All Questions
10,239 questions
7
votes
2
answers
841
views
Why is $\mathbb R^{\mathbb N}$ not high-dimensional enough?
In this paper [1], the authors consider the limiting distribution of $$S_{n,p}:=\frac{1}{\sqrt n}\sum_{k=1}^nX_k$$ for $p\rightarrow\infty$ as $n\rightarrow\infty$, where $X_1, X_2,\dots, X_n$ are ...
- 181
2
votes
0
answers
124
views
dimensionality reduction of Markov chains
Suppose that $M$ is a time-homogeneous (and, for simplicity, stationary) Markov chain on $d$ states, which induces the probability measure $P$ on paths of length $n$. I seek a Markov chain $M'$ on $d'&...
- 6,541
5
votes
0
answers
77
views
What is the maximal advantage of randomized over deterministic algorithms for approximation in the worst-case?
Let $X\subset Y$ be Banach spaces and $B_X:=\{x\in X: \|x\|_X\le1\}$ be the unit ball of $X$.
The goal is to find an approximation of every element from $B_X$ with error measured in $Y$ by using at ...
3
votes
1
answer
220
views
What we know about the function in Fefferman's Theorem
In Fefferman's many papers on Whitney's theorem he, amongst other things, constructs the existence of a smooth function $F$ which extends a function $f$ on a (say) finite set $E\subseteq \mathbb{R}^n$ ...
- 5,405
2
votes
2
answers
154
views
Closure of $C([0,1]^2)$ via weak*-topology [closed]
Let $C([0,1]^2)$ denote the set of continuous functions on $[0,1]^2$. Let $L^1([0,1]^2)$ be the set of all Lebesgue integrable functions on $[0,1]^2$.
The dual space of $C([0,1]^2)$, denoted by $C^*([...
- 349
3
votes
0
answers
130
views
A Talagrand inequality for the supremum of partial sums over function classes under dependence. (Reference request)
As a consequence to the Talagrand concentration inequality, it is well known that for a measurable space $(S,\mathcal{S})$ and an i.i.d. sample $X_1,...,X_n$ of $S$-valued random variables, if $\...
- 141
1
vote
0
answers
87
views
Supremum of sums of functions in $L^1$ taking random signs
Consider the Banach space $X=L^1([0,1])$, and let $n\gg1$ and $x_1, ..., x_n$ be any points in the unit sphere of $X$.
Is there any reasonable lower bound for $$\sup_{(\epsilon_i)_{i=1}^n \in \{-1,+1\}...
- 393
2
votes
0
answers
75
views
Pullback by surjective submersion is injective?
Denote by $\mathcal{D}'_X$ the sheaf of distributions on a smooth manifold $X$.
Let $M$ and $N$ be smooth manifolds and $\Phi: M \to N$ a submersion. Then $\Phi$ defines a unique morphism of sheaves $\...
- 261
-1
votes
1
answer
98
views
Spectrum of sum of positive and negative operators
Let $(\mathscr{H}, \langle \cdot, \cdot \rangle)$ be a separable complex Hilbert space, and let $\mathscr{D}$ be a dense subset of $\mathscr{H}$. Let $P: \mathscr{D} \to \mathscr{H}$ and $N: \mathscr{...
- 21
0
votes
0
answers
78
views
What properties does representing measure $\mu$ for $z\in\mathbb{D}^n$ has to satisfy so that $\nu=0$ is the only measure singular with respect to it?
Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$.
We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if
$$\forall_{u \in A(\...
- 63
4
votes
1
answer
214
views
Characterisation of Sobolev spaces using their Lipschitz approximations
Let $f \in W^{1, p} (\mathbb R^n)$. A classical approximation theorem (see for instance, the book by Evans and Gariepy) says that we can approximate $f$ by Lipschitz functions, in the sense that for ...
- 6,215
5
votes
2
answers
149
views
Showing an operator is (or not) closed on $L^2(\mathbb{R})$
I am linearizing nonlinear waves and get operators of the form below. Everything is considered in $L^2(\mathbb{R})$.
Consider the operator $L_1=\frac{d}{dx}$. The domain is $H^1(\mathbb{R})$ and it is ...
2
votes
2
answers
105
views
Characterization of the dual of intersection of Banach spaces
Let $U,V$ Banach spaces and define $X = U\cap Y$ endowed with the norm $\|u\|_X = \|u\|_U + \|u\|_V$. If we take $\varphi \in U'$ and $\psi \in V'$, I can prove that $\varphi|_X + \psi|_X \in X'$, ...
- 167
1
vote
0
answers
98
views
Equivalence of Sobolev norms for smooth functions with compact support
Let $f\in C^\infty_c([0,1]^n)$, then we can extend it to a $1$-periodic smooth function $\tilde f$. We define the fourier transform (series) of $f$ ($\tilde f$):$$
\hat f(\xi):=\int e^{2\pi i x\cdot \...
- 435
1
vote
1
answer
157
views
Is finding the CDF from the Laplace transform well-posed?
In my study of Dynamic Light Scattering, I came across the following inverse problem. Let $F(s):[0,T]\rightarrow[0,T]$ be the Laplace transform of a probability distribution $f(t)$ on the real line ...
- 654
2
votes
0
answers
191
views
Smoothing property of the heat kernel on the one-dimensional torus
Let $G=G(x,t)$ be the heat kernel on the one-dimensional torus $\mathbb{T}^1,$ with $x \in \mathbb{T}^1$ and $t \in (0,T].$ $G$ is given by \begin{equation}
G(x,t) = (4 \pi t)^{-1/2} \sum_{k \in \...
- 185
0
votes
1
answer
231
views
Questions on the compactness of $L_1([0,1]^2)$'s unit sphere
Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$ and $f(x,y)\geq 0: a.e. (x,y)\in [0,1]^2$. Recently in my study I need to study the compactness of $U$. By Riesz's theorem ...
- 349
0
votes
1
answer
101
views
Limit sequence of regular function in $L_1$‘s unit sphere
Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$. For any $f\in U$, we say it is regular if $\int_{x_0\times [0,1]}f=\int_{[0,1]\times y_0}f=1$ for a.e. every $x_0, y_0\in [...
- 349
4
votes
0
answers
147
views
Weakly compact sets forced to contain $0$
Let $E$ be an infinite-dimensional real normed space and let $K\subset E$ be a weakly compact set such that, for each $\varphi\in E^*\setminus \{0\}$, there exists a unique $\tilde x\in K$ such that
$$...
- 283
0
votes
0
answers
121
views
How to find the inverse of this linear integral operator?
Let $f(x): \mathbb{R}^d \rightarrow \mathbb{R}$ be a function that decays ``fastly enough'' at infinity.
We can define the following linear operator
$$L[f](x):= \int_{\mathbb{R}^d} d^d y \, \frac{f(y)}...
- 1
3
votes
1
answer
158
views
Upper and lower bounds for a Rademacher-type expectation
Suppose that $\varepsilon_i$
are independent Rademacher random variables
(that is,
$
\mathbb{P}(\varepsilon_i=-1)
=
\mathbb{P}(\varepsilon_i=1)
=1/2
$.
Fix an $a\in\mathbb{R}^n$
and define the random ...
- 6,541
5
votes
0
answers
160
views
Hartman uniform distribution of means
Background: for a discrete abelian group $G$, a character of $G$ is a homomorphism $\chi:G\to \mathbf S^1$, $\mathbf S^1$ being the circle group $\{z\in \mathbb C:|z|=1\}$ with ordinary multiplication....
- 929
-6
votes
1
answer
180
views
An analog of Anderson's result in C* algebra setting [closed]
Let $\mathcal{A}$ be a unital $C^{*}$-algebra and $S(\mathcal{A})$ denote the states space of $\mathcal{A}$.
For $a\in \mathcal{A}$ , define $W(a) =\{\phi(a):\phi\in S(\mathcal{A})\}$
It's known that $...
- 307
2
votes
0
answers
179
views
Analytic continuation of $\int_V (f(x_1,\cdots,x_n))^s dx_i$
Let $V$ be an $n$-dimensional simplex, let $f(\boldsymbol{x}) = f(x_1,\cdots,x_n)\in \mathbb{C}[x_1,\cdots,x_n]$ be a product of linear polynomials that is non-zero in interior of $V$. Also let $E(\...
- 528
1
vote
1
answer
102
views
Does $C^{k,s-k}$ function with lipschitz lower order derivatives give a certain bound on the Taylor remainder?
Let $\Omega \subseteq \mathbb{R}$ be open (not necessarily an interval). Let $ s > 0$ and $k \in \mathbb{N}_0$ be such that $s \in (k, k+1]$. Suppose that
$f \colon \Omega \to \mathbb{R}$ is an ...
- 820
2
votes
1
answer
152
views
Co-locating slowly increasing smooth functions in two different ways
This question is subsequent from my previous one.
I will write everything in detail for the sake of completeness.
Let $g_1$ and $g_2$ be smooth functions on $\mathbb{R}$, whose derivatives of all ...
- 3,477
5
votes
2
answers
256
views
On the closed convex hull of a weakly compact set
Let $H$ be an infinite-dimensional real Hilbert space and let $B$ be the closed unit ball of $H$. Let $K\subset B$ be a weakly compact set whose closed convex hull agrees with $B$. Question: does $K$ ...
- 283
2
votes
0
answers
102
views
Semiclassical limit of spectral gap of Schrödinger operators with nonsmooth potential
Let $\Omega$ be a connected compact subset of $\mathbb{R}^d$. It is well known that for a smooth potential $V:\Omega \to \mathbb{R}$ that has a unique nondegenerate minimum $V(0) = 0$, the operator $H ...
- 1,551
0
votes
0
answers
121
views
Is there a good or commonly accepted short notation for the set of differentiable, but not necessarily continuously differentiable maps?
Every once in a while I find myself in need of some short notation for the set of differentiable, but not continuously differentiable maps, say, $X \to Y$. Always having to specify "...
- 7,127
3
votes
1
answer
182
views
Tensor product of a slowly increasing smooth function and a tempered distribution converging to a co-located product
Let $T$ be a tempered distribution on $\mathbb{R}$ and $g$ be a smooth function on $\mathbb{R}$ whose derivatives of all orders are all polynomially bounded (a.k.a. slowly increasing).
For any pair of ...
- 3,477
-1
votes
1
answer
168
views
Space of distributions on $[0,1]^2$: weakly compact or not?
Let $X_1,X_2$ be distributions on $[0,1]$ and let $X=(X_1,X_2)$ be the joint distribution of $X_1,X_2$. Let $\mathcal{X}$ be the set of all such joint distribution $X$.
Question 1: Does $\mathcal{X}$ ...
- 349
1
vote
1
answer
40
views
Envelopes of functions with respect to some convex cone $\mathcal{F}$
Let's say we are given a function $f:\mathbb R ^d\to \mathbb R$ continuous. Assume that $\mathcal F$ is a convex cone of continuous functions ($\mathbb R^d$ to $\mathbb R$) closed under maxima. I am ...
- 291
4
votes
0
answers
80
views
Interpolation-extrapolation scales of H. Amann
I am currently reading H. Amann's notes titled "Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems", and I have a question regarding the abstract ...
- 145
0
votes
0
answers
41
views
Whether the constant of a fractional Sobolev inequality is universal for all cubes
Let $D\subset \mathbb{R}^d$ with $d\geqslant 1$ be a bounded open connected Lipschitz set, $\eta\in (0,1)$, $p>0$, and $\alpha>0$. Then the paper (On comparability of integral forms written by ...
- 3
1
vote
1
answer
125
views
Subtlety of identifying $W^{k,p}\bigl([0,1] \bigr)$ and $W^{k,p}(S^1)$ - from ME
I apologize for repeating the same question from ME, but it seems more subtle than I expected.
Let me fix the notations here first:
\begin{equation}
C^\infty_c(0,1):= \{ f : (0,1) \to \mathbb{C} \mid ...
- 3,477
5
votes
1
answer
183
views
What is a natural interpretation of the commutator of the conditional expectation operator?
Notation: We denote by $\mathbb E_{\mathcal F} X$ the conditional expectation of the random variable $X$ with respect to the $\sigma$-algebra $\mathcal F$.
Given two $\sigma$-algebras $\mathcal G, \...
- 6,215
3
votes
1
answer
111
views
Sobolev inequalities and Wiener algebra
It follows from the Gagliardo-Nirenberg inequality that for a locally integrable function $f$ defined on $\mathbb R^d$ (we assume $d\ge 3$)
such that $\nabla f$ belongs to $L^2(\mathbb R^d)$ and
$$
\...
- 16.2k
1
vote
1
answer
330
views
Does $\sum_{n=1}^{\infty}\frac{(-1)^n e^{\sin{n}}}{\sqrt{n}}$ converge?
I am trying to study the converge of the series
$$\sum_{n=1}^{\infty}\frac{(-1)^n e^{\sin{n}}}{\sqrt{n}}$$
But $e^{\sin{n}}$ is not monotone, and the Abel's test rule fails here. Can someone help me? ...
- 287
5
votes
1
answer
164
views
Does quadratic asymptotic growth imply log-Sobolev inequality?
Let $f : \mathbb{R}^n \rightarrow [0,\infty)$ be a smooth function and consider $h$ s.t $h(\vec{x}) = f(\vec{x}) + \lambda \Vert \vec{x} \Vert^2$.
Does this imply that irrespective of any other ...
- 617
10
votes
1
answer
314
views
Weakly metrizable sets in normed spaces
A similar question was asked on MSE without getting an answer.
In the proof of lemma 1.2 of Asplund operators and holomorphic maps the author (my attempt to contact him failed because the only e-mail ...
- 16.4k
0
votes
0
answers
60
views
Spectral analysis of Dirac operators coupled to gauge potential on $\mathbb{R}^n$
Dirac operators on compact manifolds seem to have been studied well, such as in this book and also this one.
However, I cannot easily find comprehensive treatment of Dirac operators coupled to gauge ...
- 3,477
3
votes
0
answers
196
views
Parabolic smoothing for semilinear PDE
Consider the semilinear energy-critical parabolic PDE in $\mathbb{R}^3$
\begin{align}
\partial_t u &= \Delta u + |u|^{4/(n-2)}u = \Delta u + u^5\\
u(0,x) &= u_0\in \smash{\dot{H}}^1(\mathbb{R}^...
- 537
0
votes
0
answers
78
views
Definition of Moore-Penrose inverse for unbounded self-adjoint operators?
I know there is a concept of Moore-Penrose or pseudoinverse of a matrix. I would like to know if one can define it for densely defined unbounded self-adjoint operators on Hilbert spaces as well. ...
2
votes
1
answer
108
views
Separability is an interpolation property
I'm trying to prove that certain space, which can be obtained as an interpolation space, is separable. The fact that is separable is well known but i want to simplify it via interpolation. I haven't ...
1
vote
1
answer
85
views
$H^2$-elliptic regularity (up to the boundary) for operators with lower order terms for Lipschitz/convex domains
Let $\Omega$ be a bounded domain which is Lipschitz or convex. Given an elliptic operator of the form
$$\langle Au, v \rangle = a_{ij}u_{x_i}v_{x_j} + b_i u_{x_i}v + cuv$$
are there any elliptic ...
- 93
0
votes
0
answers
45
views
Functional inequalities on neighbourhood graphs
Consider an open domain $\Omega \in \mathbb{R}^d$, say the unit disk in $\mathbb{R}^2$ with $N$ points sampled i.i.d. on it. One of the simplest possible (unnormalised) discrete Laplacian of a ...
- 111
6
votes
1
answer
170
views
Do projections in an $AW^\ast$-algebra form an orthomodular lattice?
I’m currently studying orthomodular lattices arising out of operator algebras. One of the most standard examples is the projection lattice of a von Neumann algebra - if $M$ acts on a Hilbert space $H$,...
- 2,830
3
votes
1
answer
116
views
Does a bounded positive modular sesquilinear form on a $C^\ast$-algebra induces an element of its multiplier algebra?
This is a question that originates from my attempt at this question. Specifically, for a $C^\ast$-algebra $A$, I am attempting to construct a map $\phi: A \times A \to A$ s.t.,
$\phi$ is sesquilinear,...
- 2,830
1
vote
1
answer
105
views
Constrained optimization over a set of functions
How to approach the following optimization problem:
$$\text{minimize }\int_0^1 f(x) \, dx$$
over all (integrable) real-valued functions $f$ on $[0,1]$ satisfying
$$1-x_1 x_2 \leq f(x_1)f(x_2)\text{ ...
3
votes
1
answer
109
views
Literature request: Covariance operators for Gaussian measures
I am looking to answer the question:
If $\mathcal{B}$ is a separable Banach space and $R: \mathcal{B}^*\to\mathcal{B}$ is a symmetric and positive operator, then $\phi: \mathcal{B}^*\to\mathbb{R}, \...
- 171