1+\gamma^2)^{-\frac 12} (1-t^2)^{\frac {n-4}2}( \Phi(\gamma t) +\Phi(-\gamma t))dt$$
is achived at $\gamma=0$, where
$$\Phi(\zeta)=
...

**0**

votes

**1**answer

235 views

### Independence of rotated spherical harmonics

Hi,
Consider a spherical harmonic of degree $l$, denoted by $y_l^m$. I rotate this harmonic using $2l+1$ different rotations. The set of functions I get is not an orthogonal set, but the functions ...

**11**

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 ...

**1**

vote

**1**answer

126 views

### Dominant weights appear in Discrete Series

If $\lambda$ is a Harish-Chandra paramater. Let $\pi_\lambda$ it's associated discrete series, it's known by the minimal K-type thm that every K-type of $\pi_\lambda\mid_K$ has highest weight of the ...

**1**

vote

**0**answers

207 views

### Consequence of Modified Young's inequality

Let $f\in L^1(\mathbb R^n)$. Define operator $T_f(g)=|f|\ast g$ for functions $g$ on $\mathbb R^n$. The set of measurable functions $f$ on $\mathbb R^n$, such that $T_f$ is bounded from $L^p(\mathbb ...

**2**

votes

**1**answer

384 views

### Ergodic decomposition of quasi-invariant measure

I have a reference request concerning Proposition 1.6 in the following article http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183548783
The ...

**12**

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 ...

**3**

votes

**4**answers

2k views

### Characterization of the non-negative definite functions $f(x,y)$

Hello,
The common definition of the non-negative definite functions is as follows:
Definition 1: A continuous complex-valued function $f(x)$ is called non-negative definite, if for any real numbers ...

**0**

votes

**0**answers

116 views

### Are all of compact support functions of $A(G)$ in its abstract Segal algebras?

Let $G$ be a locally compact group. We know that if $G$ is abelian and $\cal F$ implies the Fourier transform, for every Segal algebra of $G$ say $S^1(G)$, ${\cal F}S^1(G)$ is an abstract Segal ...