All Questions
451 questions
2
votes
0
answers
89
views
Link between subharmonic and subanalytic functions
Consider $\Omega$ an open set of $\mathbb{C}$ and $f : \Omega \to \mathbb{R}$ a $C_{\infty}$-function. The following two definitions are well-known (at least the first one) but I prefer to recall them ...
- 1,017
1
vote
1
answer
334
views
Orthonormal basis and decay
Edit: I added smoothness, hoping to simplify the problem with this additional assumption.
Let me motivate this question first: In signal analysis it is often of interest to understand when a certain ...
- 501
2
votes
0
answers
136
views
Equivalent statement of the Wiener-Tauberian theorem?
I would like to know why we have the equivalence between the following three statements of the Wiener-Tauberian theorem:
version 1: If $I$ is a closed ideal in $L^1(\mathbb R)$, such that the set $...
- 650
2
votes
0
answers
79
views
One-dimensional integral equation uniquely solvable?
I recently met a question similar to this one and I would like to post it here, because I basically found nothing:
We define the (possibly unbounded) integral operator $T:D(T) \subset C_0(\mathbb{R}) ...
- 329
1
vote
0
answers
124
views
Inequality about the Fourier transform: $\Vert u \Vert_{L^k} \le \Vert \mathcal{F}(u) \Vert_{L^m}$ (where $1 \le m \le 2$ and $m,k$ Holder conjugates)
How can I prove the following inequality about the Fourier transform?
$$\Vert u \Vert_{L^k(\mathbb{R}^N)} \le \Vert \mathcal{F}(u) \Vert_{L^m(\mathbb{R}^N)}$$ for $1 \le m \le 2$ and $m,k$ Holder ...
user89890
1
vote
0
answers
85
views
A kernel on the d-dimensional flat torus with smoothing properties in the $L^{\infty}$-norm
Let $\rho: \mathbb{R}^d\rightarrow \mathbb{R}_+$ be smooth, symmetric, of compact support, and satisfy $\int_{\mathbb{R}^d}\rho(x)dx=1$. For each $\epsilon>0$, set $\rho_{\epsilon}(x)=\epsilon^{-d}\...
- 11
2
votes
1
answer
1k
views
Pointwise convergence implies uniform convergence?
Let $K$ be an integral kernel of a bounded operator $S:L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n) $ defined like
$$(Sf)(x)= \int_{\mathbb{R}^n}K(x,y)f(y)dy.$$
Assume that $K\in C^{\text{bounded}...
- 329
4
votes
0
answers
349
views
Fractional integral inequality (Hardy-Littlewood-Sobolev)
I am investigating the following integral
\begin{equation}
I^*(x) = \int_{\mathbb{R}} \frac{f(y) \ln |y-x| }{|y - x|^{\mu}} \, dy
\end{equation}
where $f \in L_p(\mathbb{R})$, $ 1 < p < q <...
7
votes
1
answer
489
views
When the value of a function in a point is equal to its integral average over the point's neighborhood?
It is well-known that the harmonic functions have this remarkable Averaging Property: if $f$ is harmonic in a domain $U \subset R^n$, then, for any point $x \in U$, $f(x)$ is equal to the integral ...
- 91
3
votes
0
answers
126
views
An identity of operator norms and de Leeuw's theorem
Let $$Hf(x_1,x_2)=p.v.\int_{-\infty}^\infty f(x_1-t,x_2-S(x_1,x_1-t))\frac{dt}{t},$$
$$T_\lambda f(x)=\lim_{\epsilon\to0}\int_{|x-y|\ge\epsilon}e^{i\lambda S(x,y)}(x-y)^{-1}f(y)dy, $$ where $S(x,y)$ ...
- 187
1
vote
0
answers
113
views
Hardy $Hp$ norm of similar function
Let $f(z)=\sum_{n=0}^{\infty} \frac{c_n}{n+1}z^n$, where sequence $c_n \in S^1=\{z:|z|=1\}.$ We observe $H^p$ norm $\|f\|_{H_p}$, where $H^p$ is Hardy space, $1 \leq p < \infty$.
Question: For the ...
- 259
1
vote
0
answers
82
views
Topology of the algebra $\mathbb{C}\{A\}$ for a LCA group $A$
Let $\mathcal{A}$ be a complex associative Hausdorff topological algebra, and let $A\subset\mathcal{A}$ be a locally compact Abelian (LCA) subgroup (multiplicative). The linear span $\mathbb{C}\{A\}$ ...
- 1,959
2
votes
0
answers
136
views
To find a positive function with compact spectrum
Let
$e_1=(0,1)^T$,
$$
S=\left\{x\in \mathbb{R}^2\Big| \frac{|\langle x, e_1\rangle|}{|x|}>\delta>0\right\},
$$
is a cone in $\mathbb{R}^2$.
I want to find a non-trivial smooth function ...
- 29
3
votes
1
answer
480
views
Is there a uniform upper bound for this oscillatory integral?
I am wondering whether the following uniform upper bound holds:
$|\int_a^{2a}\frac1t\sin(N b^2t)\exp(iNbt^2)dt|\le Cab^2,$
where $0<a<b<1$, $N>N_0(a,b)\gg1$, and $C$ is a constant ...
- 187
1
vote
0
answers
202
views
Space of analytic function and sequence space $l^p$
Let $\mathbb{D} = \{z:|z|<1\}$ be open unit disc in complex plane. Define space of analytic function:
$N^p=\{f:\mathbb{D} \to \mathbb{C} | f(z)=\sum_{n=0}^{\infty} a_n z^n, \sum_{n=0}^{\infty}|a_n|...
- 259
1
vote
0
answers
864
views
A guide to the work of Timothy Gowers on Banach Spaces [closed]
I'm undergraduate student and I'm thinking of doing my graduation thesis on some of Prof. Gowers work on Banach Spaces. It is not required to produce an original result in my thesis, I'm only asked to ...
- 329
3
votes
1
answer
195
views
Boundedness of different Fourier transforms
Let $f: \mathbb{R}^n \rightarrow \mathbb{C}$ be in $L^2\cap L^1,$ then the Fourier transform is in $L^2 \cap L^\infty.$
Does this imply that we can take common norms in the sense that we can estimate ...
- 33
3
votes
0
answers
214
views
Is flatness of Wigner Ville Distribution of error function in Fourier Approximation possible? Is it required?
For a real valued function $f(t)$ I want to check the information left, after taking a Fourier partial sum/integral. Let $\hat{f}$ be its Fourier transform and let $$e_{\omega}(t) = f(t) - \int\...
- 698
1
vote
1
answer
124
views
On a weaker condition of summability for Fourier series
The Wiener algebra $W:=W(\mathbb{T}^n)$ on the torus is defined as the algebra of all continuous fonctions $f$ on $\mathbb{T}^n$ such that $(\widehat f(k))_{k\in \mathbb{Z}^n} \in \ell^1(\mathbb{Z}^n)$...
- 1,035
2
votes
1
answer
352
views
Young’s complement of $ x \mapsto x \, {\log^{+}}(x) $, $ N $-functions and Orlicz spaces
The function $ \Phi: \mathbb{R} \to \mathbb{R} $ is an $ N $-function if and only if it is continuous, even and convex with:
$ \displaystyle \lim_{x \to 0} \frac{\Phi(x)}{x} = 0 $.
$ \displaystyle \...
- 259
1
vote
0
answers
112
views
Notations - Hardy and Sobolev Spaces [duplicate]
After some confusion on my part, I wanted to know is there a profound mathematical reason why both Hardy spaces and Sobolev spaces are denoted by $H^p$(1). Is it just coincidence? Does it have any ...
- 3,574
1
vote
1
answer
460
views
Fourier transform either changes sign infinitely often far out or is continuous at $x=0$
I am reading a book "Fourier Series and Integrals" by Dym & McKean.
There is an exercise (Page 106):
Exercise: Check that if $f$ is a real, even, summable function and
if $f(0+)$ and $f(0-)$...
- 371
10
votes
0
answers
207
views
Projective tensor squares of uniform algebras
In discussion with a colleague recently (Jan 2017),
$\newcommand{\AD}{A({\bf D})}\newcommand{\CT}{C({\bf T})}$
I was reminded that if $A(D)$ denotes the disc algebra and $\iota: \AD\to \CT$ is the ...
- 25.8k
3
votes
0
answers
317
views
Best constant for maximal function for locally compact groups
Maximal functions for locally compact groups have been studied. In particular papers of K. Phillips and Phillips-Taibleson (here and here) contain nice results in this direction. It might be too much ...
- 1,583
2
votes
1
answer
183
views
is this weighted-maximal function unbounded?
The Hardy-Littlewood maximal operator
$$Mf(x)=\sup_{x\in B}\frac1{\vert B\vert}\int_B\vert f(y)\vert dy$$
where the supremum is taken over all balls $B\subset\mathbb{R}^n$ which contain $x$.
It is ...
- 43.2k
3
votes
2
answers
457
views
Integrality of complex infinite series
Let $(a_n)$ be a (double-sided) sequence of complex numbers satisfying
$$\sum_{\mathbb{Z}}\vert n\vert\,\vert a_n\vert^2<\infty, \tag1$$
$$\sum_{\mathbb{Z}}a_n\bar{a}_{n+k}=\delta_0(k), \qquad \...
- 43.2k
1
vote
0
answers
194
views
Cotlar-Stein's Lemma and the Dirichlet kernel
It is well-known that Cotlar-Stein's Lemma can be used to prove the $L^2$ boundedness of the Hilbert transform. See e.g. $L^2$ boundedness of the Hilbert transform via Cotlar-Stein Lemma. Then using ...
- 171
2
votes
0
answers
186
views
Is this simple oscillatory integral operator uniformly bounded on $L^2$?
Let $\phi(t,s)$ be a real-valued function smooth away from the diagonal, and equal to 0 on the diagonal. Assume that $0\le \phi(t,s)\le |t-s|$ for $t,s\in \mathbb{R}$. Let
$$T_\lambda f(t)=\int \frac{\...
- 171
2
votes
0
answers
125
views
Imbedding Theorems between Besov Spaces and space of continuos functions on the unit circle
I'll try to be brief.
Let us consider the Besov Space $B^{1/p}_{p, p}(\mathbb{T})$, where $1\leq p<\infty $ and $\mathbb{T}$ is the unit circle in the complex plane. I would like to know for which ...
- 141
2
votes
0
answers
183
views
Are there any improvements on the estimate of oscillatory integral with one-side folds?
Suppose $X$ and $Z$ are open sets in $\mathbb{R}^d$ and $\mathbb{R}^{d+1}$, respectively. Define $T_\lambda f:L^2(Z)\to L^2(X)$ by $$T_\lambda f(x)=\int e^{i\lambda\Phi(x,z)}a(x,z)f(z)dz,$$where the ...
- 171
3
votes
1
answer
652
views
When does an inverse PDE operator have a kernel (i.e. a fundamental solution?)
Let $L$ be an elliptic linear operator on $\mathbb R^n, n\geq3$. For simplicity, let's stick to the following Schrodinger operator
$$
Lu:=-\Delta u+V(x)u
$$
where $V\geq0$ is the electric potential, ...
- 391
4
votes
1
answer
140
views
Finding a special Banach algebra and a net of homomorphisms
If $A$ is a Banach agebra and $M$ is a Banach $A$-bimodule then a linear map $T:A\to M$ is called an $A$-module homomorphism if $$T(ab)=aT(b),\quad T(ab)=T(a)b,\qquad a,b\in A.$$ Also $A\hat{\otimes} ...
- 231
2
votes
1
answer
3k
views
A simple question about the Hardy-Littlewood maximal function
Let $f\in L^1(\mathbb{R}^n)$. It is well known that the Hardy-Littlewood maximal function $Mf\notin L^1(\mathbb{R}^n)$ (if $f \ne 0$ a.e.), though there is a weak-type (1,1) bound for this maximal ...
- 171
5
votes
2
answers
2k
views
Which functions are continuous with respect to the weak topology?
Let me first introduce the restricted setting in which this question has a nice answer. I came up with this when messing around with a homework problem in a PDE class a couple years back.
Let $\phi \...
- 579
2
votes
0
answers
85
views
There is an analogous to the inerpolation theorem of Aronszajn-Gagliardo for multilinear operators in Banach spaces?
The Aronszajn-Gagliardo theorem states that on the category $\mathcal B$ of Banach spaces, given $A$ an interpolation space with respect to the couple $\overline A$ there exists an exact interpolation ...
- 71
0
votes
1
answer
697
views
How much do we know about this "local" Hardy-Littlewood maximal function?
The "local" Hardy-Littlewood maximal function is given by $$(M_\phi f)(x)= \sup_{0<\epsilon<1}|\phi_\epsilon \ast f|(x),$$ which is similar to the classical Hardy-Littlewood maximal function : $$...
- 171
0
votes
1
answer
328
views
Discrete Calderon-Zygmund operators
I would like to know whether there exists a Calderon-Zygmund theory discrete singular kernels. In particular I am interested when the discrete operator $T$ with kernel $K(n,m)$ given by
$$(Tf)(n)=\...
- 153
4
votes
1
answer
1k
views
The convolution of a $L^1$ function and an approximate identity
It is well known that the convolution of a $L^1$ function and a Schwartz function is also in $L^1$, by Young's inequality for convolution. Let $f\in L^1(\mathbb{R}^n)$ and $\phi\in S(\mathbb{R}^n)$, ...
- 171
5
votes
0
answers
215
views
Explicit description of the Plancherel measure for $GL_n(\mathbb{R})$
Let $G:=\mathrm{GL}_n(\mathbb{R})$ and $f\in C_c^\infty(G)$. One can uniquely determine the Plancherel measure $d\mu_p$ on $\hat{G}$, the unitary (actually tempered) dual of $G$, by the equation
$$f(g)...
- 1,692
1
vote
1
answer
238
views
Does the bounded extension of the Fourier multiplier operator agrees with its original explicit definition?
We consider the Fourier multiplier operator $T_0$ defined by the explicit expression
$$(T_0f)(x)=\int_{\mathbb{R}^n}{e^{ix\cdot \xi}m(\xi)\hat{f}(\xi)d\xi}, \ f\in S(\mathbb{R}^n),$$ where $S(\mathbb{...
- 171
6
votes
2
answers
318
views
Natural map $C^*(G) \to M(A\rtimes G)$
Let $A$ be a C*-algebra, let $G$ be a locally compact group, and let $\alpha\colon G\to\mathrm{Aut}(A)$ be a (strongly) continuous action. It is well known that there is a natural map $\iota_G\colon C^...
- 717
2
votes
1
answer
172
views
Ideals of $L^1(G)$ and normal subgroups of $G$
Let $G$ be a locally compact group. Is there any correspondence between closed two-sided ideals of $L^1(G)$ and closed normal subgroups of $G$? (Especially, is there any correspondence between finite ...
- 565
5
votes
0
answers
332
views
Harish-Chandra's submersive principle on closed subsets
Harish-Chandra's submersion principle says the following. Let $X,Y$ be two manifolds of dimensions $m$ and $n$ respectively. Let $\pi: X\rightarrow Y$ be a surjective smooth map which is submersive ...
- 960
2
votes
0
answers
132
views
Wiener-Ikehara Theorem and Signal Processing
I am trying to understand the Wiener-Ikehara Tauberian theorem which can be a step to understanding the prime number theorem. Let
$$ \hat{a}(s) = \int_0^\infty e^{-us}\, da(u) $$
with $a(u)$ some ...
- 22.8k
4
votes
0
answers
114
views
Coming up with a represenation for sum of functions in the Fourier algebra
This is my first overflow question, so let me apologize in advance if this belongs on http://math.stackexchange.com.
Let $G$ be a discrete group.
Let $\lambda:G\to B(\ell^{2}(G))$ be the left ...
- 161
2
votes
0
answers
110
views
If $f_j\to f$ in $L^1(\Bbb R^n)$ then $Tf_j\to Tf$ in $L^{1,\infty}(\Bbb R^n)$
Let's define $A:=\{f\in L^1(\Bbb R^n)\cap L^2(\Bbb R^n)\;:\;f\;\mbox{has compact support}\}$. So $A$ is dense in $L^1(\Bbb R^n)$.
Given then $f\in L^1(\Bbb R^n)$; by density there exists $\{f_j\}_j\...
user76025
10
votes
1
answer
439
views
Interpolation between $L_1^0$ and $L_2^0$
Let $L_p^0$ be the mean zero functions in $L_p(G)$, where, say, $G$ is an infinite compact group endowed with normalized Haar measure. Suppose that $T$ is a bounded linear operator on $L_1$ that maps $...
- 31.5k
5
votes
3
answers
2k
views
Morrey's inequality for Sobolev spaces of fractional order
Let $H^s(\mathbb T)$, where $s\in\mathbb R$, be the space of $2\pi$-periodic functions (or distributions), $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx}$, such that
$$
\|u\|_{H^s}^2=\sum_{k\...
- 2,933
2
votes
1
answer
481
views
Ideals of $L^1(G)$
I want to study the closed ideal structure of $L^1(G)$. Is there a good paper or book which characterizes closed ideals and maximal ideals of $L^1(G)$?
- 565
1
vote
1
answer
1k
views
Does the product function $fg$, where $f$ is in $L^2$ and $g$ is in $C^{\infty}_0$ belong to hardy space $H^1$?
I am struggling to know whether the product function $fg$, where $f$ is in $L^2$ and $g$ is in $C^{\infty}_0$ belong to hardy space $H^1$.
$fg$ has compact support but I can't figure out how I can try ...
