All Questions
3,841 questions with no upvoted or accepted answers
3
votes
0
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113
views
inifinite tensor product algebra representation
For a finite integer $N$, let $A_n = \bigotimes^n M_N(\mathbb{C})$. $A_n$ embeds in $A_{n+1}$. Let $A_\infty = \cup A_n$. Are the (complex) irreducible representations of $A_\infty$ known? It is ...
- 103
3
votes
0
answers
201
views
Degree of a function in $H^{\frac{1}{2}}(\mathbb{S}^1,\mathbb{S}^1)$
We can define the degree of a function $f \in H^{\frac{1}{2}}(\mathbb{S}^1,\mathbb{S}^1)$ as
$$\mathrm{deg} \hspace{1mm} f = \frac{1}{2\pi i} \int_{\mathbb{S}^1} f^{-1} \frac{\partial f}{\partial \...
- 31
3
votes
0
answers
329
views
Trace Norm Inequality for the Discrete Fourier Transform
I am having some trouble proving an inequality involving the trace norm or the operator $f{\cal{F}}_Ng$ where $f, g$ are diagonal matrices, f is positive semidefinite and $\cal{F}_N$ is the Discrete ...
- 141
3
votes
0
answers
860
views
decreasing rearrangements: why the asymmetry of measure-preserving maps?
Ryff proved in 1970 that the decreasing rearrangement $f^*$ of a, say, continuous function $f:[0,1]\to\mathbb{R}$ admits a measure preserving map $\phi$ such that $f=f^*\circ\phi$. In general it is ...
- 16.6k
3
votes
0
answers
177
views
A Fourier analysis-type identity arising in potential theory
I have recently been learning about potential theory in the complex plane, and I would like to understand why the following proposition is true:
Suppose $f: \mathbb C \to \mathbb R$ is smooth, ...
- 437
3
votes
0
answers
151
views
Is there a Fourier transform for principal r-discrete groupoids?
I have been looking for an analog of the Fourier transform for groupoids coming from tilings (which are generally principal and r-discrete), however all the generalizations I have found assume that ...
- 1,010
3
votes
0
answers
179
views
How to use Galerkin method to obtain existence with spaces $V \subset H$ not compactly embedded
With $V \subset H \subset V'$ a Hilbert triple (separable spaces as well), let's consider
$$u' + Au = f$$
in $L^2(0,T;V')$, where $A:V \to V'$ is bounded and linear. If $V \subset H$ is not compact, ...
- 41
3
votes
0
answers
304
views
Is the fundamental group of a maximal subfactor always $\mathbb{R}_{+}^{*}$?
The fundamental group $\mathcal{F}(N \subset M)$ of a unital inclusion of II$_{1}$ factors $N \subset M$ is defined as : $\mathcal{F}(N \subset M) =\{t >0 \ | \ (N \subset M)^{t} \simeq (N \...
3
votes
0
answers
180
views
Сonvergence of the sum
This is a problem from my exam on functional analysis. I did only trivial case and now I'm just curious about another cases.
Let $T : H \rightarrow H$ is a linear continuous unitary ($T^*=T^{-1}$) ...
- 31
3
votes
0
answers
168
views
Deleting "weak homeomorphism" in a Hilbert space
It is well-known that there exists a homeomorphism $h$
from an infinite-dimensional Hilbert space $H$ to $H\setminus\{0\}$.
Does there exist a "weak homeomorphism" $g:H \to H\setminus\{0\}$,
that is, $...
- 91
3
votes
0
answers
660
views
Fourier : from analyticity to exponential decay ; what prevents optimal decay ?
Hello everyone !
In Fourier Analysis, Self-Adjointness by Reed and Simon (Methods of Modern Mathematical Physics, Vol. 2), Theorem IX.14 tells us that (I'll take dimension 1 for simplicity) :
if $...
- 319
3
votes
0
answers
172
views
Shift-invariant submultiplicative seminorms of $\ell^{\infty}$
Question: Is there a shift-invariant submultiplicative seminorm $||\cdot||$ of $\ell^\infty$ which satisfies the following property?
If $f:\mathbb{N}\rightarrow\mathbb{N}$ is an increasing function ...
- 31
3
votes
0
answers
84
views
Application and relevance of Sobolev gradients
The Sobolev gradient concept has been developed in the 1970s, with a first publication in 1985, and an introduction can be found at: Ranka
I would like to learn how strong the impact of Sobolev ...
- 5,327
3
votes
0
answers
301
views
What information about a locally compact group $G$ is encoded in $C_r^\ast(G)$ which is not in $L^1(G)$?
Let $G$ be a locally compact group and let $ C_r^\ast(G) $ denote its reduced group $C^\ast$-algebra. Many features of a $G$ can be realized from $L^1(G)$ or $C_r^\ast(G)$. For example, $G$ is ...
user23860
3
votes
0
answers
176
views
Extending a Hilbert space isometrically
Let $H$ be a Hilbert space, and let $X$ be a topological vector space.
Under what conditions on the topologies of $X$ and $H$ does there exist an injective, continuous linear map $f : H \to X$?
...
- 8,512
3
votes
0
answers
382
views
Extension divergence-free, curl-converging vector field
Hi.
Consider a smooth open Set $\Omega\subset\mathbb{R}^3$ and a bounded sequence of vector fields $(u_n)_n \in L^2(\Omega)$ having $0$ divergence. I know how to extend this sequence to the whole ...
- 3,425
3
votes
0
answers
145
views
Growth of inner functions on the disk
Recall that an inner function on the disk $D$ is a bounded analytic function on $D$ having radial limits of modulus one almost everywhere.
There has been many works on the growth of the inner ...
- 769
3
votes
0
answers
183
views
Is the construction of ring C*-algebra functorial?
Cunz and Li defined defined C*-algebras for arbitrary rings with of course some condition. One can look at their article (http://arxiv.org/abs/0905.4861). My question: is the construction functorial? ...
- 95
3
votes
0
answers
642
views
question about bidual normed space
Consider a Banach $\mathbb{R}$-space E and an element $u\in E''$ such that :
for all sequence $\phi_n\in E'$ which $\sigma(E',E)$-converges to $\phi \in E'$, one has $\lim u(\phi_n)=u(\phi)$.
Is it ...
3
votes
0
answers
185
views
spectrum of a polygon and zeta function
Let $\Delta(x,y) = 1,0$ according to whether $(x,y)$ is in some polygon (symmetric with respect to the diagonal axis).
E.g. The convex hull of three points (taken from a paper on dominoes)
$$ \...
- 22.8k
3
votes
0
answers
163
views
Isometric automorphism of $c_0$ different than coordinate permutation
Does there exist an isometric automorphism of $c_0$ which is not a permutation of coordinates?
- 311
3
votes
0
answers
269
views
Continuous selection given both upper and lower hemicontinuity
Suppose that $X,Y$ are compact metric spaces, and $g:X\rightarrow Y$ is a multivalued mapping that is both upper and lower hemicontinuous. Is there a single valued continuous selection of $g$? If it ...
- 203
3
votes
0
answers
209
views
A maximum of a function
When studying the $\text{UMD}$ constants of spaces like $L_{p_1}(L_{p_2}(\cdots (L_{p_n})\cdots))$, I encounter the following question: Let $\alpha > 0$, define $$C(\alpha) : = \sup_{a > 0, b>...
- 769
3
votes
0
answers
396
views
Norm estimate for Moore-Penrose pseudo-inverse of $i^\ast T i$
Let $G$ and $H$ be Hilbert spaces, let $i : G \rightarrow H$ be an isometric inclusion (so $G$ is a subspace of $H$) and let $T : H \rightarrow H$ be a bounded linear operator with closed range.
That ...
- 5,327
3
votes
0
answers
302
views
Dense subalgebras of von Neumann algebras and increasing nets
[Question previously asked on Math.SE]
Let $N$ be a von Neumann algebra, and $A$ be a dense $∗$-subalgebra of $N$ (in the ultraweak topology) with $A''=N$. Is it true that:
For any $x∈N^+$, there ...
- 33
3
votes
0
answers
254
views
Ways to establish equality of measures on locally compact spaces
Let $M$ be a locally compact space and $\mu$ be some probability measure on $M$. Let $y^\ast \in M$, $f(x,y)$ be a real continuous bounded function $M \times M \to \mathbb{R}$. Consider an equality
$$
...
- 1,329
3
votes
0
answers
409
views
Continuous function sort
If you have a real-valued function f(x), positive, continuous and bounded on some interval, then what kind of transform would convert this to a monotonic function g(x) on that interval analogously to ...
- 529
3
votes
0
answers
217
views
Is this integral operator about Stokes' Flow compact?
Consider the following integral operator $\mathcal{A}$ on [EDITED: continuous vector function $f=(f_i):\partial S\to{\mathbb R}^3$]:
$$
({\mathcal A}f)_j(x_0):=\int_{\partial S}\sum_{i=1}^3 f_i(x)G_{...
user14319
3
votes
0
answers
188
views
Does the existence of an asymtpotic density imply the existence of a measure on infinite dimensional (path) space?
This question is related to the following question
Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?
A couple of authors have observed that composing a ...
- 265
3
votes
0
answers
415
views
Pointwise bounds for Dirichlet kernel over truncated lattice
In 1 dimension, a "one-sided" Dirichlet kernel $D_N(x)=\sum_{k=0}^{k=N}e^{\frac{2\pi}{N}ikx}$ has its module sharply peaked around points corresponding, roughly, to the "dual lattice" $N\cdot\mathbb{Z}...
- 3,323
3
votes
0
answers
456
views
Morphism of von Neumann Algebras
Hello,
Is there a counterexample to the following statement:
let $A,B$ two von Neumann algebras, every morphism $A \rightarrow B$ of $C^* $-algebras is a $W^*$-homomorphism ?
( a $W^* $-...
- 663
3
votes
0
answers
498
views
PDES - from Vector fields whose inner product with their vector Laplacian equals norm of the vector field
Let $g(x_{1},........,x_{n}) = \sum_{i=1}^{n}g_{i}(x_{1},\cdots,x_{n})e_{i}$ be a function in $\mathbb{C}^n$ ($e_{i}$ are the standard bases).
Let $\nabla^{2}$ be the vector Laplacian. Let $<\cdot,...
- 800
3
votes
1
answer
1k
views
Borel-Cantelli lemma for general measure spaces (those with infinite measure)
The Borel-Cantelli lemma is often stated for a probability space or spaces with finite measure.
But it seems to me that it still holds if the space $X$ is of infinite measure. I seem to be able to ...
- 31
3
votes
0
answers
385
views
Off-diagonal asymptotic expansion of the Bergman kernel on hyperbolic Riemann surfaces
Let $X$ be a compact Riemann surface of genus at least 2. Let $K$ denote the canonical line bundle, and $E$ be any vector bundle. Let $P^{(m)}$ be the projection map from the space $L^2(X,K^mE)$ of ...
user2529
3
votes
0
answers
637
views
Fixed point theorem for convex, closed multivalued mapping
There is well-known fixed point theorem theorem for multivalued l.s.c. maps, based on Michael selection theorem:
Suppose, that $X$ is compact, convex and metrizable in locally convex Hausdorff ...
- 696
3
votes
0
answers
251
views
What is the origin of the metrization problem for compact convex sets?
The following is an ``old question in analysis:''
Is it true that every perfectly normal compact convex subset of a locally convex topological vector space is metrizable?
Here perfectly normal means ...
- 3,547
3
votes
0
answers
356
views
Stability of convex sets w.r.t. integration over [0,1]
In the preprint on pages 19−20, first using Hahn−Banach, one proves
Lemma 38. For any closed convex set $U$ in a real Hausdorff locally convex space $E$ and for any Riemann integrable $\gamma:[0,1]\...
- 3,584
3
votes
0
answers
223
views
Extension of positive operators and Bauer-Namioka
When $X$ is a vector subspace of an ordered vector space $A$, any positive linear functional $f: X \to R$ extends to all of $A$ as a positive linear functional provided one can find a nonvoid, ...
- 31
3
votes
0
answers
1k
views
weak regularity conditions for regions to assure boundary of measure zero
Let $\Omega \subset \mathbb{R}^d$ be a region ( bounded, simply connected, open set ). What are some regularity conditions to assure the boundary $\partial\Omega$ is a set of (lebesgue-)measure zero? ...
- 131
3
votes
0
answers
130
views
Positive block matrices over tensor algebras
Let $A$ be a unital C*-algebra. A positive block matrix in $M_2(A)$ must have the form
$$ \begin{pmatrix} a & a^{1/2} x b^{1/2} \\ b^{1/2} x^* a^{1/2} & b \end{pmatrix}, $$
where $a,b$ are ...
- 18.7k
3
votes
1
answer
355
views
convex function with distributional Hessian $D^2 f \leqslant \lambda$, $\lambda$-concave?
Let $f:R^n \to R$ be convex (may not $C^1$), $$[D^2f]=[D^2f]_{ac}+[D^2f]_s=[h_{ij}] L^n+[D^2f]_s$$ is the Lebesgue decomposition of the Hessian matrix. Where $[h_{ij}]$ is the density w.r.t the ...
- 471
3
votes
1
answer
902
views
Is the integral always nonzero?
Let
$$I_{n,p,a}:=\int_0^{\infty-} \frac{g_{n,a}(t)}{t^{p+1}} \, dt,$$
where
$$(*)\qquad\qquad\qquad n\in\mathbb N,\quad -\infty<a<\infty,\quad p_{n,a} < p < n,\qquad\qquad\qquad\qquad\...
- 128k
3
votes
1
answer
310
views
Do cyclic product vectors generatating irreducible representation of a Lie group come from a unique orbit?
Consider a Hilbert space $\mathcal{H}$ which is a carrier space of a unitary, irreducible and strongly continuous representation $\Pi$ of a Lie group $G$. Let $\Pi\otimes \Pi$ denote the corresponding ...
- 965
3
votes
1
answer
342
views
When does $\lVert T_n \rVert_p \to \lVert T_0 \rVert_p$ imply $\lVert T_n - T_0 \rVert_p \to 0$?
maybe this question is too elementary but I could not find any results in the functional analysis textbooks I own:
For separable Hilbert spaces $H$, $K$, let $S_p(H,K)$ be the $p$th Schatten class of ...
- 617
3
votes
1
answer
791
views
Real part of eigenvalues and Laplacian
I am working on imaging and I am a bit puzzled by the behaviour of this matrix:
$$A:=\left(
\begin{array}{cccccc}
1 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 &...
user136033
2
votes
0
answers
120
views
Uncertainty principle: minimize $\int_{-\infty}^\infty |t| |\widehat{f}(t)|^2 dt$ for $f$ of compact support
This is a question of uncertainty-principle type stemming from Eigenvalue of a convolution and a restriction?
Let $f:\mathbb{R}\to \mathbb{R}$ be even, absolutely continuous and supported in $[-\frac{...
- 20.2k
2
votes
0
answers
82
views
Function that is (essentially) a self-convolution but not a multiple of a self-convolution
Call a function $F:\mathbb{R}\to C$ nice if it is of the form $F = f\ast \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$. (Of course nice functions are precisely those whose Fourier transform is ...
- 20.2k
2
votes
0
answers
230
views
A deceptively simple regularity problem for functions on the plane
By various meanderings and toying with simpler problems, my current research has lead me to the following quite straightforward question, which I am wholly unable to answer:
Consider a twice ...
- 989
2
votes
0
answers
118
views
Error in discrete FFT
I am interested in taking an FFT of an image which is periodic in space (does not decay) across a finite window of size $L\times L$. The image has triangular symmetry; for simplicity one could imagine ...
- 21
2
votes
0
answers
69
views
Bessel spaces and Triebel Lizorkin
It is known that bessel potential spaces $H^{s,p}$ coincide with Triebel-Lizorkin spaces $F^{s}_{p,2}$ for $s\in \mathbb{R}$ and $1<p<\infty$. Im wondering what can be said por $p=1$ and $p=\...
