All Questions
451 questions
4
votes
0
answers
264
views
Is the Gelfand transform strictly continuous?
Let $M$ be the Banach algebra of measures on the circle with $L_1$ naturally sitting as a closed ideal of $M$. Then $M$ carries the strict topology implemented by the family of seminorms $\|\mu\|_f = \...
- 113
10
votes
2
answers
926
views
Isomorphisms between spaces of test functions and sequence spaces
I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest ...
- 799
1
vote
2
answers
227
views
Number theory on Banach space $L^2(\mathbb R)$ meets linear independence?
Consider an orthonormal basis $(\varphi_k)$ of $L^2(\mathbb R)$ with Lebesgue measure.
I came along a nice number theoretic question in analysis:
Write $$f_k(x):=\int_{\left\lvert y \right\rvert \...
- 54.2k
5
votes
1
answer
171
views
Invariant subspace in infinite dimensions
Let $A(t)$ be a family of skew self-adjoint operator defined on some Hilbert space $H$ with common domain $D(A).$
The dependence on $t$ is in the strongly continuous sense, i.e. for all $x \in D(A)$ ...
- 12.6k
1
vote
1
answer
737
views
$L^2$ function in Schwartz space?
Let $f:\mathbb R^n \rightarrow \mathbb R$ be a smooth function whose derivatives are all polynomially bounded and $f \in L^{\infty}.$
Such a function has the property that when multiplied with any ...
- 5,169
1
vote
1
answer
52
views
Infinitely many independent functions that are only frequency localized?
A function $f \in L^2(\mathbb R^d)$ will be called $K$-frequency localized if the following inequality holds
$$\int_{\mathbb R^d} \lvert \widehat{f}(x) \rvert^2 x^2 \ dx \le K \int_{\mathbb R^d} \...
- 5,169
3
votes
1
answer
213
views
About understanding manifold structure on WAP compactification of $\Bbb{C} \rtimes \Bbb{T}$
Let $G$ be a locally compact topological group. A continuous bounded function $f$ on $G$ is called (weakly) almost periodic if the set $L_Gf$ of left translates is relatively compact in the (weak) ...
- 2,391
2
votes
1
answer
127
views
Are the Prolate Spheroidal Wave Functions absolutely integrable?
I would like to know if the Prolate Spheroidal Wavefunctions (PSWFs, defined below) are in $L^1(\mathbb{R})$. I know that they are square integrable, but cannot decide about absolute integrability.
...
- 91.8k
2
votes
0
answers
418
views
Hölder-Zygmund spaces of negative order
In the equation (1.1.17) in Proposition 1.1.6 (ii) in Alazard and Delort's Sobolev Estimates for Two Dimensional Water Waves, there appears a norm named $C^{-1}$, but in Chapter 6 (Appendix) of the ...
- 60.5k
14
votes
1
answer
514
views
Generalizing the Fourier isomorphism between Sobolev spaces and weighted $L^2$ spaces to (locally) compact groups?
Motivating examples:
Let $V$ be a real vector space with Haar measure $dv$. The fourier transform induces the following topological isomorphism: $$H^s(V,dv) \cong L^2(V^*,(1+|v^*|^2)^sdv^*)$$
The ...
CommunityBot
- 1
6
votes
2
answers
469
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 ...
CommunityBot
- 1
1
vote
0
answers
237
views
On the bound of the Stein-Wainger oscillatory integral
Let $\lambda\in \mathbb{R}$, $\phi\in C^\infty(\mathbb{R})$. We define the Stein-Wainger oscillatory integral by
$$I=p.v.\int_\mathbb{R} e^{i\lambda\phi(t)}\frac{dt}{t}.$$
Stein-Wainger [1] showed ...
- 11
2
votes
1
answer
311
views
Differentiation on $[0,1]$
EDIT:
Perhaps a more reasonable question after thinking about the answer I got would have been.
Is there a set $N$ of measure $1-\varepsilon$ and a disjoint partition of that set $N$ with finitely ...
- 536
4
votes
0
answers
238
views
Does Novikov condition imply BMO martingale?
Let $(\Omega,\mathbb{F},P)$ be a complete probability space, equipped with a filtration $\mathcal{F}_t, 0 \le t < \infty$. Consider a continuous local martingale $(X_t, \mathcal{F}_t)$ such that $...
- 195
3
votes
0
answers
180
views
When is a minimal immersion holomorphic?
Let $(X,g_X)$ be a Riemann surface and $(Y,g_Y)$ a Kahler manifold. Let:
$\phi\colon X\to Y$
be a minimal immersion, that is, a conformal harmonic smooth map with respect to $g_X$ and $g_Y$. If I am ...
- 2,816
4
votes
0
answers
211
views
Inclusion of Hardy spaces
It is well-known that any convergence in $L^p$ for $p \in [1,\infty]$ implies convergence in $L^1_{\text{loc}}$ by Hölder's inequality.
It is also known that for $p>1$ it holds that $L^p(\mathbb R)...
11
votes
1
answer
691
views
Reference request: Fourier transform on the multiplicative group of real numbers
Let us consider the three groups $(\mathbb{R},+)$, $(\mathbb{Z}/2\mathbb{Z},+)$ and $(\mathbb{R}^\times,\cdot)$ (where $\mathbb{R}^\times := \mathbb{R} \setminus \{0\}$). We endow $\mathbb{R}$ with ...
- 31.5k
1
vote
1
answer
168
views
Sampling set: relatively dense and uniformly discrete
The Paley-Wiener space of a domain $\Omega\subset\mathbb{R}^d$ is the set $$PW_\Omega:=\{f\in L^2(\mathbb{R}^d):\text{supp}\widehat{f}\subset\Omega\}.$$
We say that a discrete set $\Lambda\subset\...
CommunityBot
- 1
4
votes
2
answers
807
views
Completion of $\mathcal{S}(\mathbb{R})$ for a given norm
Assume that $\lVert \cdot \rVert$ is a norm on the space of rapidly decaying functions $\mathcal{S}(\mathbb{R})$. Under which conditions on the norm can we say that the completion $\mathcal{X}$ for ...
- 3,085
2
votes
0
answers
70
views
Can the STFT decrease arbitrarily quickly near the origin?
For $f,g \in L^2(\mathbb{R}^d)$ we can define the Short Time Fourier Transform (STFT) $V_gf \in C_0(\mathbb{R}^{2d})$ as $$V_gf(x, \omega) = \int_{\mathbb{R}^d} f \overline{g(t - x)} e^{-2 \pi i t \...
- 1,010
5
votes
0
answers
119
views
Characterizing Herz-Schur multipliers using coefficient functions of uniformly bounded representations
Let $G$ be a group and let $c > 1$ be a constant. We denote by $B_c(G)$ the space of all coefficients of the representations of $G$ which are uniformly bounded by $c$; more precisely, a function $f:...
- 299
1
vote
0
answers
107
views
Laplacian on squashed spheres
Is anything known about the Laplacian on squashed spheres $S^{2n-1}_\omega$, where the ambient $C^n$ coordinates satisfy
$$ 1= \sum_{i=1}^n \omega_i |z_i|^2 $$
for fixed real numbers $\omega_i$? for ...
- 533
7
votes
1
answer
2k
views
Regularity of Fourier transforms of $L^p$ functions for $2<p\le\infty$
I was recently reading about the Mikhlin and Hörmander Multiplier Theorems, which give conditions for a measurable function $m:\mathbb R^d\to\mathbb C$ to be an $L^p$ multiplier, i.e. for there to ...
- 7,817
5
votes
1
answer
472
views
Embedding theorem for anisotropic Sobolev spaces
Let $d=d_1+d_2$, $s_1,s_2>0$, $p>1$ and $(x_1,x_2)\in \mathbb{R}^{d_1}\times \mathbb{R}^{d_2}$, $(\xi_1,\xi_2)\in \mathbb{R}^{d_1}\times \mathbb{R}^{d_2}$. Define
$$
W^{s_1,s_2}_{p}:=\left\{f: ...
CommunityBot
- 1
3
votes
0
answers
168
views
Zak transform and VMO
The Zak transform of a function $f\in L^1(\mathbb R)\cap L^2(\mathbb R)$ is defined as follows:
$$
Zf(x,\omega) := \sum_{k\in\mathbb Z}f(x+k)e^{-2\pi i k\omega},\quad (x,\omega)\in Q_0 :=(0,1)^2.
$$
...
- 63.9k
-1
votes
1
answer
153
views
$\ell^q$ analog of square function
It is a classical result in harmonic analysis that
$$ \|\|P_kf\|_{\ell^2_k}\|_{L^p_x}\approx\|f\|_{L^p} $$
for $p\in(1,\infty)$, where $P_k$ is the Littlewood-Paley decomposition onto frquency $\...
- 873
2
votes
1
answer
244
views
Smoothness of distributions defined by oscillation integrals
In M.A. Shubin's book Pseudodifferential Operators and Spectral Theory, we have the following statement.
Let $X\subset\mathbb{R}^n$ be an open set, and fix a symbol $a\in S_{\varrho,\delta}^m(X\...
- 1,247
2
votes
1
answer
358
views
Existence of an integrable representation
An irreducible continuous unitary representation $\pi$ of $G$ is said to be integrable, if the map $\phi(x)=\langle\pi(x)\zeta,\zeta\rangle$ is integrable on $G$, where that $\zeta\in H(\pi)$.
...
- 25.8k
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 $L^1(G)$?
CommunityBot
- 1
2
votes
1
answer
181
views
On a paper by Adams and Frazier
I am reading a paper by Adams and Frazier (namely Adams, Frazier, Composition operators on potential spaces. Proc. Amer. Math. Soc. 114 (1992), no. 1, 155–165, available here), whose main purpose is ...
- 3,146
3
votes
3
answers
768
views
Pointwise convergence of Fourier series, Fefferman's article
This is the first time I ask a question here, so sorry if I make any mistake in the way I ask it. I'm studing Fefferman's article Pointwise Convergence of Fourier Series, and I have two questions:
1)...
- 11
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}) ...
- 473
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
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 ...
- 17.2k
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 $\phi\in S(\mathbb{R}^n)$, ...
- 3,146
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
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 \...
CommunityBot
- 1
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 ...
CommunityBot
- 1
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 <...
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}...
- 24.2k
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 $$e_{\omega}(t) = f(t) - \int\...
- 698
3
votes
0
answers
126
views
An identity of operator norms and de Leeuw's theorem
Let $$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, $$ where $S(x,y)$ ...
- 187
7
votes
1
answer
489
views
When the value of a function in a point is equal to its integral average over the point's neighborhood?
It is well-known that the harmonic functions have this remarkable Averaging Property: if $f$ is harmonic in a domain $U \subset R^n$, then, for any point $x \in U$, $f(x)$ is equal to the integral ...
- 91
1
vote
0
answers
82
views
Topology of the algebra $\mathbb{C}\{A\}$ for a LCA group $A$
Let $\mathcal{A}$ be a complex associative Hausdorff topological algebra, and let $A\subset\mathcal{A}$ be a locally compact Abelian (LCA) subgroup (multiplicative). The linear span $\mathbb{C}\{A\}$ ...
- 1,959
1
vote
0
answers
113
views
Hardy $Hp$ norm of similar function
Let $f(z)=\sum_{n=0}^{\infty} \frac{c_n}{n+1}z^n$, where sequence $c_n \in S^1=\{z:|z|=1\}.$ We observe $H^p$ norm $\|f\|_{H_p}$, where $H^p$ is Hardy space, $1 \leq p < \infty$.
Question: For the ...
- 259
2
votes
0
answers
136
views
To find a positive function with compact spectrum
Let
$e_1=(0,1)^T$,
$$
S=\left\{x\in \mathbb{R}^2\Big| \frac{|\langle x, e_1\rangle|}{|x|}>\delta>0\right\},
$$
is a cone in $\mathbb{R}^2$.
I want to find a non-trivial smooth function ...
- 29
5
votes
0
answers
332
views
Harish-Chandra's submersive principle on closed subsets
Harish-Chandra's submersion principle says the following. Let $X,Y$ be two manifolds of dimensions $m$ and $n$ respectively. Let $\pi: X\rightarrow Y$ be a surjective smooth map which is submersive ...
- 960
2
votes
0
answers
132
views
Wiener-Ikehara Theorem and Signal Processing
I am trying to understand the Wiener-Ikehara Tauberian theorem which can be a step to understanding the prime number theorem. Let
$$ \hat{a}(s) = \int_0^\infty e^{-us}\, da(u) $$
with $a(u)$ some ...
- 13.9k