All Questions
3,629 questions with no upvoted or accepted answers
0
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89
views
Cousin of Fourier transform for rescaling and translating functions
Is there a cousin of the Fourier transform which obeys the following property: if I rescale and translate a function, then its transformed form, at each frequency, only depends on the original ...
0
votes
0
answers
171
views
Explanation of a step in a preprinted work
I have been studying this preprinted paper and the references therein. I believe that there are some typos; However, since I am not an expert yet, I would like to make sure I am correct.
I do not ...
- 159
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0
answers
126
views
On the convergence of operators and their spectra
We consider a sequence of operators $\{L_n\}_{n=1}^\infty$. Each operator $L_n$ is a densely defined (possibly unbounded) closed linear operator on a real Hilbert space $H_n.$ The domain of $L_n$ is ...
- 721
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225
views
Evaluate the projection operator norm with respect to maximum norm
Consider the normed space $\left(\mathbb{R}^{3},\|\cdot\|\right)$ where $\|\cdot\|=|\cdot|_{\infty}$ is the usual maximum norm. Consider the 2 -dimensional vector subspace $\left \{ (x,y,z):x+y+z=0 \...
- 111
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129
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Dirac operator on 4-dimensional rectangle with the periodic boundary conditions is self-adjoint? What is its spectrum?
Let us think of the Euclidean Dirac operator $iD^k \gamma_k$ on the rectangle $[-1,1]^4$ with the periodic boundary conditions.
The covariant derivative $iD^k$ carries a gauge potential term and we ...
- 3,477
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0
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83
views
Partial derivative of the Bessel's operator
Let $J^s = (I- \Delta)^{\frac{s}{2}}$ where $\Delta$ is the Laplacian, and $w(x,y) \in L^2(\mathbb{T}^2)$. During my study to the paper, https://arxiv.org/pdf/1809.02027.pdf, the author stated that
$$\...
- 159
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answers
71
views
Sufficient condition to be increasing, following a vector field
Let $f\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R})$ $(n\geq 1)$ be an observable, and let $v^1,v^2\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ be two vector fields such that for any $(x^1_t)_{t\geq 0}$ ...
- 449
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0
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241
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Can a non-reflexive space embed into a reflexive space?
My question is inspired from the concept of super-reflexivity which was defined by James here: https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/superreflexive-banach-...
- 11
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57
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Existence of measure-preserving Lagrange flow for inhomogeneous transport equation
I asked this question on stackexchange:
Let us consider the Cauchy problem for the transport equation
$$ \partial_t \varphi + b\cdot \nabla \varphi= f \text{ in } (0,T)\times\mathbb{R}^3,\\ \varphi(0,...
- 173
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122
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Sharp, salient and opposite cones
I have been reading about star shaped sets and support cones from this article.
Can anyone please help me with examples the difference between a sharp and dull cone.
How come a salient cone has a ...
- 27
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202
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Regarding definition of convex cone and apex
I have been reading about star shaped sets and support cones from this article. I am wondering about the definition of the cone as to why is it defined this way? Why is it $C-a$? I am familiar with ...
- 27
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162
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Compact embedding of anisotropic Sobolev space
I am wondering if the embedding from $W^{2,1}_p(\Omega \times [0,T])$ to $C^{\alpha,\alpha/2}(\Omega \times [0,T])$ is compact, for some suitable domain, $p$ and $\alpha$. I have found some results. I ...
- 54
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94
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Is the space of affine continuous functions a Baire space
Let $\Omega$ be a compact convex set in q linear normed space. Let $A(\Omega)$ be the space of affine continuous real-valued functions. My question is whether the space $A(\Omega)$ is a Baire space? ...
- 675
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143
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Minimax problem : uniqueness of a solution
Let $n\geq2$. Is it true that the minimax problem:
$$
\min_{p\in\mathcal{P}}\max_{H\in\mathcal H}p^tH\bar{p},
$$
where
$\mathcal H\subset\mathcal{M}(n)$ is a strictly convex bounded subset of ...
- 4,034
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0
answers
113
views
Existence of distance-preserving mappings for general norm in vector space
We say a mapping $f:\mathbb R^n\to \mathbb R^n$ be 1-Lipschitz with respect to a norm $\|\cdot\|$ if $\|f(x)-f(z)\|\le\|x-z\|$ holds for all $x,z\in\mathbb R^n$. Such a mapping are sometimes called a ...
- 601
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52
views
How to know if two special functions are related by an elementary function?
Suppose I have two special functions $f_1$ and $f_2$. Is there an algorithm which can tell me whether there exists elementary $g$ such that $f_1 = g\circ f_2$? Furthermore, is there any possibility to ...
- 333
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540
views
The definition of essential spectrum for general closed operators
I've asked this problem in MSE several days ago, see here. But there is no reply up until now. Maybe I wrote things too complicated there and so I'll write a very clean problem here. For background ...
- 1
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43
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Minimal condition on set for an optimisation problem
We fix $\Omega \subset \mathbb{R}^{2}$ an open set. My question is: what are the minimum conditions we need on $E \subset \Omega$ such that the following optimisation problem:
$$
\sup\{ \int_{E}(\...
- 181
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210
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Reed-Simon Vol. IV: Question regarding convergence of eigenvalues
I am reading through Chapter XIII.16 of Reed and Simon's Methods of Modern Mathematical Physics IV: Analysis of Operators about Schrödinger operators with periodic potentials. Since the topic is kind ...
- 91
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answers
21
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Estimatives for elliptic systems involving the laplacian
Considering the problem
\begin{equation}
\left\{
\begin{array}[c]{11}
\Delta(\Delta \chi -\chi) = 0 & \text{in } \Omega, \\
\Delta \chi -\chi = h_2 - h_1, & \text{on } \partial\Omega \\
\end{...
- 101
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0
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75
views
Extracting the point mass measure of some type of positive measures
Let us consider the measure algebra $M(\mathbb{R})$ consisting of all Radon measures on the reals.
Let $\delta_0$ be the point mass measure concentrated on 0, which is also the multiplicative ...
- 4,058
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votes
0
answers
80
views
Cyclicity of composition operators
Let $E=C([0,1]^m,\mathbb{R}^n)$ where $K=[0,1]^m$ where $E$ has the compact convergence topology. Recall that for a function $f:[0,1]^m\rightarrow [0,1]^m$ the associated composition operator $C_f$ ...
- 5,405
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votes
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answers
116
views
Is there a proper term for a "continuum-convex" set?
Let $X$ be a Banach space, and for any compactly supported Borel probability measure $\mathbb{P}$ on $X$, define the mean $\mu_\mathbb{P}$ by $\mu_\mathbb{P}=\int_X x \, \mathbb{P}(dx)$.
I want to say ...
- 708
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votes
0
answers
40
views
Time regularity of traces
I have a question about the time regularity of the traces in one dimension.
Suppose I have a function space $$X = C^1([0,T];L^2(0,1))\cap C([0,T],H^1(0,1))$$ and I define an operator $E$ on $X$ by $(...
- 11
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votes
0
answers
70
views
Normal vector to a level set and fractional Laplacian
Let $U=\{u\le0\}$ and $\partial U=\{u=0\}$. Suppose $\nabla u$ does not vanish on $\partial U$. Then the (canonical extension of the) normal vector field to $\partial U$ (pointing to the interior of $\...
- 99
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112
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How to prove or disprove $ |r'(a)| \leq c_1 \sup_{x \in [a-1/2,a+1/2]} |r(x)|$?
Let $ A = \begin{bmatrix}
a & 1 \\ 0 & a
\end{bmatrix}$ be a Jordan matrix with $ -1 < a < 1 $.
Let $r(z) = \frac{p(z)}{q(z)}$ be an irreducible rational function, where $p(...
- 101
0
votes
0
answers
141
views
Extending an unbounded dense linear functional
Let $H$ be an infinite dimensional separable Hilbert space over $\mathbb{C}$
Let $V \subset H$ be a dense subspace of $H$
Let $f : V \to \mathbb{C}$ be a unbounded functional linear
My question is:
Is ...
- 433
0
votes
0
answers
103
views
A question on the Haar basis for $L_{1}[0,1]$
Let $(x_{n})_{n=1}^\infty$ be a basis for a Banach space $X$. It is important to know the exact expression of the norm of $\|\sum_{i=1}^{n}a_{i}x_{i}\|$ for all $n$ and all scalars $a_{1},a_{2},\ldots,...
- 3,343
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112
views
Fixed point of a contraction map
This question is a continuation of Is this a contraction mapping for small $T$?
Set, for $T, m>0$, $H^m_T:=\{h:[0,T]\to [0,m]:~ h,~h' \mbox{ are both continuous on } [0,T]\}$ endowed with the norm
$...
- 1,334
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votes
0
answers
148
views
About the theorem of Weierstrass?
Is $E=Vect\{1,x,x^2,...,x^{2^n},...\}$ dense in $C([0,1])$ for the uniform norm?
While looking for a short proof for Weierstrass' theorem, I came across this justification(*) (which shows this result)...
- 4,074
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votes
0
answers
84
views
Determining the tails of a convolution from its behavior on a compact set
Let $p$ be a smooth (say, $C^\infty$, but this is not crucial) density on the interval $I=[0,1]$ and $g_\sigma$ be the density of $N(0,\sigma^2)$. Define $f=p\ast g_\sigma$. To what extent does the ...
- 19
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56
views
Existence of minimal subset of dual ball such that the intersection of kernels is trivial
Let $(V, \lVert \cdot \rVert)$ be a separable Banach space and let $B_{V^*}$ denote the closed ball in the dual $V^*$. Suppose we have a family $C \subseteq B_{V^*}$ such that $\bigcap_{\Lambda \in C} ...
- 820
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votes
0
answers
198
views
eigenvalues of the product of a unitary with a diagonal
In $M_n(\mathbb{C})$, suppose $U$ and $D$ are a unitary and an invertible diagonal matrix with eigenvalues $\{e^{i\theta_1},\cdots,e^{i\theta_n}\}$ and $\{e^{i\eta_1},\cdots,e^{i\eta_n}\}$ ...
- 4,058
0
votes
0
answers
176
views
A convergence question in $L^2$ construction of Brownian motion
I feel confused with a particular step in the $L^2$ consturction of Brownian motion.
Let $\{\xi_n \sim N(0,1)\}_{n\geq 1}$ be a sequence of i.i.d Gaussian random variables on some probability space $(\...
- 227
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votes
0
answers
165
views
Compact embedding of Lipschitz continuous functions
Let $(X,d,\mu)$ be a metric measure space, not necessarily with $\mu(X)<\infty$. I would like to study the embedding of $W^{1,2}(X)\cap \mathrm{Lip}(X)$ into $L^2(X)$. Are there simple conditions ...
- 3,388
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votes
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109
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Generator problem for reduced group C*-algebra
(Not sure if it is appropriate or not, if no I will delete the post)
Recently I am concerned about the number of generator of $C^{*}_{r}(\mathbb{F}_{k})$, the reduced group algebra of the free group, ...
- 523
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votes
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199
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Is the kernel $\vert d_X - d_Y \vert^p$ conditionally negative definite?
Given two finite metric spaces $(X,d_X)$ and $(Y,d_Y)$, for $p > 0$, define the kernel ($4$-D tensor) $K$ on $(X \times Y)^2$ by:
$$K\big( (x_i, y_k), (x_j, y_l) \big) = \vert d_X(x_i, x_j) - d_Y(...
- 111
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votes
0
answers
109
views
The non empty set of accumulation points of a bounded linear operator is the spectrum of another operator
Let $X$ be an infinite dimensional Banach space, and let $T \in L(X)$ such that the set of accumulation points of $T$ is non empty, i-e $\mbox{acc}\,\sigma(T)\neq 0.$\
Is there a Banach space $Y$ ...
- 1
0
votes
0
answers
129
views
Bounding trace operator from below
In a paper, I've read the following thing. Here $\Omega$ is a smooth domain
From the standard trace theorem we know there exists a bounded linear operator $$\gamma: H^1(\Omega) \rightarrow H^{\frac{1}...
- 111
0
votes
0
answers
186
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Critical exponent in Sobolev scale and singular continuous measure
Recently I started the following discussion about Sobolev spaces where I was interested in finding the maximal $s \in \mathbb{R}$ such that given function/distribution belongs to the Sobolev space $H^...
- 9,330
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72
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Does $L^{\infty}[0,1]$ admits infinitely many densely defined derivations in weak* topology?
To clarify the question. First we define what is densely defined derivation. A densely defined derivation $\delta:D(\delta):\rightarrow L^{\infty}[0,1]$ where $D(\delta)$ is a dense subalgebra( in ...
- 523
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votes
0
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96
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Books on limiting properties of matrices with growing size
This question has been posted on Math-Se previously.
I am studying asymptotic properties of the Projection Matrix
$$
H_n=X'(X'X)^{-1}X
$$
By the Gerschgorin disc theorem, the bounds on the ...
- 101
0
votes
1
answer
431
views
Approximation of the product $(\bar{z} - a)^{-1} \cdot (z - b)^{-1}$
$\def\zbar{\smash{\overline z}\vphantom z}$I would like to construct an approximation of the product
\begin{equation}
f(z) = \frac{1}{\zbar-a} \frac{1}{z-b},
\end{equation}
where $a, b \in \mathbb{C}$,...
- 11
0
votes
0
answers
197
views
Link between a categorical and an algebraic characterization of (infinite-dimensional) Hilbert space
On one side, a very recent paper of Chris Heunen and Andre Kornell "Axioms for the category of Hilbert spaces" (Arxiv:2109.7418v1 latest Arxiv version) offers a characterization of the ...
- 2,655
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votes
0
answers
85
views
How can we define $\chi_{\Omega}(A)$?
I was reading Spectrum and dynamics where Paolo Facchi discusses projection-valued measures and integrals. The discussion and constructions are all based on the fact that one can define characteristic ...
- 1,305
0
votes
0
answers
95
views
Empirical estimation of Brenier map from data
Let $f:\mathbb R^d \to \mathbb R$ be a "nice" (say, continuous) function define $A = A_f := \{x \in \mathbb R^d \mid f(x) \ge 0\}$ and $B =B_f:= \{x \in \mathbb R^d \mid f(x) \le 0\}$, and ...
- 6,853
0
votes
0
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114
views
Analysis, matrix exponential, infimum and limit
I was working in this problem for a long time and I didn't have success.
Someone could help me, please?
The problem:
Let $f: \mathbb{R}^{n^2} \times \mathbb{Z}^{n} \longrightarrow \mathbb{R}$ defined ...
0
votes
0
answers
85
views
An amenable operator algebra has the total reduction property
This is from
https://www.cambridge.org/core/services/aop-cambridge-core/content/view/CB20539885C03522D141C34024707702/S1446788700014026a.pdf/div-class-title-operator-algebras-with-a-reduction-property-...
- 101
0
votes
0
answers
109
views
Operator algebra on an invariant subset
In Rickart, page 50 Theorem 2.2.1, the statement is made: A linear subspace $\mathfrak{M}$ of the algebra $\mathfrak{A}-\mathfrak{L}$ is invariant with respect to the representation $a{\rightarrow}A_a^...
- 109
0
votes
0
answers
291
views
Operator norm on tensor product of trace classes is multiplicative
Given Hilbert spaces $\mathcal H_1,\mathcal H_2,\mathcal K_1,\mathcal K_2$ and bounded linear maps $S_i:\mathcal B^1(\mathcal H_i)\to\mathcal B^1(\mathcal K_i)$, $i=1,2$ between the respective trace ...
