All Questions
10,233 questions
2
votes
1
answer
151
views
Distance between convex hulls in a bounded closed convex set
Let $X$ be an infinite-dimensional Banach space and $C\subseteq X$ be a bounded closed convex subset. Let $\{z_i\}_{i\in\mathbb{N}}$ be a sequence of linearly independent points in $C$ and for each $n\...
- 307
4
votes
1
answer
356
views
How Does One Solve the Functional Root of $xe^x$
Given the function $xe^x$ is there a way to solve the functional root, meaning solve for the function that satisfies the equation $f(f(x))=xe^x.$ I know that there may not be one unique solution that ...
4
votes
0
answers
146
views
Infinite dimensional homology theory for submanifolds of Hilbert and Banach spaces
Is there a version of homology theory for spaces for which explicitly infinite dimensional "cells" are allowed?
The spaces in question include e.g.
\begin{equation}
X = (x: x \in l_2: p_i(x) ...
- 593
4
votes
2
answers
781
views
Is there any bilinear Poincaré/Sobolev inequality?
Is the following, I call it bilinear Poincaré inequality, true?
Let $\Omega$ be an open bounded set in $\mathbf R^n\DeclareMathOperator{\dL}{d\!}$. There exists $C > 0$ such that for any $u, v \in ...
- 185
8
votes
2
answers
644
views
Uniqueness of the uniform distribution on hypersphere
I'm looking for a uniqueness-type result for the following problem, which is related to the uniform distribution in the hypersphere $\mathbb{S}^{p-1}$. Suppose $f$ is a sufficiently smooth function on ...
- 81
1
vote
0
answers
144
views
Estimator for the conditional expectation operator with convergence rate in operator norm
Let $X$ and $Z$ be two random variables defined on the same probability space, taking values in euclidian spaces $E_X$ and $E_Z$, with distributions $\pi$ and $\nu$, respectively.
Let $L^2(\pi)$ ...
- 111
1
vote
0
answers
126
views
Artin approximation for Banach rings
Let $B$ be an integral Banach ring, meaning it is complete with respect to a norm and it is an integral domain. Let $F$ be its fraction field. Let $\widehat{F}$ be the completion of $F$ with respect ...
- 39
3
votes
0
answers
131
views
Do all nuclear Frechet spaces have Schauder basis?
When I read an article recently, I see this statement as a conjecture.
So I want to ask, do all nuclear Frechet space have Schauder basis?
Is this still open? Or has been solved? I’ll appreciate any ...
- 61
6
votes
0
answers
187
views
Gaussian lower heat kernel bounds on non-convex bounded domain
I am looking for a proof the following theorem.
Let $U \subset \mathbb{R}^n$ be a bounded domain with $C^2$ boundary and $p(x,y,t)$ be the Neumann heat kernel. Then there exist a constant $C>0$ ...
- 61
3
votes
0
answers
60
views
Noncommutative maximal weak $L_1$ norms with respect to sub algebra
Let $(\mathcal M,\tau)$ be a von Neumann algebra with normal finite faithful trace $\tau.$ For any sequence $(x_n)_{n\geq 1}\in \mathcal M$ define $\|(x_n)\|_{\Lambda_{1,\infty}(\mathcal M;\ell_\infty)...
- 1,879
2
votes
1
answer
145
views
Orthonormal bases in RKHSs via interpolating sequences
Definitions and setting
Let $\mathcal{H}$ be a separable, infinite-dimensional, reproducing kernel Hilbert space on a nonempty set $X$. As usual, denote the reproducing kernel on $\mathcal{H}$ by $K$ ...
- 5,405
1
vote
0
answers
73
views
Is $L^p_\text{loc} (Y)$ dense in $(L^0(Y), \hat \rho)$?
Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(Y, \...
- 657
1
vote
0
answers
50
views
Is there $r>0$ such that the norm $[f]_r:=\sup_{n \ge 1} \|1_{B(y_n, r)} f\|_{L^p}$ is equivalent to $\|f \|:=\sup_{y \in Y}\|1_{B(y,1)}f\|_{L^p}$?
Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(E, |\...
- 657
0
votes
0
answers
47
views
Is the embedding $i: (L^p_\text{loc} (Y), \| \cdot \|_{L^p_\text{loc}}) \to (L^0(Y), \hat \rho)$ continuous or Borel measurable?
Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(Y, \...
- 657
3
votes
2
answers
102
views
Reference for Wiener type measure on $C(T)$ when $T$ is open
I'm considering Gaussian process on open domain $T$ in $\mathbb{R}^n$ and I tried to follow the abstract Wiener space construction of Gross. Since my sample paths are meant to be continuous, I thought ...
- 163
0
votes
0
answers
113
views
The set of measurable functions together with convergence in measure is a completely metrizable abelian topological group
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the space of $\mu$-simple ...
- 657
2
votes
0
answers
160
views
Understanding the Bochner space $W^{1,q}\bigl([0,T], L^p(U) \bigr)$ in terms of the Fréchet derivative
In the context of linear parabolic equations, the Sobolev space $W^{1,q}\bigl([0,T], L^p(U) \bigr)$ appears all the time. Here, $U$ is some bounded region of $\mathbb{R}^n$ and $1<p,q<\infty$.
...
- 3,477
1
vote
1
answer
265
views
Is there a version of dominated convergence theorem for local $L^p$ spaces?
Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let $\tilde L^p (\mathbb R^d)$ be the space of ...
- 825
2
votes
0
answers
58
views
Extension of averaged nonexpansiveness for mappings that are not self maps
Let $\mathcal{H}$ be a Hilbert space and let $\alpha \in (0,1)$. We say that an operator $f:\mathcal{H} \rightarrow \mathcal{H}$ is
Nonexpansive if $\|f(x)-f(y)\|_{\mathcal{H}} \le \|x - y\|_{\...
- 21
3
votes
1
answer
293
views
How to find the inverse of a product of two integral equations
Problem
I am trying to invert an equation of the form:
$R(l_0)=(\int_{0}^{l_0} \rho(x) \, dx)(\int_{l_0}^{l} \rho(x) \, dx)$
where $0\leq l_0 \leq l$
I.e. I want to find $\rho(x)$ given $R(l_0)$ via ...
- 33
0
votes
0
answers
96
views
Closed formula for Laguerre
Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$.
Assume $0<\beta<1$. Is there a closed formula for this sum
$$\sum^\infty_{j=0} \frac{1}{(b+j)^{1-\beta}}L^{m}_j(x)$$
where $b>0$ and $...
- 531
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
0
votes
1
answer
118
views
For any smooth function $f$ on $[0,1]$, do we have $\lVert P_N f \lVert_{1} \leq \lVert f \rVert_1$ for $P_N$ defined by Fourier expansion?
Let $C^\infty[0,1]$ be the space of periodic smooth functions on $\mathbb{R}$ with the period $1$. It is well-known to be a Frechet space with the uniform convergence of all derivatives.
Also, $\{ e^{...
- 3,477
20
votes
7
answers
5k
views
Why do infinite-dimensional vector spaces usually have additional structure?
On Mathematics Stack Exchange, I asked the following question: Why are infinite-dimensional vector spaces usually equipped with additional structure? Although it received one good answer, I feel that ...
- 961
1
vote
0
answers
82
views
Injective envelopes of 1-extensible spaces
Please read this post as a naive follow up on a previous question.
Let $X$ be a Banach space and let $(I(X),\alpha)$ denote its injective envelope (e.g., CohenLacey1969). A low hanging fruit is the ...
- 2,605
1
vote
1
answer
119
views
Extremally disconnected rigid infinite Hausdorff compacta(?)
Question: does there exist an extremally disconnected infinite Hausdorff compact space $\ X\ $ such that the only homeomorphism
$\ h: X\to X\ $ is the identity homeomorphism
$\ h=\mathbb I_X:\ X\to X\...
- 4,786
0
votes
1
answer
165
views
For $\mu$-a.e. $x \in X$, the sequence $(f_n(x, \cdot))_n$ is Cauchy in $L^1 (Y)$. Then $(f_n)$ is Cauchy in $L^1 (X \times Y)$
Below we use Bochner measurability and Bochner integral. Let
$(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces,
$(E, | \cdot |)$ a Banach space,
$S (X)$ the ...
- 825
1
vote
1
answer
74
views
Subspaces generated by the orbits of the group of isometries on $C(K)$
Let $X$ be an extremally disconnected compact Hausdorff space with no open points, and $f:X\to\mathbb{C}$ be a non-constant continuous function. Let $D_f$ be the linear span of the functions of the ...
- 2,605
0
votes
2
answers
125
views
Is there a modification of $f$ on a null set such that $F: [0, T] \to L^p ({\mathbb R}^d), t \mapsto f(t,\cdot)$ is Bochner measurable?
Let $T>0$ and $p \in [1, \infty)$. Let $f \in L^p ([0, T] \times {\mathbb R}^d)$. By a theorem in this thread, there is a Lebesgue null subset $N$ of $[0, T]$ such that $f(t, \cdot)$ is Lebesgue ...
- 825
1
vote
1
answer
195
views
Reference request: Inequalities involving convex sets and Gaussian variables stated in a paper by Talagrand
I'm looking for references for two facts that are stated without proof in the paper:
Talagrand, M., Are all sets of
positive measure essentially convex?, Lindenstrauss, J. (ed.) et al.,
Geometric ...
- 484
1
vote
0
answers
105
views
Applications of finite speed of propagation property
Consider the Laplace operator $\Delta:=\sum_{j=1}^{n}\partial_{x_{j}}^{2}$ on $\mathbb{R}^n$. Let $E_{\lambda}$ be the spectral resolution of $\Delta$, and
$$ H_{t}[f]:=\cos{(t\sqrt{-\Delta})}f=\int_{...
- 287
1
vote
1
answer
97
views
Choosing a net of projections from a given collection
Let $\mathcal M$ be a von Neumann algebra. Let $S$ be a nonempty set. Consider $\Omega$ to be the collection of all finite subsets of $S$. Suppose for all $\omega\in\Omega$ we have $I_\omega$ a ...
- 1,879
2
votes
1
answer
101
views
Hyperexpectations from injective subfactors of a type $II_1$ factor
Let $M$ be a type $\rm{II}_{1}$ factor with trace $\tau$, acting by the GNS representation on $B(L^{2}(M,\tau))$. Let $R\subset M \subset B(L^{2}(M,\tau))$ be a hyperfinite $\rm{II}_{1}$ subfactor of $...
- 7,067
2
votes
0
answers
153
views
Riesz’s representation theorem in a weak form
Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^N$ $(N\geq 3)$, $\phi\in H_0^1(\Omega)$ is a solution of $$
\begin{cases}\Delta \phi+ \phi=h & \text { in } \Omega, \\ \phi=0 &...
- 471
1
vote
0
answers
109
views
PDE coupled with the pronic numbers (related to triangular numbers)
I am studying the linear PDE:
$$ t^2\frac{\partial^3}{\partial t^3}\sum_{n=1}^\infty \Psi_n(t,s)=s^2\frac{\partial}{\partial s}\sum_{n=1}^\infty \Psi_n(t,s)+\sum_{n=2}^\infty b(n)\frac{\partial}{\...
- 383
1
vote
0
answers
44
views
Characterizing some similarity invariant homogeneous log-superharmonic functions of matrices
Let $L:M_n(\mathbb{C})^r\rightarrow[0,\infty)$ be a function that satisfies the following properties:
$\log(L)$ is plurisubharmonic.
$L$ is homogeneous in the sense that $L(\lambda A_1,\dots,\lambda ...
4
votes
2
answers
228
views
(Reference request) higher order Hölder spaces on riemannian manifolds
I am looking for a reference regarding the higher (than the first) order Hölder spaces on Riemannian manifolds. I am aware that defining Hölder spaces of form $C^{0,\alpha}$ is not an issue even ...
- 820
8
votes
3
answers
1k
views
Are all positive eigenfunctions principal eigenfunctions?
In a given domain $\Omega$, we have: $\Delta u=-\lambda u$ with $u>0$. Does this mean that $u$ is a principal eigenfunction for $\Delta$ in $\Omega$?
Also, more generally, does this also apply for $...
- 421
7
votes
1
answer
246
views
A notion of restricted injectivity for Banach spaces
I apologize in advance if this is well-known.
Let $X$ be a Banach space. Let's call only for this post that $X$ is self-injective if for every closed subspaces
\begin{equation}
A\subseteq B\subseteq X ...
- 2,605
1
vote
0
answers
111
views
Residues of analytic operators
Suppose we have analytic operators $P_{z}: C^1[0,1]\to C^1[0,1]$, where $z \in \mathbb{C}$, and the spectrum of $P_{z_0}$ possesses an isolated eigenvalue $1$ (assuming multiplicity is 1 and $I-P_z$ ...
- 11
4
votes
1
answer
305
views
Holomorphic extension of the Fourier transform of a measure
If an entire holomorphic function $f(z)$ is given by the analytic continuation of $f(x)=\int_\mathbb{R}e^{-ix\xi}\,d\mu(\xi)$ with a finite Borel measure $\mu$ on $\mathbb{R}$, then $g(x):=\int_\...
- 63
0
votes
0
answers
145
views
Why is this function in $L^1$?
I had a question about a claim made in the paper "Group Invariant Scattering" and why it is true. Consider the function $h_j(x) = 2^{nj}\psi(2^jx)$, where $\psi$ is a function such that $\...
- 101
0
votes
1
answer
254
views
Is the space $L^p_{\text{loc}} (\mathbb R^d)$ separable w.r.t. the norm $\|f\|_{\tilde L^p} := \sup_{x \in \mathbb R^d} \|1_{B(x, 1)} f\|_{L^p}$?
Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let ${\tilde L}^p (\mathbb R^d)$ be the space of ...
- 825
2
votes
1
answer
170
views
Ultralimit of $w^*$-continuous maps
Let $\omega$ be a free ultrafilter on $\mathbb N.$ Let $(\mathcal M_n)$ be a sequence of finite von Neumann algebras. Let $\mathcal N$ be another finite von Neumann algebra and we have maps $\phi_n:\...
- 1,879
2
votes
1
answer
143
views
Decomposition of weak* convergent nets into positive weak* convergent nets
Let $F$ be an order unit Banach space with order unit $e$ and topological dual space $F^*$ ordered by the dual cone. Let $E\subset F^*$ be a closed subspace that separates points of $F$ and such that ...
- 71
1
vote
2
answers
188
views
Does a measurable $F :[0, T] \to L^p (\mathbb R^d; \mathbb R_{\ge 0})$ have a "flattened" measurable version?
Let $d \in \mathbb N^*,p \in [1, \infty]$ and $T>0$. Let
$$
F :[0, T] \to L^p (\mathbb R^d; \mathbb R_{\ge 0}), t \mapsto F_t
$$
be measurable. I would like to ask if there is a measurable function ...
- 825
1
vote
1
answer
2k
views
Product of Dirac delta function
The following equation may be meaningful, but how can we make it well-defined
$$\delta(x-a)\cdot\delta(x-b)=0$$
Question: How do we defined this equation? Or more broadly define product between ...
- 13
5
votes
1
answer
158
views
Backwards stable factors
A factor $R$ is called stable if $M_n(R)\cong R$ for all $n>0$. For the sake of this question, we call a factor backwards stable if $R\cong M_n(S)$ implies $S\cong R$ where $S$ is allowed to be any ...
- 769
0
votes
0
answers
149
views
Validity of Hölder inequality for the homogeneous Besov spaces $\dot{B}^0_{1,2}(\mathbb{R}^n)$ and $\dot{B}^0_{2,2}(\mathbb{R}^n)=L^2(\mathbb{R}^n)$
I am looking at Corollary 1. in p.244-245 of the book
"Sobolev Spaces of Fractional Order,
Nemytskij Operators,
and Nonlinear
Partial Differential Equations" (1996) by Thomas Runst
Winfried ...
- 3,477
2
votes
0
answers
325
views
Examples of RKHS that are "classical"
Among the so-called "classical" Hilbert spaces ($L^2$, Sobolev, Hardy, Bergman, etc.), which are very well-studied, which are RKHSs?
It is easy to construct example of RKHSs by applying the ...
- 21