All Questions
10,936 questions
4
votes
2
answers
340
views
A space isometric to $\ell_\infty^2$
Consider a norm on $\mathbb C^2$ as $\|(z_1,z_2)\|:=\max\{|z_1|,|z_2|,\frac{1}{\sqrt{2}}|z_1+iz_2|\}.$
Question. Is $(\mathbb C^2,\|.\|)$ linearly isometric to $(\mathbb C^2,\|.\|_{\infty})$ where $\|(...
- 1,879
0
votes
1
answer
294
views
Does $L_{p}(\mu, X)^*=L_{q} (\mu, X^*)$ hold for $\sigma$-finite measure spaces?
I'm reading Theorem 1 at page 98 of Vector Measures by Joseph Diestel, John Jerry Uhl.
THEOREM 1. Let $(\Omega, \Sigma, \mu)$ be a finite measure space, $1 \leq p<\infty$, and $X$ be a Banach ...
- 825
2
votes
0
answers
57
views
Regularity on $\mathbb{T}^3$ of the "functional average" of a map $S : C^\infty(\mathbb{T}^3, \mathbb{R}) \to L^2(\mathbb{T}^3, \mathbb{R})$
For simplicity, let $C^\infty(\mathbb{T}^3, \mathbb{R})$ be the real Frechet space of periodic smooth functions on $\mathbb{R}^3$. Here, $\mathbb{T}^3$ is the $3$-dimensional torus.
For a fixed ...
- 3,477
15
votes
2
answers
1k
views
Analytical origins of the Stone duality
I've asked this question in the HSM community, but by the nature of my question, some user told me to ask this question here.
This is the original post https://hsm.stackexchange.com/q/13087/14296
...
- 281
0
votes
0
answers
175
views
Does l2 projection of sequences preserve l1 norm convergence?
Let $\ell^2$ denote the set of square summable sequences with inner product $\langle x,y\rangle=\sum_{i=1}^{n}x(i)y(i)$ and $\ell^2$ norm $\|x\|_2=\sqrt{\langle x,x\rangle}$. Let $\|x\|_1=\sum_{i=1}^{\...
- 45
0
votes
0
answers
63
views
Inequality for normed power n, m
Let $ B (H) $ indicate the set of all bounded linear operators on a complex separable Hilbert space $ H $.
Let $ A \in B(H) $, where $ A $ is a positive semi-definite operator in $ H $ (i.e. $ \langle ...
5
votes
1
answer
212
views
Two dimensional oscillatory integral
I am having a little confusion in verifying the two dimensional oscillatory integral in Lemma 2.1 in This paper, namely
$$I_t (x,y) = \int_{\mathbb{R}^2} |\xi|^{\epsilon + i \beta} e^{i t(\xi^3 + \xi \...
- 159
3
votes
1
answer
181
views
General form of bounded linear functionals on Banach spaces
It is a famous result due to Riesz that every bounded linear functional $f$ on a Hilbert space $\mathcal{H}$ is of the form $f(x)=\langle x,z \rangle$ for a unique $z\in H$.
On p.188 of Introductory ...
- 709
4
votes
0
answers
152
views
Maximally fine topologies on $B(H)$ making the unit ball compact
Let $H$ be a Hilbert space, and $B(H)$ its algebra of bounded operators. One of the reasons the Ultraweak topology is (in a way) more useful than the weak operator topology is that the Ultraweak ...
- 190
12
votes
5
answers
1k
views
Examples of metric spaces with measurable midpoints
Given a (separable complete) metric space $X=(X,d)$, let us say $X$ has the measurable (resp. continuous) midpoint property if there exists a measurable (resp. continuous) mapping $m:X \times X \to X$ ...
- 6,853
0
votes
0
answers
113
views
Is this formula for 2D Fourier integral of diffraction kernel correct?
Well I have a function parametrized by $z$
$$g_z(x,y) = \frac{z}{i \lambda r^2} e^{i k r}, \quad r = \sqrt{x^2+y^2+z^2},$$
where $\lambda > 0$ is real constant and $k = \frac{2\pi}{\lambda}$. This ...
- 151
5
votes
0
answers
167
views
Computing sums with linear conditions quickly
Let $f:\{1,\dotsc,N\}\to \mathbb{C}$, $\beta:\{1,\dotsc,N\}\to [0,1]$ be given by tables (or, what is basically the same, assume their values can be computed in constant time). For $0\leq \gamma_0\leq ...
- 20.2k
1
vote
0
answers
216
views
Is there an approximate formula for this summation function?
Consider the function $$\sum_{n=1}^\infty \frac{\cos(nx)}{n^r},$$ where $r\in\mathbb{N}$. Is there any approximate formula (closed form possibly avoiding this type of summation) for this function? I ...
- 59
4
votes
2
answers
298
views
Takesaki volume II chapter VII lemma 1.15
Consider the following fragments from Takesaki's second volume "Theory of operator algebras" in chapter VII on weight theory, lemma 1.15. Here $\mathcal{M}$ is a von Neumann algebra and $\...
- 175
1
vote
0
answers
136
views
Banach spaces in which every DP-set is a limited set
Let $X$ be a Banach space and $A\subseteq X$ be a bounded subset.
$A$ is a Dunford-Pettis set if every weakly null sequence $(f_n)$ in $X^*$ converges to $0$ uniformly on $A$, that is $$ \lim_{n\to\...
- 2,605
5
votes
1
answer
357
views
Are the polyhedral cones the only examples of cones that remains closed when they are added to vector subspaces?
Let $C \subset \mathbb{R}^{n}$ be a closed convex cone. If one wants to know whether the linear map $T:\mathbb{R}^{n} \to\mathbb{R}^m$ sends the closed set $C$ to another closed one, $T(C)$, it is ...
- 225
0
votes
0
answers
92
views
Proof of the isomorphism of the Toeplitz algebra and the algebra generated by the element and the relation
Please tell me where can I see the proof of this well-known fact?
enter image description here
1
vote
1
answer
230
views
Why we have $f=0$
Define the Fourier transform for a suitable function $f\in L^1(\Bbb R)$ by $\widehat{f}(\xi)=\int_{\Bbb R}f(x)e^{-ix\xi} dx$.
Assume the condition $$\int_{\Bbb R}\int_{\Bbb R}|\widehat{f}(\xi)f(x)|^...
- 531
1
vote
0
answers
94
views
Convex combination of positive mean-ergodic operators
Let $T_1,T_2:L^1([0,1],\mathrm{d}x)\to L^1([0,1],\mathrm{d}x)$ be positive mean-ergodic operators such that:
For every $h:[0,1]\to \mathbb{R}_+$ we have that
$$\int_0^1 T_1 h(x)\mathrm{d}x = \int_0^1 ...
- 333
8
votes
3
answers
3k
views
Bounding supremum norm of Lipschitz function by L1 norm
Consider $f:[0,1]^d \to \mathbb{R}$. Suppose that $f$ is $L$-Lipschitz w.r.t. the Euclidean norm. Can we provide an upper bound on $\|f\|_\infty$ in terms of $\|f\|_1 := \int_{[0,1]^d} |f(x)|dx$ ?
In ...
- 301
2
votes
0
answers
67
views
Künneth formula and continuity of the isomorphism
In the book Sheaf Theory, by Bredon (edition from 1997), Theorem 14.1, he writes a natural exact sequence, which, in some nice cases, leads to the Künneth formula. Do we have any reason to believe ...
12
votes
3
answers
2k
views
Looking for sufficient conditions for positive Fourier transforms
I am looking for some sufficient conditions for an even, continuous, nonnegative, non-increasing, non-convex function to be non-negative definite. In other words
$$
\int_0^\infty f(x)\cos(x\omega) \, ...
- 178
1
vote
0
answers
104
views
Localize functions in the Hardy space $\mathcal H^1(\mathbb R^n)$
Let $f$ belong to the Hardy space $\mathcal H^1(\mathbb R^n)$, $B\subset \mathbb R^n$ be the unit ball. Does there exist a $\bar f\in \mathcal H^1(\mathbb R^n)$ with compact support such that $\bar f=...
- 435
5
votes
0
answers
204
views
Given a totally ordered system of Banach spaces, can we we always change the norms to get isometric embeddings?
Given a real vector space $V$ which is the union of a totally ordered family of vector subspaces $V=\bigcup_{i\in I} V_i$. By that I mean that we assume that $(I,\leq)$ is a totally ordered set and ...
- 609
1
vote
1
answer
404
views
Is Schwartz space $\mathbb R^n$ contained in every fractional Sobolev space on $\mathbb R^n$? [closed]
Can someone kindly confirm that the Schwartz space $\mathcal S(\mathbb R^n)$ made of all infinitely-differentiable functions $f:\mathbb R^n \to \mathbb R$ with rapidly decreasing derivatives of all ...
- 6,853
11
votes
1
answer
339
views
What is an example of two Banach spaces $X,Y$ such that $X$ embeds isometrically but not linearly into $Y$?
By a result of Godefroy and Kalton if $X,Y$ are separable Banach spaces and $X$ embeds isometrically into $Y$, then $X$ embeds with a linear isometry into $Y$.
Is this result known to fail for ...
1
vote
1
answer
89
views
Does an isometric automorphism of $L_p (X,\mu, E)$ preserve pointwise convergence?
Let $(X, \mathcal A, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Here we use the Bochner integral. Let $p \in [1, \infty)$ and $q \in (1, \infty]$ such that $p^{-1}+q^{-...
- 825
4
votes
2
answers
257
views
A functional inequality which calculates the limitation of human eyes
Find all pair of function $f^-,f^+:[0,1]\rightarrow[0,1]$ such that:
(1)$f^-(x)\leq x\leq f^+(x)$.
(2)$f^-(x)+f^+(1-x)=1$.
(3)$f^-(x)f^-(y)\leq f^-(xy)\leq f^-(x)f^+(y)$.
(4)$f^+(x)f^-(y)\leq f^+(xy)\...
- 1,462
1
vote
0
answers
133
views
Does the Gaussian Poincare inequality hold for infinite dimensional measure metric spaces?
This is a question subsequent to the one:
Does the Gaussian Poincare inequality hold for $p=1$ as well as $p=2$?
There, I received a very helpful answer that the Gaussian poincare inequality for any ...
- 3,477
0
votes
0
answers
104
views
Can the best constants in harmonic analysis be approximated in principle?
Consider the trivial example of Holder's inequality $\|f\|_p\,\|g\|_q\geq |fg|_1$ if $\frac{1}{p}+\frac{1}{q}=1, p,q\geq 1$ and $f,g$ are functions on $\mathbb{R}^n$. Let's suppose we don't know how ...
- 265
3
votes
1
answer
162
views
Approximation in Bochner spaces
Is there any result like the Bramble-Hilbert lemma for Bochner spaces?
More specifically: let $H$ be a (e.g.) Hilbert space, $I\subset \mathbb R$ a bounded interval, and $L \in \mathcal L(H^k(I;H), Y)...
- 235
0
votes
0
answers
48
views
A direct proof for non-zero limit points of weighted backward shifts
Fix a sequence $(w_1,w_2,\ldots)$ of positive reals such that the linear operator $T: \ell_2\to \ell_2$ given by
$$
T(x_1,x_2,x_3,....)=(w_2x_2,w_3x_3,\ldots) \text{ for all sequences in } \ell_2
$$
...
- 1,511
3
votes
0
answers
143
views
Extrapolated Integral operator (compactness)
I am studying the compactness of some convolution operators. Let the convolution with extrapolation
$$ \Gamma: X\longrightarrow X; x\mapsto\int_0^t T_{-1}(t-s)B(s)x\mathrm{d}s. $$
Here $T(\cdot)$ is a ...
- 299
22
votes
5
answers
1k
views
Rigorous justification for this formal solution to $f(x+1)+f(x)=g(x)$
Let $g\in C(\Bbb R)$ be given, we want to find a solution $f\in C(\Bbb R)$ of the equation
$$
f(x+1) + f(x) = g(x).
$$
We may rewrite the equation using the right-shift operator $(Tf)(x) = f(x+1)$...
- 1,245
6
votes
1
answer
177
views
Properties of non-integer powers of the Hodge Laplacian
Consider a complete smooth Riemannian manifold $(M,g)$.
I think that it is not difficult to prove that the $k$ Hodge Laplacian is essentially selfadjoint in the relevant $L^2$ space of $k$ forms, ...
- 205
32
votes
19
answers
23k
views
Good books on theory of distributions
Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.
0
votes
1
answer
120
views
A property of the canonical dual frame in a Hilbert space
Let $\{ g_n \} $ be a frame in a separable Hilbert space $H$. Then the frame operator $S:H\to H$ defined as
\begin{equation} S f := \sum_{n=1}^\infty (f,g_n)g_n \end{equation}
is a Hilbert space ...
1
vote
1
answer
173
views
Taut string algorithm and TV-minimization equivalence
Given real numbers $y_i's$, consider the following convex optimization problem:
$$
\min_{x_i's} \sum_{i=1}^N(y_i-x_i)^2 + \lambda\sum_{i=1}^{N-1}|x_{i+1}-x_{i}|.
$$
The paper A Direct Algorithm for 1D ...
- 1,216
14
votes
1
answer
830
views
Spectrum of matrix involving quantum harmonic oscillator
The quantum harmonic oscillator relies on two classical objects, the so-called creation and annihilation operator
$$a ^* = x- \partial_x \text{ and }a = x+\partial_x.$$
Fix two numbers $\alpha,\beta \...
- 192
5
votes
0
answers
137
views
A list of properties of $(\bigoplus \ell^1_n)_{\ell^p}$, $1<p<\infty$
The Banach space $E=(\bigoplus_{n=1}^{\infty}\ell^1_n)_{\ell^p}$ for $1<p<\infty$ shows up in various places in the literature to construct counterexamples. The purpose of this post is to ...
- 2,605
1
vote
0
answers
95
views
Are the sum and product of nonlinear compact operators compact?
In the 'interactive' proof of Lomonosov's Theorem about hyperinvariant subspaces (in the book Hilbert Space Operators, A Problem Solving Approach), one is asked to prove the compactness of the ...
- 11
2
votes
1
answer
332
views
Are there "pathological convex sets" over ultravalued fields of char 2?
In their book Topological Vector Spaces (2nd ed.) Lawrence Narici and Edward Beckenstein generalise convex sets for TVS over ultravalued field $K$ as $K$-convex sets. The definition goes as following:...
- 519
2
votes
1
answer
149
views
On a core for Neumann Laplacian on $C(\overline{D})$
Let $D \subset \mathbb{R}^d$ be a bounded $C^1$ domain. We consider a reflected Brownian motion $X=(\{X_t\}_{t \ge 0},\{P_x\}_{x \in \overline{D}})$ on $\overline{D}$. Let $\{p_t\}_{t>0}$ denote ...
- 721
2
votes
1
answer
76
views
The eigenvalues of the product $WD$ for some particular matrices
Let $D$ be a diagonal matrix in $M_{2n}(\mathbb{R})$ such that $D^2=I$ and Trace$(D)$=0
Suppose that $e_k$s are the standard vectors in $\mathbb{R}^{2n}$, that is
$$e_k=(0,\cdots 0,1,0,\cdots,0)^t$$...
- 4,058
5
votes
1
answer
421
views
Ring of continuous functions is a Jacobson ring
Let $X$ be an infinite discrete topological space. Is $$C_b(X)=\{ f \colon X \to \mathbb{R} \text{ bounded }\}$$ a Jacobson ring ?
0
votes
0
answers
145
views
A possible generalization of Pitt's theorem
Inspired by Pitt's theorem and this post we ask the following question:
First we remind the Pitt's theorem and also introduce a particular Banach space as averaging of $\ell^p$ spaces for $1\leq p ...
- 366
5
votes
0
answers
236
views
Proof of density of smooth functions in $H(\operatorname{curl})$ with $L^2$ tangential trace
I am trying to understand the proof of the following statement that is presented in the book “Finite Element Methods for Maxwell's Equations” by Peter Monk. The original source of the proof is a 1997 ...
1
vote
0
answers
89
views
Let $T\in B(H)$. Then prove that $T$ is a contraction if and only if the closed unit disc is spectral set for $T$
Let's first recall the definition of Spectral set for a bounded operator $T$ on a hilbert space $H$. Let $X\subseteq \Bbb{C}$ be compact such that $\sigma(T)\subseteq X$. Define $$\mathcal{R}(X):=\...
- 111
2
votes
0
answers
62
views
On a real smooth version of white noise distribution theory
In white noise analysis, one starts with a real Gelfand triple $\mathcal{N}\subset \mathcal{H} \subset \mathcal{N}^{*}$ and produces out of it, using complexifications along the way, the complex ...
- 557
4
votes
1
answer
245
views
How to unperiodise a function
We know that given a sufficiently regular function $f: \mathbb{R} \to \mathbb{R}$, then its periodisation (say to period $1$) is given by
$$
\begin{align}
F(x) := \sum_{n\in\mathbb{Z}} f(x + n).\tag{$...
- 595