The harmonic-analysis tag has no wiki summary.

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### A microlocal representation for quantum operator dynamics

In Maciej Zworski's book $\textit{Semiclassical Analysis}$, an important step in proving $L^p$ bounds on quasimodes is deriving a microlocal oscillatory integral representation formula for families of ...

**5**

votes

**2**answers

286 views

### Operators from $L^{\infty}$ to $L^{\infty}$

If $T$ defined as $T(f)(x)=\int K(x,y)f(y)dy$, where $K(x,y)$ is locally integrable, is bounded from $L^{\infty}$ to $L^{\infty}$, how can we show that $\|\int|K(\cdot,y)|dy\|_{L^{\infty}}\le ...

**2**

votes

**2**answers

141 views

### Non-global oscillation of banded Fourier transform

Can we say something like monotonicity, growth rate and oscillation of the Fourier transform of a banded function $f$ with support $[0, N]$
$$\mathcal{F}f(\xi) = \int_{0}^N f(x)e^{-ix\xi}dx.$$
Of ...

**3**

votes

**1**answer

96 views

### To give an estimate for the maximal function associated to the Schrödinger group by using a measurable selector function

I am consulting some papers (references below) about the Carleson's problem for the pointwise convergence of the SchrÃ¶dinger group
\begin{equation}
S_t=e^{i t \Delta}.
\end{equation}
In this context ...

**2**

votes

**1**answer

143 views

### Interpolating delta like functions by trigonometric polynomials of bounded modulus and fast decay

Consider a grid of points $T=\{t_0,t_1,\ldots,t_m\}$ with $0\le t_i\le 1$. I would like to find a function $f(t):[0,1]\rightarrow \mathbb{C}$ of the form
\begin{equation*}
f(t)=\sum_{k=-n}^n c_k ...

**2**

votes

**1**answer

205 views

### bounding the absolute value of a trigonometric polynomial

Consider a function $f:[0,1]\rightarrow \mathbb{C}$ and points $t_0,t_1,\ldots,t_n\in[0,1]$
\begin{equation*}
f(t)=\prod_{k=1}^n\frac{(e^{2\pi i t}-e^{2\pi i t_k})}{(e^{2\pi i t_0}-e^{2\pi i t_k})}
...

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votes

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290 views

### Topology on the Unitary Dual

Suppose I have a locally compact topological group G. The unitary dual of G is the set of equivalence classes of irreducible unitary representations of G. Now, it seems to me that the sensible way of ...

**0**

votes

**1**answer

167 views

### Can any compactly supported continuous function be written as a linear combination of functions with small support

Does anyone have a reference for the following result? I am pretty sure that it is true, and should not be hard to prove, but i would surprise me if it is not already proven in many places:
Let ...

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votes

**4**answers

411 views

### What are the best settings for the large scale geometry of locally compact groups?

My current research involves locally compact groups and from time to time I am tempted to check whether certain notions and statements of geometric group theory of finitely generated groups are still ...

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votes

**3**answers

178 views

### Speed of convergence for Weyl's Equidistribution theorem

If $f$ is a continuous periodic fonction on $[0,1]$ and $a\not\in\mathbb{Q}$, the Weyl's equidistribution theorem states that
$$\frac{1}{n}\sum_{k=0}^{n-1}f(ak)\rightarrow \int_0^1 f(x)dx.$$
Can we ...

**10**

votes

**1**answer

269 views

### Do circular pipes maximize flow rate?

Suppose that $U \subset \mathbb{R}^2$ is nonempty, open, connected and bounded. Consider a Poisseuille flow in the pipe $U \times \mathbb{R}$. That is: a time-independent incompressible flow of the ...

**1**

vote

**1**answer

169 views

### Convergence of a sum to the integral

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a 1-periodic function. I am looking about the conditions on $(a,b)\in\mathbb{R}^2$ such that we have the property :
...

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votes

**1**answer

174 views

### Finite index subgroups of locally compact groups

Let $G$ be a locally compact (Hausdorff) group and let $H$ be a finite index subgroup of $G$. Can we say that $H$ has to be a closed subgroup of $G$? If it is not correct, do you know any ...

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votes

**0**answers

98 views

### How to bound Haar coefficients in terms of total variation?

I'm trying to get the basic idea behind the proof of Theorem 8.1 of this paper, but I'm having difficulty. Specifically, it says:
We shall show that there is a set $\Lambda_n\subset\mathcal{D}$ ...

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votes

**0**answers

112 views

### A question about multiplier algebra of $C_0(G)\otimes C_b(G)$ for a locally compact group $G$

Let $G$ be locally compact group. How we can show that
$$
M(C_0(G)\otimes C_b(G))=C_b(G,C_b(G)).
$$
($M(C_0(G)\otimes C_b(G))$ is the multiplier algebra of $C_0(G)\otimes C_b(G)$)

**3**

votes

**1**answer

134 views

### Plancherel formula for non-second-countable (non-unimodular) groups

The Plancherel formula for unimodular, second-countable, type 1 groups can be found in A Course in Abstrict Harmonic Analysis by Gerald Folland (theorem 7.44) or here. It states that we can get a ...

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votes

**0**answers

143 views

### Laplacian type operator on compact Lie group

Consider the operator $S = \sum_{i,j} X_{ij}^2$ on $L^2(SO(n+1))$, where $X_{ij}$ generates the rotation of the sphere $S^n$ in the $ij$-plane keeping the $(n - 2)$ complementary directions fixed. ...

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votes

**1**answer

232 views

### Littlewood-Paley theory and norm estimation

In the paper "A Convolution Inequality Concerning Cantor-Lebesgue Measures", the Littlewood-Paley theory is used to estimate the norm of multiplier operator in Lemma 1.
It is claimed that Lemma 2 is ...

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votes

**1**answer

275 views

### Could we interpolate the compactness of compact operators?

Classical theorems of Marcinkiewicz and Riesz and their extensions to general Banach spaces by CalderÃ³n, Lions, Peetre, et al. allow us to interpolate the continuity of two operators, viz., the ...

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votes

**0**answers

97 views

### Question about a oscillatory integrals on manifold

Let $M$ be a compact oriented Riemannian manifold without boundary.
Set $f(x)=a(x)+\sqrt{-1}b(x)$ be a complex-valued function on $M$,
where $a(x),b(x)$ are real-valued function on $M$.
Then, how to ...

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votes

**2**answers

387 views

### For what spaces is the Hardy-Littlewood maximal operator of strong type $(p,p)$ if and only if $p > p_0 > 1$?

(This is essentially a continuation of my previous question, here.)
Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. Further assume (though you ...

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vote

**1**answer

154 views

### Description of Bessel potential spaces

Hi, let $1 < p <\infty $, $ 0 < \alpha < 1$, and $ \mathscr{L}^p_\alpha(R^n) $ be the usual Bessel potential space defined by
$$
\mathscr{L}^p_\alpha = (1-\triangle)^{-\alpha/2}L^p(R^n).
...

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votes

**0**answers

150 views

### Notation for a functional L2 matrix norm

Hi,
Let $v(z)$ be a 2x2 matrix depending on a complex variable z, defined on an oriented contour $\Sigma$ in the complex plane. Can anyone tell me the meaning of the notation:
...

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votes

**2**answers

707 views

### Is there a “right” proof of Riemann's Theta Relation?

Let $\theta$ denote the usual Jacobi Theta function (with auxiliary parameter $\tau = i$, for simplicity), i.e.
$$
\theta(z) = \sum_{n \in \mathbb{Z}} \exp(-\pi (a + n)^2 + 2 \pi i n z) \ .
$$
I'm ...

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votes

**0**answers

218 views

### Harmonic analysis on the Heisenberg group

It is well known that:
Theorem 1. For $f\in L^{2}(\mathbb H_{n}=\text{The Heisenberg group of dimesion } 2n+1)$ we have the expansion
$$f(z, s)= (2\pi)^{-n} \sum_{k=0}^{\infty} \int_{0}^{\infty} f ...

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votes

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554 views

### Image of L^1 under the Fourier Transform

The Fourier Transform $\mathcal{F}:L^1(\mathbb{R})\to C_0(\mathbb{R})$ is an injective, bounded linear map that isn't onto. It is known (if I remember correctly) that the range isn't closed, but is ...

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147 views

### variant of Haar measure

I found a certain trig identity in a discussion on Lie groups
\[ \frac{\prod_{i < j} 2 \sin (\mu_i - \mu_j)
\prod_{i< j} 2 \sin (\nu_i - \nu_j) }{\prod_{i, j} 2 \cos (\mu_i - \nu_j) }
= ...

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244 views

### What information about a locally compact group $G$ is encoded in $C_r^\ast(G)$ which is not in $L^1(G)$?

Let $G$ be a locally compact group and let $ C_r^\ast(G) $ denote its reduced group $C^\ast$-algebra. Many features of a $G$ can be realized from $L^1(G)$ or $C_r^\ast(G)$. For example, $G$ is ...

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141 views

### The polynomial Freiman-Ruzsa conjecture for the special case when $f$ is a bijection

The polynomial Freiman-Ruzsa conjecture states that there exists $k > 0$, such that for all $\epsilon > 0$, and for all large $n$ and all functions $ f: \mathbb{F}_2^n \mapsto\mathbb{F}_2^n$, ...

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### Dependence between $\langle f(x), g(y) \rangle$ and $\langle x, y \rangle$.

Let $p$ be a large prime and $n \geq 3$ be an integer. Let $f$ and $g$ be two arbitrary bijections from $\mathbb{F}_p^n$ to $\mathbb{F}_p^n$. I want to find a condition on $f$ and $g$ such that the ...

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147 views

### Inferring asymptotic behaviour from the dominant pole of the Laplace transform

Hi,
I am reposting the following question with the hope that a more detailed description will lead to a more descriptive response:
dominant pole in the laplace transform
I have a vector function ...

**2**

votes

**0**answers

169 views

### Gauss circle problem and Jacobi-type estimates for higher dimensions

Hello everyone, I was doing some late night random reading and I got to wonder about some stuff about the Gauss circle problem.
To begin with, consider a circle in $\mathbb{R}^{2}$ with centre at the ...

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votes

**1**answer

212 views

### About the boundedness of a multiplication operator.

Let be $f$ a $2\pi-$periodic function and $\hat{f}(k)=\frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-ikx}dx$. Consider the operator:
\begin{equation}
Tf(x)=\sum_{k\in\mathbb{Z}}sign(k)\ \hat{f}(k)\ e^{ikx}.
...

**5**

votes

**1**answer

234 views

### For which metric measure spaces is the Hardy-Littlewood maximal operator not of weak type (1,1)?

Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. I'll denote the Hardy-Littlewood maximal operator - either centred or uncentred, I don't mind ...

**0**

votes

**1**answer

235 views

### An interpolation inequality.

For all $s>0$ define for $\epsilon\in(0,1)$ the function:
\begin{equation}
g(\epsilon)=\sum_{k=0}^{\infty}(1+k)^s(\sqrt{1-\epsilon})^k.
\end{equation}
Prove that $\exists C>0$ and $\phi(s)$ such ...

**3**

votes

**1**answer

241 views

### Invariant measures for Cellular automata

An easy question that I have never been able to answer.
Suppose we have the CA on $\{ 0,1,2 \}^{\mathbb{N}}$ with local rule given by $f(x,y)=A_{x,y}$ and $A$ the $3\times 3$ matrix ...

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votes

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191 views

### Pencils of circles and Liouville's theorem

Is there any relation (maybe implicit) between the conformal geometry in the space of circles and spheres and the study of harmonic functions?
In the original question I was musing whether the ...

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votes

**1**answer

123 views

### Maximal function associated to convex bodies

Let $B$ be a centrally symmetric convex body in $\mathbb R^n.$ The maximal function associated to $B$ is defined by
$$
Mf = \sup_{r>0}(\chi_{B})_{r}*|f|.
$$
Bourgain ...

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102 views

### Growth of inner functions on the disk

Recall that an inner function on the disk $D$ is a bounded analytic function on $D$ having radial limits of modulus one almost everywhere.
There has been many works on the growth of the inner ...

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votes

**2**answers

252 views

### Sufficient condition for $L^p$ multiplier on the torus

Hello,
I am searching for a reference of an " easy " sufficient condition insuring that a bounded sequence $(b_{\mathbf{n}\in\mathbb{Z}^d})\in\ell^\infty$ defines a bounded operator from ...

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votes

**2**answers

239 views

### “geometric” description of the algebra of central functions on a Lie group

I am looking for a a description of the algebra of continuous central functions on a group, say a compact simple Lie group $G$, as the algebra of all continuous functions on a "nice" compact Hausdorff ...

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238 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 ...

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votes

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248 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 ...

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votes

**1**answer

245 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 ...

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votes

**1**answer

244 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 ...

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votes

**0**answers

99 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

**2**answers

249 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) = ...

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votes

**2**answers

310 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 ...

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votes

**1**answer

142 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

**2**answers

359 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 ...