The harmonic-analysis tag has no wiki summary.

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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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**0**answers

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 ...

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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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**2**answers

418 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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447 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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votes

**1**answer

552 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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**2**answers

796 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

440 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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367 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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200 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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vote

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403 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

372 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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**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 $f$ ...

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vote

**1**answer

303 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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votes

**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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vote

**1**answer

195 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 ...

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**0**answers

458 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 ...

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**0**answers

373 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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votes

**1**answer

318 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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votes

**1**answer

356 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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871 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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208 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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votes

**1**answer

424 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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votes

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283 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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389 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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380 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

167 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 ...

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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 ...

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176 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)=
...

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votes

**1**answer

243 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 ...

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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 ...

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**1**answer

127 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 ...

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216 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 ...

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**1**answer

396 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 ...

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**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 ...

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votes

**4**answers

3k 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 ...

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117 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 ...

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318 views

### Basic question on minimal flows

I know that minimal flows are actions for which no proper closed invariant subsets exist, but I am unclear how to understand this concept.
If a coset flow on a quotient space Gamma/S is ergodic, ...

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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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442 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

451 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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148 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

160 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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263 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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554 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

633 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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votes

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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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587 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. ...