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3
votes
1answer
136 views

Sets $E$ in $\mathbb{Z}$ such that any $l^2$ function with support on $E$ comes from Fourier of a continuous function

Are there infinite sets $E\subset\mathbb{Z}$ such that any $f\in l^2(\mathbb{Z})$ with support on $E$ comes from the Fourier transform of a continuous function on $\mathbb{T}$ ? If yes, is there a ...
0
votes
0answers
116 views

Weak amenability and quasi central bounded approximate identity

Let $\cal A, \cal B$ be a non commutative Banach algebras, and $\cal A$ be weakly amenable and has a quasi central bounded approximate identity. Let $T:\cal A\to \cal B$ be an algebra ...
4
votes
1answer
168 views

eigenfunctions of the Fourier transform in the Schwartz space

Recall that $L^2(\mathbb R)$ decomposes into the direct sum of the eigenspaces of the Fourier transform corresponding to its four eigenvalues, namely the four fourth roots of unity. If $f\in ...
1
vote
0answers
86 views

Differentiability of $f*g$ on the circle, for integrable f, bounded g, and some decay of the Fourier coefficients of f

If $f\in L^1(\mathbb{T})$ and $g\in L^\infty(\mathbb{T})$ where $\mathbb{T}$ is the circle, such that $\hat{f}\in L^{p}(\mathbb{Z})$ for some $1\leq p<\infty$, do we have that $f*g$ is ...
7
votes
0answers
148 views

Why is it hard to obtain improved $L^6$ bound of eigenfunction of Laplacian on 2-dimensional compact Riemannian manifold?

Consider the $L^p$ estimate of the Laplacian on a compact boundaryless Riemannian manifold, suppose that $-\Delta_ge_{\lambda}=\lambda^2e_\lambda(x), x\in M$. C.D. Sogge proved that we have the ...
2
votes
1answer
243 views

Beurling density and interpolation

Let $\Lambda=\{\lambda_n\}_1^\infty$ a set of points on the real line. We denote by $\bar{n}(r)$ the largest number of points in any interval $[x,x+r]$, $r>0$. Define the upper uniform density ...
0
votes
1answer
365 views

Existence of Green's function and the Dirichlet problem

Let $U \subset \mathbb{R}^n$ be a bounded domain, and consider the following problem : $$\left\{ \begin{array}{lcr} -\Delta u = 0 & & \text{in } U, \\ u = g & & \text{on } \partial U, ...
1
vote
2answers
224 views

Is there a Calderon-Zygmund decomposition for $L^p$ function

The Calderon-Zygmund decomposition for a $L^1$ function is well known, which says for any $f\in L^1$, then we can decompose $f$ into a good term $f$ and a bad term $\sum b_k$, such that for any ...
1
vote
0answers
124 views

A problem concerning measures on locally compact spaces

I am stuck on a question for quite sometime now, although in the text it is said to be "apparent". The problem goes as the following : Let $X$ and $Y$ be locally compact Hausdorff spaces. Then $M(X)$ ...
1
vote
1answer
168 views

Examples of functions in $W^{k,p}(\Omega)$ with exact smoothness

Please give, explicitly, a function $f:\Omega\mapsto\mathbb{R}$ such that $f\in W^{k,p}(\Omega)$ but $f\notin W^{s,p}(\Omega)$ for $s>k$. Here $\Omega$ can be a subset of $\mathbb{R}^n$ with ...
2
votes
2answers
228 views

The Szego projector, the dual disc bundle $\overline{D}$ and representation of $S^1$ on $H^2$($\overline{D}$)

This construction arises when constructing the Szego projector. Let's consider the dual disc bundle $\overline{D}$ of a positive Hermitian line bundle ($L$,$h$) over a compact Kahler manifold $M$, ...
2
votes
2answers
367 views

$L^p$-norm of Fourier series in terms of coefficients, $p \neq 2$

It is known that the $L^2$-norm of a Fourier series equals the $l^2$-norm of the coefficients. Are there similar results in the case of $L^p$-norm for $p\neq 2$? Can it be expressed explicitly in ...
0
votes
0answers
83 views

Applying Anderson's theorem to Spherically symmetric distribution in Stein estimation

The question appears Example 3.1 of the paper "Stein Estimation for Spherically Symmetric Distributions: Recent Developments" ...
2
votes
0answers
67 views

What are the boundary asymptotics of harmonic symmetric transverse traceless rank-s tensors on $\mathbb{H}^n$ in the Poincare upper-half-space model? [closed]

This question is motivated by the results in this paper, http://calvino.polito.it/~camporesi/JMP94.pdf In this paper some of its most important results about the asymptotics of symmetric traceless ...
1
vote
1answer
198 views

A proof of energy functional appearing in the regularity of elliptic and parabolic equations

I have got trapped in this problem for nearly two years when I dealt with regularity of solutions of elliptic and parabolic equations. I have not found a nice proof to support this assertion. Now I am ...
1
vote
0answers
115 views

Tauberian measures on a locally compact abelian group

Given a locally compact abelian group $G$ with Haar measure $m$, it is well-known that there exist measures $\mu\in M(G)$ which are singular (with respect to $m$), but the convolution product ...
4
votes
1answer
182 views

sum of integral part of n/k

Is there any direct formula or algorithm better than the brute force (O(n) algorithm by iterating from 1 to n) way to calculate the sum \begin{equation} S = \sum\limits_{i=1}^n [{\frac{n}{i}}] ...
3
votes
1answer
170 views

Does the topological Varopoulos algebra consist of functions that are continuous and Varopoulos norm bounded?

Let $X_1,\dots,X_n$ be compact Hausdorff spaces. Let's define the Varopoulos algebra as the projective tensor product: $$V(X_1,\dots,X_n) := C(X_1) \hat{\otimes} \dots \hat{\otimes} C(X_n),$$ i.e. the ...
5
votes
0answers
132 views

Which cocompact subgroups of $G$ do contain a cocompact normal subgroup of $G$?

Let $G$ be a locally compact group and let $H$ be a cocompact (or more generally, a cofinite) subgroup of $G$. Is there any criterion to determine whether $H$ contains a cocompact normal subgroup of ...
2
votes
0answers
73 views

Fourier multiplier with a singularity on a convex curve

Let $h$ be a strictly convex function such that $h(0) = h'(0)=0$. Let $\Phi: \mathbb{R}^2 \to \mathbb{R}$ be a $C^{\infty}$-function with compact support (say, $\Phi$ is supported on ...
8
votes
1answer
562 views

How differentiable is the convolution of two continuous functions?

The question is really simple: Given $$ f, g\in C^\alpha_c(\mathcal{R}^d) $$ is $$ f*g\in C^d_c? $$ I came up with a formal argument using the decay of the Fourier transform of continuous functions, ...
2
votes
1answer
235 views

iwaniec's conjecture

Does anyone know whether there is any geometric applications of the iwaniec's conjecture on $ l^p $ bound of beurling alfhors transform( or the complex hilbert transform). One application could have ...
5
votes
1answer
290 views

Poisson Summation Formulas for Cut and Project Quasicrystals

In Lagarias' paper "Mathematical Quasicrystals and the Problem of Diffraction" http://www.math.lsa.umich.edu/~lagarias/doc/diffraction.pdf he discusses various ways one might get Poisson summation ...
2
votes
1answer
117 views

The measure on the harmonic spectrum from Selberg trace formula

One can see the following two equations, Theorem 6.1 (Selberg Trace formula) on page 26 of these notes. Equation 3.19 and 3.20 on page 11 of this paper. I vaguely feel that these two are the ...
8
votes
0answers
179 views

when do norm-continuous unitary representations separate points of a group?

Recently I found in the web a discussion on the following question: ...
0
votes
1answer
118 views

Application of Toms- Stein restriction theorem for Strichartz estimates

The initial value problem for one dimensional Shrödinger equation is $$iu_{t}+u_{xx}=0,$$ $$u(x, 0)= f(x),$$ where $u:\mathbb R \times \mathbb R \rightarrow \mathbb C$ is a complex valued ...
1
vote
1answer
103 views

An example for the Convolution on not compact topological groups

Let $G$ be a locally compact but not compact topological group. I'm looking for an example to show that for an arbitrary $p>1$, there exist some $f,g\in L^{p}(G)$ such that $f*g\notin L^{p}(G)$.
1
vote
0answers
220 views

Obtaining a pointwise bound on the convolution of two singular measures

I am confused about a passage in the paper by T. Tao A sharp bilinear restriction estimate for paraboloids. We are in Section 7, near equation (34) (pag.16 of the arxiv). Notations and ...
4
votes
1answer
718 views

Riemann hypothesis and Kakeya needle problem

The question may be a bit vague. I noticed an analogy that both Riemann hypothesis and Kakeya needle problem has been proved in finite fields. Can somebody shed light on why finite field analogues are ...
7
votes
1answer
266 views

Kakeya and Nikodym maximal functions

I've been working through part of Terry Tao's 1999 article "The Bochner-Riesz Conjecture Implies the Restriction Conjecture." (It appeared in the Duke Mathematical Journal.) A little more ...
3
votes
1answer
116 views

Matched pair of locally compact groups

In measure theoretic language there is a notion of matched pair of locally compact (l.c.) groups due to Baaj-Skandalis-Vaes. A pair $(G_{1}, G_{2})$ is called a matched pair of l.c. groups if there ...
1
vote
2answers
157 views

Tangent vectors on the algebra of trigonometric polynomials

Let $G$ be a compact real Lie group and ${\sf Trig}(G)$ the algebra of trigonometric polynomials on $G$ (defined in the Hewitt-Ross, Abstract harmonic analysis, (27.7)), i.e. the algebra of functions ...
0
votes
1answer
98 views

local moments of measures whose Fourier transform vanish in an interval

Assume h is a measure whose Fourier transform vanishes in an interval $[-\Omega,\Omega]$. I'm interested in obtaining inequalities of the form \begin{equation*} \int_{-\delta}^{+\delta}|h|(dt)\le ...
3
votes
2answers
185 views

On lower bounds of exponential frames in l1 norm

Let $\{t_k\}_{k=-\infty}^\infty$ be a sequence of real numbers. I'm interested in finding the largest number A such that \begin{equation*} \int_{-\Omega}^\Omega|\sum_{k=-\infty}^{+\infty}c_ke^{2\pi i ...
8
votes
2answers
694 views

the convolution of integrable functions is continuous?

The question is simple but I still can't prove it or contradict it. Here it goes: Suppose $f$ and $g$ are defined on the circle (or, equivalently, $2\pi$ periodic functions) and Lebesgue ...
3
votes
4answers
335 views

How does one show the existence of discrete and complementary series for SL(2,R)?

In his book on $\mathrm{SL}(2,\mathbb{R})$, Lang shows that any nontrivial irreducible unitary representation of this group is infinitesimally isomorphic to an irreducible admissible subrepresentation ...
2
votes
0answers
119 views

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
2answers
321 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
2answers
143 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
1answer
108 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 Schrödinger group \begin{equation} S_t=e^{i t \Delta}. \end{equation} In this context ...
2
votes
1answer
151 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
1answer
217 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})} ...
10
votes
3answers
352 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
1answer
179 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 ...
10
votes
5answers
505 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 ...
2
votes
3answers
197 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
1answer
284 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
1answer
174 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 : ...
3
votes
1answer
186 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 ...
0
votes
0answers
103 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 $\Lambda_n\subset\mathcal{D}$ ...