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3
votes
1answer
242 views

An integral estimate over rotations of the dyadic grid

I'm currently reading the paper Rectifiable Sets and the Traveling Salesman Problem (link) by Peter Jones (Invent. math. 102, 1-15 (1990)), and am having trouble understanding an integral estimate ...
1
vote
0answers
105 views

Do the irreducible unitary representations of a locally compact group form a separating set for the Radon measures on the group?

Let $\mu$ and $\nu$ be two Radon measures on a locally compact group $G$. For every irreducible unitary representation $\pi$ of $G$ and vectors $u$ and $v$ from the corresponding Hilbert space $H_\pi$ ...
7
votes
1answer
222 views

C* algebras of Almost Periodic Functions

Suppose we take, for example, the $C^*$-algebra which is the sup norm closure of the exponentials $e^{2 \pi i ax}$ where $a \in \mathbb{Z} + \theta \mathbb{Z}$ for $\theta$ an irrational number. This ...
2
votes
0answers
97 views

Justification of the convolution operation of $L^1(G)$ functions where $G$ is a LCA group (measurability)

Suppose $G$ is a locally compact abelian Hausdorff group (LCA), and $\lambda$ is the Haar measure on it. We all know the convolution of two $L^1(\lambda)$ functions $f$ and $g$ on $G$ is defined as ...
2
votes
0answers
80 views

What is the source of this simple lemma in harmonic analysis over compact groups?

I recently needed this simple fact of harmonic analysis: Let $G$ be a discrete abelian group and $X\subset G$ be a finite subset. Then for any $\epsilon>0$ the set $\Gamma_\epsilon=\{ ...
2
votes
0answers
136 views

A decomposition of a representation via characters of a normal compact subgroup

This is connected to my question here. Let $K$ be a normal compact subgroup in a locally compact group $G$, $\widehat{K}$ the dual object for $K$, and $\mu_K$ the normed Haar measure on $K$ ...
5
votes
2answers
224 views

When is the group algebra $L^1(G)$ semisimple?

Let $G$ be locally compact group. Define group algebra as $$L^1(G)=\{f\colon G\to\Bbb{C}\mid\int\lvert f(x)\rvert\, dx<\infty\}$$ with convolution product. When is the group algebra $L^1(G)$ ...
6
votes
2answers
139 views

Spherical functions for sl(2,Q_p)

I kindly would like to ask you the following- I am refering to page 175 in the Book by Gelfand, Graev, Shapiro, etc, on "Automorphic forms ..." My question to which I would kindly ask you to answer ...
0
votes
0answers
40 views

harmonic functions on hyperbolic plane x Real line

I am looking for references of harmonic functions(no growth condition) on the product of hyperbolic plane $\mathbb H^2$ with the real line $\mathbb R$. The metric is just the product one. Especially ...
3
votes
1answer
82 views

Is the annihilator of the intersection of two subgroups of a (countable) discrete abelian group generated by the annihilators of the two subgroups?

Let $G$ be a (countable) discrete abelian group and denote by $\hat{G}$ its Pontryagin dual, i.e. the compact abelian group of group homomorphisms $\chi:G \longrightarrow \mathbb{T}$. Recall that, for ...
3
votes
0answers
113 views

Estimates of a bilinear oscillatory integral

Consider the operator $T_\lambda f(x)= \int_{\mathbb{R}^3}{\frac{\sin\lambda|x-y|}{|x-y|}\phi(x-y) f(y) dy}$, where $\phi\in C_0^{\infty}$, $\phi(x)=1$, when $|x|<1$ . My question is that do we ...
13
votes
2answers
266 views

Can a continuous function on a compact group $G$ be interpreted as the sum in $C(G)$ of its Fourier series?

I asked this in math.stackexchange, but it disappeared from the "main list" almost immediately, so I hope it will be appropriate as a separate question in MO. For a given function $f\in C(G)$ on a ...
5
votes
3answers
318 views

Decomposition of $L^2(\Gamma \backslash G)$

Let $G$ be a semisimple Lie group, and $\Gamma$ be an lattice (arithmetic) - typical examples I am thinking about would be $(SL_2(\mathbb{R}), SL_2(\mathbb{Z})$, or $(SL_2(\mathbb{C}), PGL_2(O_F))$ ...
2
votes
1answer
144 views

To understand integral :$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$

I wants to understand the integrals of the form $$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$$ where $\mu(\alpha)$ is a non decreasing function ...
2
votes
0answers
132 views

Sum of Squares and Harmonic Functions

Let $a_0, a_1, \dots a_k$ be non-negative reals. Any homogeneous polynomial $p$ of degree $2k$ in $\mathbb{R}^{d}$ can be decomposed as $$ p(x)=\sum_{i=0}^{k}c_i|x|^{2(k-i)}p_{2i}(x) $$ for some ...
18
votes
2answers
2k views

Is this statement which relates the Fourier transform of a function to its singularities correct?

I am working on a problem, which would possibly relate the Fourier transform/series with the jump singularities of the function where the function itself or one of its derivatives jump. ((some kind of ...
2
votes
0answers
133 views

pointwise limit of uniformly bounded sequence in $A(\mathbb T)$ is again in $A(\mathbb T)$?

Let $\mathbb T$ be a circle group, and $\hat{f}(n)= \frac{1}{2\pi}\int_{0}^{2\pi} f(t) e^{-int} dt;$ $(n\in \mathbb Z, f\in L^{1} (\mathbb T)).$ Put $A(\mathbb T)= \{f\in C(\mathbb T): \hat{f}\in ...
3
votes
1answer
256 views

Can the Fourier series of a continuous function diverge on an uncountable set of measure zero?

I know that there exist continuous function $f: [0,2\pi]\rightarrow\mathbb{R}$ whose Fourier series diverges at all rational points of $[0,2\pi]$(c.f. Katznelson).We also know that the set of ...
1
vote
0answers
44 views

Special case of Forced Harmonic Oscillator

$$ \frac{d^2l}{dt^2} + c \frac{dl}{dt} +k= F_0\sqrt{1-l^2}$$ Either a closed form or something like given initial conditions, the max value of $l$. For small $l$, this boils down to simple step ...
9
votes
2answers
315 views

Entropy for Haar measure on $O(n)$

Let $G$ be a locally compact group. A measure $\mu$ is the right-Haar measure on $G$ if for every $g\in G$ and $E\subseteq G$ Borel set $\mu(Eg)=\mu(E)$. It is known that every locally compact group ...
3
votes
0answers
115 views

$f, \hat{f} \in L^{p}\cap L^{\infty} \implies f\in B(\mathbb R)$ (algebra of Fourier- Stieltjes transforms )?

For a bounded complex Borel measure $\mu$ on $\mathbb R$, we define, its Fourier-Stieltjes transform, $\hat{\mu}(y)= \int_{\mathbb R} e^{-2\pi ix\cdot y} d\mu(x); (y\in \mathbb R).$ Let $1\leq p \leq ...
1
vote
1answer
74 views

$L^p$ estimate for (powers of) a Laplacian with inverse square potential

I need an estimate of the form $$ \|v\|_{L^p} \le C \|(K-\Delta- c|x|^{-2})^s v\|_{L^p} $$ where $K>0$ can be large if necessary, $c$ is positive but below the Hardy constant $(n-2)^2/4$, where $n$ ...
2
votes
0answers
95 views

Tauberian theorem from generalized Gelfand transform

Wiener's theorem gives the necessary and sufficient conditions for the set of translates of a set of functions to be dense in $L^1(\mathbb{R}^n)$, which translates algebraically into a statement about ...
5
votes
0answers
104 views

Special elements in $L^{\infty}(G)^*$

Let $G$ be a locally compact group. Let $M(G)$ denote the measure algebra and $L^1(G)$ denote the group algebra on $G$. Then $M(G)$ acts on $L^1(G)$ by convolution. So by duality $M(G)$ acts on ...
0
votes
1answer
53 views

properties of frequency-uniform decomposition operator $\square_{k}^{\sigma}$

Let $\rho \in S(\mathbb R^{n})= \text{Schwartz space}, \ \rho:\mathbb R^{n}\to [0,1]$ be smooth radial function verifying $\rho(\xi)=1$ for $|\xi|_{\infty}\leq \frac{1}{2}$ and $\rho(\xi)=0$ for ...
1
vote
2answers
223 views

Lattices in general totally disconnected locally compact groups

Besides automorphism groups of trees and buildings, I was wondering if the lattices in general totally disconnected locally compact groups have been studied in the literature? I appreciate if you ...
0
votes
1answer
171 views

Closed sets in the space of Fourier transforms $\mathcal{F}L^{1}$

Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ...
0
votes
0answers
49 views

Hermite coefficients of a positive density

It is well known that a necessary condition for a function in $L_2$ to be a.e. positive is that its fourier transform is positive-definite (in fact, due to Bochner's theorem, this is also a sufficient ...
0
votes
0answers
44 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
vote
0answers
70 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
100 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
vote
0answers
43 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
votes
1answer
222 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
votes
0answers
25 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
votes
0answers
153 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
votes
0answers
168 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
vote
2answers
112 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
votes
1answer
171 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
86 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
vote
0answers
97 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
votes
1answer
125 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
98 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 ...
6
votes
1answer
127 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
votes
0answers
120 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
vote
0answers
59 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
202 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
votes
2answers
121 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
votes
0answers
58 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
votes
1answer
165 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
votes
1answer
172 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 ...