All Questions
13,944 questions
0
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78
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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
0
answers
99
views
Closed form for $\int_0^{+\infty} \ln^p(t) \frac{\sin^q(t)}{t^r}dt$
Do you know if there exists a closed form for the integral :
$$I_{p,q,r} = \int_0^{+ \infty} \ln^p(t) \frac{ \sin^q (t)}{t^r} dt$$
where $p$, $q$, $r$ are natural integers such as this integral ...
- 69
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 ...
0
votes
0
answers
73
views
An example of a groupoid that satisfy the following hypothesis
In the paper titled, 'Tannaka–Krein duality for compact groupoids I, Representation theory', the author proves the Peter Weyl theorem on compact groupoids. In the statement, he gives the hypothesis ...
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
2
votes
1
answer
133
views
Points of differentiability of convex functions
Let $U$ be an open neighbourhood of $0 \in \mathbb{R}^2$ and $f\colon U \to \mathbb{R}$ a convex (and bounded) function. Denote by $D \subset U$ the set of points on which $f$ is totally ...
- 2,998
0
votes
0
answers
60
views
Criteria for log-absolute-monotonicity
Consider a function $f: [0,1] \rightarrow \mathbb R$ defined by a power series $f(x) = a_0 + a_1 x + a_2 x^2 + \dots$, where all $a_i$ are positive.
Is there are any criterion in terms of the ...
- 3,475
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
7
votes
1
answer
271
views
Sequential continuity and the Axiom of Choice
It is well-known that ZF cannot prove the following:
"for a function $f$ from reals to reals and any real $x$, $f$ is continuous at $x$ if and only if $f$ is sequentially continuous at $x$."...
- 4,359
23
votes
4
answers
2k
views
Identity for an infinite product
Here is an experimental "result" exhibiting the difference of two (formal) infinite products that "almost factorizes".
QUESTION. Is this true?
$$\prod_{n\geq1}(1+x^{2n-1})^{24} - \...
- 43.2k
1
vote
1
answer
105
views
Constrained optimization over a set of functions
How to approach the following optimization problem:
$$\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{ ...
2
votes
0
answers
86
views
Higher cohomology groups for the trivial action of the reals on themselves
For a freely generated countable abelian group $A$ with the trivial action on itself ($a\cdot b = b$) the resulting cohomology groups are well-known and eventually vanish (see e.g. here). Coming from ...
- 1,411
3
votes
1
answer
109
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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 $\phi: \mathcal{B}^*\to\mathbb{R}, \...
- 171
5
votes
1
answer
379
views
Nuclear vs Banach spaces: compactness properties
A question about the meaning from following excerpt from german wikipedia adressing interesting crucial feature of nuclear spaces opposing them from Banach spaces (transl.):
While normed spaces, ...
- 6,028
2
votes
0
answers
47
views
Norm density of evaluation functionals in the space of weak$^*$ continuous multilinear functionals on products of dual Banach spaces
Let $K$ be a compact (metrizable) space and let $C(K)$ be the Banach space of continuous real-valued functions on $K$, equipped with the supremum norm. It is then well known that the dual space $C(K)^*...
- 56
-1
votes
1
answer
61
views
Asking for some references on correlations of joint optimization problems
Here are two problems that I am trying to understand, and it would be nice if someone could provide references on whether there is some structure theorem for these problems that have been studied in ...
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 ...
- 356
1
vote
0
answers
40
views
relatively weakly compact sets in the projective tensor product of $\ell_p $ and a Banach space $X$
We will use the notation in [1].
A sequence $(x_n)$ in $X$ is called weakly $p$-summable ($p\ge 1$) if $(x^*(x_n))\in \ell_p$ for each $x^*\in X^*$. Equivalently, a sequence $(x_n)$ in $X$ is ...
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(...
- 565
0
votes
0
answers
46
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What's the problem in using spanning Bessel sequences that are not frames to decompose vectors?
This is related to a question I recently asked on math.SE.
Consider a subset $G\equiv \{g_k\}_{k\in\mathbb{N} }\subseteq\mathcal H$ in a separable Hilbert space $\mathcal H$, and suppose $G$ spans the ...
- 342
0
votes
0
answers
50
views
About extreme case on complex interpolation
I'm trying to prove some equality of spaces via complex interpolation with the usual Calderon functor $[,]_\theta$. If $(E_0,E_1)$ is a compatible couple, it is known that $$[E_0,E_1]_j, j=0,1,$$ is a ...
1
vote
0
answers
86
views
Gamma convergence via density argument: Looking for references
I am looking for a reference or result dealing with Gamma via density argument.
Let me elaborate more my wish. I am actually trying to establish the Gamma convergence (precisely only the liminf) of a ...
- 1,101
2
votes
0
answers
83
views
3/2 Sobolev Norm on the boundary of a bounded open subset of $\Bbb R^n$
Let $\Omega\subset\mathbb{R}^{n}$ be a open bounded set and $\partial\Omega$ be the boundary of $\Omega$.
Following the reference text by Alois Kufner, Oldřich John and Svatopluk Fučík, Function ...
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 $f : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$:
$$
f(\mathbf{v},\mathbf{w}) = \|\mathbf{v}\|\|\mathbf{w}...
- 6,351
1
vote
0
answers
174
views
Interpolation of Sobolev spaces with constraints
Let us consider a real interval $[0, L]$, with $a\in (0, L)$, and let $I_1=(0, a)$ and $I_2=(a, L)$. We denote by $H^k(I_1)$ and $H^k(I_2)$ the usual Sobolev spaces, defined for $k\in \mathbb{N}$. Now,...
- 81
3
votes
1
answer
67
views
Infinite direct sum decomposition of the heat semigroup on $\mathbb R$
This question is based on a very similar question posted by me yesterday. A very nice solution was provided by Aleksei Kulikov. Here I modify my question slightly.
Let $Q_t$ be the heat semigroup on $...
- 407
0
votes
0
answers
16
views
Representing a periodic strip operator as a tensor product of operators
I hope this question is not trivial, but here goes. I want to consider a bounded operator on $\mathcal{H}=\ell^2(\mathbb{Z}\times \{0,...,N-1\})$ that is a discrete Schrodinger like operator.
...
- 633
7
votes
1
answer
346
views
Mean Cauchy sequences
Let $X$ be a complete metric space. Suppose a sequence of elements $x_n$ is Cauchy in mean, in the sense that
$$\lim_{K \to \infty} \limsup_{N, M \to \infty} \frac{1}{NM} \sum_{i = K+1}^{K + N} \sum_{...
- 6,215
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
$$\oplus_{...
- 407
2
votes
1
answer
315
views
Are surjective homogeneous maps open at zero?
I'm asking this question as a follow-up inspired by this one: An open mapping theorem for homogeneous functions?
I'm actually wondering whether there exists an homogeneous map $f:\mathbb R^n\to\mathbb ...
- 311
7
votes
1
answer
224
views
Does the decomposability of $\mathbb{R}$ imply analytic LLPO?
By "BISH" I mean constructive mathematics without axiom of countable choice.
By $\mathbb{R}^f$ I mean real numbers as fundamental sequences of rational numbers and by $\mathbb{R}^d$ I mean ...
3
votes
1
answer
240
views
Solutions and asymptotics of the ODE $ f''=f^{-\alpha} $
Consider the ODE $ f''=f^{-\alpha} $, where $ \alpha>1 $ and $ f>0 $ in $ \mathbb{R} $. Assume that for $ [f]_{\frac{2}{\alpha+1}}\leq A $, where $ A>0 $ is a constant and
$$
[f]_{\frac{2}{\...
- 1,219
0
votes
1
answer
117
views
Validity of approximation method for von Mangoldt function
I'm working on a problem involving the pointwise almost everywhere convergence of multilinear ergodic averages with the von Mangoldt function inspired by this paper. Specifically, I'm looking at ...
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 ...
- 21
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 ...
- 151
0
votes
1
answer
153
views
Lebesgue measure of the level set of sum of two nonnegative functions
Let $f, g:\mathbb{R}^n\to \mathbb{R}$ be nonnegative functions such that $g$ is a strictly positive homogeneous function. As commented by Fedor Petrov below, one may not have that for any $\lambda>...
- 407
3
votes
1
answer
79
views
Closed linear span of the range of $\boldsymbol f$ and Pettis integrals of $\boldsymbol f$
Let $X$ be a noncompact locally compact topological space, let $H$ by a complex Hilbert space and let $\boldsymbol f:X\to H$ be a continuous function vanishing at infinity whose support is equal to $X$...
- 407
5
votes
0
answers
608
views
What is the correct $L^\infty$ limit of this strange variational problem, and what does it encode?
1. On the $L^\infty$ calculus of variations:
The field known as the $L^\infty$ calculus of variations is a relatively new field that concerns itself with minimising functionals involving the supremum ...
- 6,215
4
votes
1
answer
249
views
Does this functional admit an absolute minimizer?
This is a close relative of the following problem.
Let $\Omega$ be an open, bounded subdomain of $\mathbb R^n$ with smooth boundary, and $f_i \in W^{1, \infty} (\Omega)$ a sequence of functions ...
- 6,215
3
votes
2
answers
614
views
Should coffee machines be placed at the region's boundary?
This is a continuation of Should coffee machines be deconcentrated?
Recall that some region is denoted by convex and compact $E\subset \mathbb R^2$. $N\ge 1$ coffee machines are provided for the ...
- 1,399
1
vote
1
answer
122
views
distance in the matrix algebra w.r.t. the nuclear norm
Let $\varphi\in\mathcal{M}_n(\mathbb{C})$ and let $Z:=\mathbb{C}\cdot I=\{zI\colon\,z\in\mathbb{C}\}$ be the one-dimensional subspace spanned by the identity matrix $I$. Let moreover $\|\cdot\|_{\...
- 375
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 ...
- 407
-1
votes
1
answer
122
views
Divergent summation [closed]
Let $(x_i)_{i=0}^\infty$ be a sequence such that $0<x_i<1\ \forall i \in \mathbb{N} \cup {0}$.Consider the following series:
$$\sum_{i=1}^\infty \frac{x_i}{\left(\sum_{k=0}^{i-1} x_k \right)^2}.$...
- 21
8
votes
0
answers
115
views
optimal regularity for the Neumann heat equation on Lipschitz domains
$\newcommand{\R}{\mathbb R}$Let $\Omega\subset\R^d$ be a bounded Lipschitz domain, possibly non-convex (but not too nasty either, whatever that means). I am looking for well-posedness and optimal ...
- 5,380
1
vote
0
answers
72
views
How to understand "sparse graph limits"
For an $n$-vertex graph $G$, we say it is a sparse graph if $e(G)=o(n^2)$. Otherwise if $e(G)=\theta (n^2)$, we say it is a dense graph.
For a sequence of dense graphs $G_1,G_2,\dots,$ we know that it ...
- 349
7
votes
1
answer
179
views
More on the Gram matrix of $6$ unit vectors in $\Bbb R^3$
Let $G=(g_{ij}\colon i,j=1,\dots,6)$ be the $6\times6$ Gram matrix of $6$ unit vectors in $\Bbb R^3$. Let
$$u:=\sum_{1\le i<j\le 6}g_{ij}^2,\quad v:=\sum_{1\le i<j<k\le 6}g_{ij}g_{ik}g_{jk}.$$...
- 128k
0
votes
0
answers
96
views
Sufficient condition for weak convergence in Banach spaces
The question is quite elementary but nonetheless no proof or counter example comes to mind immediately.
Suppose that $X$ is a Banach space and $\{x_n\}$ is a sequence in $X$ such that $(x_n,y)$ ...
2
votes
1
answer
231
views
Is Boltzmann entropy well-defined for arbitrary probability density function?
$\newcommand{\bR}{\mathbb{R}}\newcommand{\diff}{\mathop{}\!\mathrm{d}}$ We define a continuous function $\varphi : \bR_+ \to \bR$ by
$$
\varphi (s) :=
\begin{cases}
0 &\text{if} \quad s =0 , \\
s \...
- 835
1
vote
0
answers
55
views
Characterizing one-sided M-projections on real C*-algebras
Let $A$ be a real C*-algebra, and let $P: A \to A$ be a bounded linear projection. We say that $P$ is a left M-projection if the map
$$
v_P: A \to C_2(A), \quad x \mapsto \begin{pmatrix} P(x) \\ x - P(...
- 11