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0 votes
0 answers
68 views

Family of separable Hilbert spaces over locally compact form a continuous field of Hilbert space?

Let $\{H_{x}\}_{x\in G^{0}}$ be a family of separable Hilbert spaces and $G^{0}$ be a locally compact second countable topological space. Let $\mathbb{B}_{x}$ be the orthonormal basis of $H_{x}$. If ...
1 vote
0 answers
176 views

If $f \in L^p(\Omega)$, then $f \in L^q(\Omega)$ for some $q < p + \epsilon$?

Loosely speaking, I would like to know whether membership in some Lebesgue space $L^p$ is stable under small perturbations of the exponent $p$. Let $\Omega \subseteq \mathbb R^n$ be a bounded domain ...
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 $E\subseteq \mathbb{R}^n$ ...
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 \...
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 ...
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 ...
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'&...
1 vote
2 answers
2k views

Bounding the norm of the Laplacian of the gradient of a function having Lipschitz continuous Hessian

It seems that the following claim is true, but I did not manage to prove it neither to find a reference. Claim Let $f:\mathbb R^p\to\mathbb R$ be a three times differentiable function such that its ...
45 votes
1 answer
2k views

Existence and uniqueness of Haar measure on compacta; a cohomological approach

I am trying to use a modification of group cohomology to prove the existence and uniqueness of Haar measure on a compact Hausdorff group. I think the best way of introducing the idea I am pursuing is ...
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^*([...
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\}...
5 votes
1 answer
774 views

Question/References on the Skorokhod M1 topology

Let $D(0,T)$ be the space of right continuous functions with left limits defined on $[0,T]$. Consider the Skorokhod M1 topology on $D(0,T)$, see e.g. S. Ledger, Skorokhod’s M1 topology for ...
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 $\...
2 votes
1 answer
106 views

Dual space and conditions for weak convergence in Orlicz Space not having $\Delta_{2}$ property

I am interested in conditions for weak convergence on Orlicz spaces where the corresponding Young function, $\Phi:[0,\infty) \rightarrow [0,\infty)$, does not have the $\Delta_{2}$ condition, i.e. ...
2 votes
1 answer
547 views

Shift-invariant spaces

We can define a shift-invariant space as $$V_{\varphi}(\mathbb{Z}):=\left\{\sum_{k\in\mathbb{Z}}c_k\varphi({\cdot}-k):(c_k)\in \ell_2\right\},$$ where convergence of the series is taken to be in $L^2(\...
-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{...
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(\...
7 votes
0 answers
619 views

Lavrentiev Phenomenon

Does there exist a (onedimensional) integral functional of calculus of variations $$ F(y)=\int_a^b f(t,y(t),y'(t))\,dt
 $$ such that not only $$ \inf_{y\in\operatorname{Lip}([a,b])}F(y)>\inf_{y\in ...
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 $\varphi|_X + \psi|_X \in X'$, ...
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 \...
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 ...
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 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 ...
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 $$...
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 [...
0 votes
0 answers
118 views

Find the maximum of an expression under the logconcave assumption

Let $F(v)$ be a cdf over $\left[0,v_{max}\right]$, $1-F(v)$ is logconcave. The corresponding density function is $f(v)$. Let $p^m$ solve $1-F(v)-f(v)v=0$ (it is a FOC of a profit maximization problem)....
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(\...
2 votes
1 answer
670 views

Integral on level sets

Let $g_\epsilon : K \subset \mathbb{R}^d \rightarrow \mathbb{R}$ (more regularity can be assumed if necessary) be defined on a compact set (with regular boundary) $K \subset \mathbb{R}^d$, and the ...
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)}...
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....
4 votes
1 answer
800 views

Bound the operator norm of the Fréchet derivative of a Lipschitz function in this setting

I want to find a bound for the operator norm of the Fréchet derivative of a Lipschitz continuous function in the following setting: Let $E$ be a $\mathbb R$-Banach space; $v:E\to[1,\infty)$ be ...
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 votes
1 answer
490 views

Space derivative of flow of ODE with monotone source

Consider the ODE $$ \begin{cases} \partial_t\Phi(t,x) = f(t,\Phi(t,x)), &\ t>0, \ x \in \mathbb R \\ \Phi(0,x) = x, & x \in \mathbb R \end{cases} $$ where $f$ is function which is a non-...
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 ...
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 ...
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 $K$ ...
8 votes
1 answer
390 views

Order bounded version of monotone complete $C^*$-algebras

Let $A$ be a $C^*$-algebra with self-adjoint part $A_{\operatorname{sa}}$. Then $A$ is called monotone complete if every increasing norm bounded net in $A_{\operatorname{sa}}$ has a supremum (with ...
7 votes
1 answer
394 views

Inverse limit in the category of $C^{\ast}$-algebras or operator spaces

Does the inverse limits (projective limits) exist in the category of $C^{\ast}$-algebras or operator spaces? I tried to search but could not find a proper reference. Any reference or comments about ...
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 "...
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, \...
-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 $\mathcal{X}$ ...
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 ...
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 ...
19 votes
5 answers
16k views

What does "kernel" mean in integral kernel?

In functional analysis, there is the term "integral kernel". Examples are Possion kernel, Dirichlet kernel etc. In algebra, the term kernel of a homomorphism refers to the inverse image of the zero ...
0 votes
0 answers
42 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 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}^...
5 votes
1 answer
320 views

Is $\mathscr{S}_h'$ a complementary subspace for $\mathscr{S}'/\mathscr{P}$, the space of tempered distributions modulo polynomials?

Recall that in many Fourier Analysis texts, given a function $\Psi$ such that $\hat{\Psi}\in\mathcal{D}(\mathbb R^d)$, $\hat\Psi\ge0$ is supported in an annulus, and $\sum_{j\in\mathbb Z}\hat\Psi(2^j\...
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 ...
6 votes
1 answer
323 views

Hartogs' theorem in Banach spaces

In complex analysis one learns Hartogs' theorem: Let $U\subseteq \mathbb{C}^n$ open and $f: U \rightarrow \mathbb{C}$ a function. Then $f$ is analytic iff for all $1\leq i \leq n$ $$ z \mapsto f(...