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40 views

Explicit constructions of fundamental systems for spherical harmonics

For definitions and notations on the theory of spherical harmonics I refer to www.cis.upenn.edu/~cis610/sharmonics.pdf‎ Let $n,k\geq 0$, and let $S^n$ be the unit sphere on $\mathbb{C}^{n+1}$. Let ...
1
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0answers
69 views

Intuition about poisson summation formula? [duplicate]

Is there any one, who have: any intuition about poisson summation formula any sence about that why poisson summation formula should be true please don't give usual proof(s). poisson summation ...
2
votes
1answer
96 views

What is the multiplicative unitary for SU_q(2) (or other quantum groups)?

Consider a (von Neumann algebraic) locally compact quantum group $(M, \Delta, \phi, \psi)$ where the von Neumann algebra $M$ is realized as operators on the Hilbert space $H$. There is a ...
1
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0answers
39 views

Invertibility of Hankel operators?

Let $D$ be the unit disc in the complex plane and $P$ the Bergman projection mapping $L^2(D)$ onto the closed subspace $A^2(D)$ of holomorphic square-integrable functions (w.r.t. Lebesgue measure). ...
6
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1answer
185 views

van der Corput lemma for oscillatory integrals

My question is about the van der Corput lemma for $$ \int_a^b e^{i t \phi(x)} \psi(x) dx $$ The version you find everywhere, e.g. on ...
0
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0answers
24 views

A question about the convention for the Plancherel measure on $\mathbb{H}^n$

Say I have to calculate the quantity, $Log Tr [ -\Delta - \frac{1}{4} + m^2]$ on $H^n$. Then looking up the spectral measure $\mu(\lambda)$ and the eigenvalues of the Laplacian ($= -\Delta = - ...
4
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0answers
148 views

$f,g , |f|f, |g|g \in A(\mathbb R) \ \text{(Banach algebra)} \implies \left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$?

Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$ and consider Fourier algebra $$A(\mathbb R):= \{f\in ...
2
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0answers
156 views

Scattering for rapidly decaying solutions of NLS

Cazenave and Weissler proved in their paper "Rapidly Decaying Solutions of the Nonlinear Schrödinger Equation" the following property. Given the problem \begin{equation} \left\{ \begin{array}{rl} ...
1
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2answers
105 views

Numerical calculation of Fourier transform with a nice error bound

I'd like to have an algorithm for a numerical calculation of Fourier transform with a nice error bound. To be precise, if $f$ is a function from $L_1(R)$, $F[f]$ is it's exact Fourier transform and ...
0
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1answer
168 views

Can we expect $\left\||f|^{2}f-|g|^{2}g\right\|\leq C ||f-g||$ in the Banach algebra $A(\mathbb T)$ ?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb T):= \{f\in ...
0
votes
1answer
76 views

uniqueness for Poisson equation in R^d with mildly regular data

I'm interested in Poisson's equation $-\Delta u=f$ set in the whole space $R^d$ (let's say $d\geq 3$ for simplicity) when $f$ has very little integrability, specifically $f\in L^{1+\varepsilon}$ for ...
1
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0answers
93 views

How to use, $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in a Banach space $C([0,T]; M^{p,1})$?

(May be this is very basic question for MO) (For details or this question you may see the paper page no. 9, MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; ...
2
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1answer
122 views

Fourier inversion

I have certain doubts about a classical Fourier inversion theorem. According to it (this is a theorem from "Panorama of Harmonic Analysis" by Krantz), if $f$ and $\hat{f}$ are both in $L_1(R)$ and ...
3
votes
1answer
91 views

Siegel domains and cuspidal functions

Let $F$ be a number field and $\mathbb{A}$ the ring of adeles over $F$. We consider $P_{n}$ the mirabolic subroup of $GL_{n}$. Do we have a analog of Siegel subset for the quotient ...
5
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0answers
82 views

Oscillatory integrals of algebraic functions

Consider an algebraic function $\phi$ on $R^{d}$. By this I mean that there exists a polynomial $P$ with coefficients in $R[x_1,...,x_d]$ (coefficients are polynomials!) such that $P(\phi) = 0$ Let ...
0
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0answers
112 views

An inequality for Fourier transform

Assume that $\lambda_1$ is the smallest eigenvalue of the Dirichlet Laplacean for the domain $\Omega\subset \mathbf{R}^n$ and let $0<\alpha\le 1$. Is the following statement well-known? Let $f\in ...
1
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0answers
57 views

Counting frequencies of occurrence of patterns within a sequence using harmonic analysis?

Assume that we are given a sequence $\mathbf x := X_1,\dots,X_n \in \mathbb N^n$ for some $n \in \mathbb N$. I am interested in calculating the frequency of occurrence of some fixed sequence $\mathbf ...
2
votes
2answers
200 views

Defining a Measure on Quotient Spaces

Let $G$ be a locally compact Hausdorff group with a left invariant Haar measure $\mu$ and a closed subgroup $H$. It is well-known (and not hard to prove) that $G/H$ possesses an invariant measure if ...
2
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2answers
120 views

Seeking a class of functions for which sums approximate integrals well

Is there a "natural" class of integrable functions $f: {\mathbb R} \rightarrow {\mathbb R}$ for which it is true (and, preferably, not too hard to prove!) that $\sup_{0 \leq a < h} |h S(a,h) - I|$ ...
0
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0answers
56 views

How to choose negative definite function $\lambda (x)$, so that $\lambda^{-1} \in L^{1}(\mathbb R)$?

We define, $$\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha), (\mu(0)=0) $$ where $\mu (\alpha)$ is a non-decreasing function such that the integral converges for ...
-3
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1answer
157 views

$L^{1}(\mathbb R) \cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R)\subset H_{1}(\mathbb R)$?

Put, $C_{0} (\mathbb R)=\{f:\mathbb R \to \mathbb C: f \text { is continuous on} \ \mathbb R \ \text {and } \lim_{|x|\to \pm \infty}f(x)=0 \}$(= Continuous functions on $\mathbb R$ vanishing at ...
0
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1answer
154 views

Does Hilbert Transform commute with Function Multiplication modulo Compact on $L^p(R)$?

Define Hilbert Transform (HT) as the convolution with the function $1/x$. E. Stein proves in his book Singular Integrals and Differentiability Properties of Functions that HT, when understood as a ...
0
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0answers
15 views

Measurable FD for lattices in lcag

I have a locally compact Abelian group $G$ and a lattice $L \subset G$. Is it always true that we can find a pre-compact measurable fundamental domain for $L$? If $G$ is separable, the answer is ...
1
vote
1answer
130 views

Result of Beurling concerning absolute convergence of Fourier series of |f|

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$ and we put, $$A(\mathbb T):= \{f\in ...
1
vote
1answer
256 views

When one can expect $\widehat{(fg)} = \hat{f} \ast \hat{g}$; $f, g\in L^{1} (G)$?

Let $f, g \in L^{1}(\mathbb T)= L^{1} ([-\pi, \pi))$. We define, the Fourier transform of $f$ as follows: $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt, \ (n\in \mathbb Z).$$ It is ...
8
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0answers
194 views

Laplace Transform in the context of Gelfand/Pontryagin

Question: Do quasi-characters properly generalize the Laplace transform or decompose functions in some setting in a way similar to how characters generalize the Fourier transform and decompose $L^1$ ...
0
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1answer
120 views

How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)? [closed]

We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and ...
1
vote
1answer
118 views

Absolute convergence of multi-dimensional Fourier series

For a Lipschitz function $f$ defined in $[0,2\pi]^d$ for $d>1$, is that true that the multi-dimensional Fourier series converges absolutely? In other words, $\sum_{k\in ...
0
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0answers
58 views

Does there exists $f\in A_{\mathbb R}(\mathbb T)$ with $||f||=r$ such that $||e^{if}||=e^{r}$?

Let $\mathbb Z$, the set of integers, be a group with respect to addition and its dual group is the $\mathbb T = \{z\in \mathbb C : |z|=1\}$ , one dimensional torus. Put, $\ell^{1}(\mathbb Z)= ...
0
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0answers
68 views

Harmonic/functional analysis question: Uniform bounds for $(1 - \varepsilon\Delta)^{-1}$ as $\varepsilon\to 0$?

Is there a space $X$, compactly embedded in $H^{-1}(R^2)$, such that the operators $(1 - \varepsilon \Delta)^{-1}$ are bounded from $L^1(R^2)$ into $X$, with operator norms that are in turn bounded by ...
2
votes
1answer
334 views

Is every distribution a linear combination of Dirac deltas?

My question is whether Dirac-type distributions over an Abelian group define a basis of the Schwartz-Bruhat space $\mathcal{S}(G)^\times$ of tempered distributions on $G$, so that any distribution ...
3
votes
1answer
84 views

Number of small projections

Suppose $X$ is a finite subset of the plane and for $0\leq \theta<\pi$, let $l_\theta$ denote the line through the origin having angle $\theta$ with the positive $x$-axis. For how many values of ...
8
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0answers
233 views

Carleson's Theorem on Manifolds

Let $M$ be an oriented, compact, differentiable manifold with some Riemmanian metric $g$, so that $(M,g)$ has a nice volume form and one can define $L^2(M,g)$ as the completion of $C^\infty(M)$ under ...
0
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0answers
55 views

Factorization in Fourier Algebra and its properties

By Schwartz-inequality and Riesz–Fischer theorem, one can deduced that, ($\ast$ stands for a usual convolution ) $$L^{2}(\mathbb T) \ast L^{2}(\mathbb T) = A(\mathbb T)(:= \{f\in L^{1}(\mathbb T): ...
3
votes
1answer
115 views

Is any finite collection of points contained in a cut and project set with $\mathbb{R}^d$ as internal space?

In "Meyer Sets and their Duals" Moody proves that any Meyer set union a finite number of points is again a Meyer set. Additionally, any Meyer set is contained in a finite union of model sets whose ...
0
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0answers
69 views

Better bound for Hardy-Littlewood maximal function

I have one question about Theorem 1 on page 3 in this notes http://www.cims.nyu.edu/~chou/notes/harmonic.pdf Is there any better bound $\frac{A}{\alpha^{t}}\|f\|_{1}$ for some $t>1$ in b)? What is ...
3
votes
1answer
114 views

A problem on the boundedness of maximal operator by using linearization method

We know that the maximal operator is bounded on $L^{p}(\mathbb{R}^{n})$ where $n\geq 1$ and $1<p<\infty$ and the proof would be contained in many classical harmonic analysis books. Here I find a ...
1
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0answers
114 views

How Fourier transform behaves if we kills the oscillation?

Let $a, b \in \mathbb R$ such that $ab> 1$ ; put $$L^{1}_{a}(\mathbb R)= \{ f:\mathbb R\to \mathbb C \ \text {measurable} : ||(1+|x|)^{a}f||_{L^{1}(\mathbb R)}< \infty \},$$ and ...
2
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1answer
178 views

Understanding Bruhat's notion of Schwartz function

I am trying to understand Bruhat's generalized Schwartz functions over (Hausforff) locally compact Abelian groups [1], following this paper [2] by Osborne. There, the Schwarz-Bruhat space ...
1
vote
1answer
126 views

Strong decomposition tools from Harmonic Analysis in other fields

I would like to know more about the tools in Harmonic analysis, but the ones that give a really good results in other theories. One of them are decompositions, like Whitney, Calderon-Zygmund etc...The ...
0
votes
1answer
86 views

How Fourier-Lebsgue spaces operates functions?

Let $a, b \in \mathbb R$ such that $ab> 1$ ; put $$L^{1}_{a}(\mathbb R)= \{ f:\mathbb R\to \mathbb C \ \text {measurable} : ||(1+|x|)^{a}f||_{L^{1}(\mathbb R)}< \infty \},$$ and ...
3
votes
2answers
341 views

Ultrafilter-based Fourier-Walsh-like Functions

Here is a (little wild) question about Boolean functions with countably many variables and a wild analog for Fourier-Walsh functions and analysis based on them. Let $x_1,x_2,\dots,x_n,\dots$ be ...
2
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0answers
168 views

Does anyone know what is the right reference for the following simple lemma from harmonic analysis?

The lemma says that given $\lambda\geq 1$, $p\geq 1$, $a_j\geq 0$, for a collection of balls $\{B_j\}_{j\in\mathbb{N}}$ in $\mathbb{R}^n$, it holds $$\|\sum_j a_j\chi_{\lambda B_j}\|_p\leq ...
1
vote
2answers
96 views

When is a collection of exponentials dense in $L^2(K), |K|<\infty$

Suppose we have a relatively dense collection of points $\Lambda \subset \mathbb{R}^d$ and $K \subset \mathbb{R}^d$ where $K$ is compact and measurable. When will the linear span of the collection of ...
1
vote
1answer
97 views

Compactly supported smooth function with Laplace transform bounded on a cone

My question is if it is possible to find a compactly supported smooth function $\varphi:\mathbf{R}\to \mathbf{R}$ s.t. the following integration $\int_{\mathbf{R}}\varphi(t)e^{itx}e^{tx}dt$ stays ...
2
votes
0answers
60 views

Associated Legendre/Gegenbauer functions with complex degree at larger order

I am interested in approximating the associated Legendre function (also known as conical function) \begin{equation} P_{-1/2 + i p}^{\frac{2-N}{2}}(x) \end{equation} when $N \to \infty$. The real ...
1
vote
0answers
79 views

A bound for a product in BMO

The question: Let's consider $f\in L^\infty(\mathbb{T})$ and $g\in BMO(\mathbb{T})$. I'm trying to figure out if the following inequality is true $$ \|fg\|_{BMO}\leq C\|f\|_{L^\infty}\|g\|_{BMO}. $$ ...
0
votes
0answers
234 views

What is $p$-adic Fourier series?

Q1: Can we define Fourier series for a function $\mathbb{Z}_p\to \mathbb{Q}_p$? Q2: There are (in a real case) Bernoulli polynomials which have the most simple Fourier expansion: ...
1
vote
1answer
100 views

A small question about the necessary conditon of linear Kakeya conjecture

For $0<\delta\ll 1$ we define a $\delta$-tube to be any rectangular box $T$ in $\mathbb{R}^d$ with $d-1$ sides length $\delta$ and one side of length $1$, observe that such tubes have volume ...
2
votes
1answer
134 views

Rate of convergence of the average of an equidistributed sequence

Let $f : \mathbb R\to\mathbb C$ be an $1$-periodic and sufficiently smooth function, which has zero average, and let $\alpha$ irrational. We know the following: a. ...