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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 \...
kumquat's user avatar
  • 185
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 ...
Isaac's user avatar
  • 3,477
4 votes
1 answer
132 views

Direct characterization of finite-dimensional $1$-injective Banach spaces

It follows from Kelley's Theorem that the only finite-dimensional $1$-injective Banach spaces are $\ell^\infty_n$, $n\in\mathbb N$. Is there a simple direct proof of this fact, without having to talk ...
Martin Argerami's user avatar
5 votes
1 answer
379 views

Nuclear vs Banach spaces: compactness properties

A question about the meaning from following excerpt from german wikipedia adressing interesting crucial feature of nuclear spaces opposing them from Banach spaces (transl.): While normed spaces, ...
user267839's user avatar
  • 6,038
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 [...
tom jerry's user avatar
  • 359
-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{...
d'Alembert's user avatar
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 ...
K N SRIDHARAN NAMBOODIRI's user avatar
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 ...
Isaac's user avatar
  • 3,477
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 ...
Mario Ullrich's user avatar
5 votes
1 answer
184 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, \...
Nate River's user avatar
  • 6,311
0 votes
0 answers
57 views

Double-periodic functions with (possible) poles

Consider the set of double-periodic function $f:\mathbb C/(\mathbb Z+i \mathbb Z) \setminus \{z_0\} \to \mathbb C$, where $z_0$ is a fixed point inside $\mathbb C/(\mathbb Z+i \mathbb Z),$ that have a ...
António Borges Santos's user avatar
-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 $\mathcal{X}$ ...
tom jerry's user avatar
  • 359
4 votes
0 answers
148 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 $$...
Biagio Ricceri's user avatar
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'&...
Aryeh Kontorovich's user avatar
1 vote
0 answers
45 views

Existence of optimal entropic weights for empirical modeling

Let $\mathcal{X} = [0,1]^n$ be the input space and $\mathcal{Y} = \{1, ..., n_c\}$ be a discrete output space. Let $D = \{(x_i, y_i)\}_{i=1}^N \subset \mathcal{X} \times \mathcal{Y}$ be a training ...
Damien's user avatar
  • 111
5 votes
0 answers
162 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....
John Griesmer's user avatar
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 ...
Kacper Kurowski's user avatar
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(\...
pisco's user avatar
  • 528
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\}...
HHN's user avatar
  • 393
1 vote
0 answers
39 views

About Carleson measures on the Hardy space on the bidisc

I have been reading the paper "Carleson Measures in Hardy and Weighted Bergman Spaces of Polydiscs" by F. Jafari and there are a few things that going on that I am not entirely convinced of. ...
an_ordinary_mathematician's user avatar
5 votes
1 answer
165 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 ...
Student's user avatar
  • 617
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(\...
S-F's user avatar
  • 63
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 $H$,...
David Gao's user avatar
  • 2,830
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 \...
Tian LAN's user avatar
  • 435
0 votes
0 answers
53 views

Spectral theory of compact operator for quasi-Banach spaces

Let $X$ be a Banach space and let $Y\subset X$ be a quasi-Banach space (with compact inclusion). Suppose $T:X\to X$ is a compact operator such that $1$ is not its eigenvalue and $T|_{Y}:Y\to Y$ is ...
Liding Yao's user avatar
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 $\...
psl2Z's user avatar
  • 331
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,...
David Gao's user avatar
  • 2,830
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}^...
Student's user avatar
  • 537
0 votes
0 answers
122 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 "...
M.G.'s user avatar
  • 7,127
2 votes
0 answers
104 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 ...
Lwins's user avatar
  • 1,551
1 vote
1 answer
215 views

Compactness with respect to topology induced by total-variation distance

I've been working on a problem and at some point in the proof I need to show that the following set $$\left\{\mu \in \mathcal{P}_{ac}(\mathbb R^d): \int \varphi(x)\mu(\mathrm{d}x)\leq C\right\}$$ is ...
Andrew Luo's user avatar
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}, \...
ChocolateRain's user avatar
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 ...
Michelangelo's user avatar
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{ ...
Arkadi Predtetchinski's user avatar
3 votes
1 answer
375 views

Dimensionality reduction for total variation

Let $P_i,Q_i$, $i\in[n]$, be distributions on a finite set $\Omega$. We will use $P^\otimes_{i\in[n]}$ to denote $n$-fold products of distributions. For each $i\in[n]$, define the dimensionally-...
Aryeh Kontorovich's user avatar
6 votes
1 answer
335 views

Existence of pairwise quasi-complementary but not complementary subspaces

Let $𝑋$ be an infinite-dimensional Banach space (complex or real). A subspace of $𝑋$ means a closed linear submanifold. Subspaces $M$ and $N$ of $X$ are quasi-complementary if $M\cap N=\{0\}$ and $M+...
Janko Bracic's user avatar
4 votes
1 answer
180 views

Analytic function with values in $L^1$

Suppose that $(\Omega, \Sigma, \mu)$ is a measure space. Let $D$ be the unit open disk and $F : D \rightarrow L^1(\mu)$ be an analytic function. Is it true that for a.e. $w \in \Omega$ the function $F(...
Seven9's user avatar
  • 565
1 vote
0 answers
174 views

Interpolation of Sobolev spaces with constraints

Let us consider a real interval $[0, L]$, with $a\in (0, L)$, and let $I_1=(0, a)$ and $I_2=(a, L)$. We denote by $H^k(I_1)$ and $H^k(I_2)$ the usual Sobolev spaces, defined for $k\in \mathbb{N}$. Now,...
rebo79's user avatar
  • 81
1 vote
1 answer
128 views

Infinite direct sum decomposition of the heat semigroup on $\mathbb{R}^n$

Consider the heat semigroup $Q_t$ on $L^2(\mathbb{R}^n)$ generated by the Laplace operator $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x^2_i}$. Does there exist a direct sum decomposition $$\oplus_{...
Ribhu's user avatar
  • 407
12 votes
1 answer
402 views

Boundedness of sequences and cardinality

Let $X$ be a set of sequences of real numbers that converge to zero with the property that for any unbounded sequence of real numbers $(y_n)$, there is a sequence $(x_n)$ in $X$ for which the ...
Chris Stuart's user avatar
2 votes
0 answers
331 views

What is the spectrum of this differential operator?

My self-adjount differential operator $L$ is defined by $$L f(x) \equiv u(x) \frac{\partial^2}{\partial x^2} \left( u(x) f(x) \right)$$ where $u(x)$ is a known but arbitrary smooth function that ...
Peter A's user avatar
  • 151
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 ...
Guillermo García Sáez's user avatar
0 votes
1 answer
114 views

Geometric interpretation of a Grammian-like function

Let $\mathbf{v}, \mathbf{w} \in \mathbb{R}^n$ and consider the following function $f : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$: $$ f(\mathbf{v},\mathbf{w}) = \|\mathbf{v}\|\|\mathbf{w}...
Nathaniel Johnston's user avatar
0 votes
0 answers
141 views

The tensor product of two Fredholm operators

What can be said about the tensor product $T\otimes S$ of two Fredholm operators $T:X_1\to Y_1$ and $S:X_2 \to Y_2$ where $X_1,X_2,Y_1, Y_2$ are Banach spaces and tensor product of ...
Ali Taghavi's user avatar
1 vote
1 answer
143 views

Operator norm of some type of discrete Fourier matrix

Let $N$ be a natural number and let $w$ be a complex number. We define the $N\times N$ matrix $C_w=(a_{k,l})_{k,l=1}^N$ as follows, $$ a_{k,l}=\begin{cases}1 & l=k+1\\ w &...
ABB's user avatar
  • 4,058
0 votes
0 answers
55 views

Compactness and Leray-Schauder degree

What's the relationship between compactness of solutions in partial differential equations (PDEs) and the Leray-Schauder degree?
Davidi Cone's user avatar
3 votes
1 answer
288 views

Expectation comparison inequality for concave function of symmetric random variables

Suppose that $X_i$, $i\in[n]$ are independent symmetric random variables. I think the conjectured result holds in greater generality, but we can additionally assume that each $X_i$ takes the values $\...
Aryeh Kontorovich's user avatar
2 votes
1 answer
204 views

A continuous analogue of the notion of Hilbert basis

Let $X$ be a locally compact space, let $H$ be a Hilbert space and let $\beta:X\to H$ be a continuous function such that the linear subspace of $H$ spanned by $\beta(X)$ is dense in $H$. I would like ...
P. P. Tuong's user avatar
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 ...
J.R.'s user avatar
  • 291
9 votes
2 answers
418 views

Reference request: Parabolic Equations

I am a PhD student working mainly on Elliptic Equations. With the other PhDs of my department, we organised a reading group, meaning that we agreed on a book we were all interested in, we meet weekly ...
Falcon's user avatar
  • 452