All Questions
12,105 questions
4
votes
3
answers
241
views
An uncountable measurable subset of $\Bbb R$ containing no nonempty perfect set
$\newcommand\R{\Bbb R}$Assuming the axiom of choice, is there an uncountable Lebesgue-measurable subset $S$ of $\R$ that contains no nonempty perfect set?
Of course, such a set $S$, if it exists, ...
- 13.4k
1
vote
0
answers
45
views
Measurability of a map involving probability measures
Let $X$ be a metrizable topological space and $\mathscr B_X$ the Borel $\sigma$-algebra on it. Let $\Delta X$ denote the set of probability measures on $(X,\mathscr B_X)$, and let $\mathscr B_{\Delta ...
- 405
1
vote
1
answer
312
views
Proof of the Dunford-Pettis theorem in the context of probability spaces
I'd like to know if there's a proof of the Dunford-Pettis theorem without using relatively advanced theorems of functional analysis such as Eberlein–Smulian Theorem. Since I'm only interested in ...
CommunityBot
- 1
3
votes
1
answer
74
views
How irregular can the set of points of non-differentiability for an L1 function's primitive F get, before the FTC fails?
A Fundamental Theorem of Calculus for Lebesgue Integration, J. J. Koliha begins with the passage
Lebesgue proved a number of remarkable results on the relation between integration and differentiation....
- 831
25
votes
6
answers
15k
views
Does every distribution define a Radon measure?
On the one hand, Wikipedia suggests that every distribution defines a Radon measure:
http://en.wikipedia.org/wiki/Distribution_(mathematics)#Functions_as_distributions (revision from February 2010, ...
- 12.9k
9
votes
1
answer
292
views
What are the points of the algebra of polynomial functions on an arbitrary vector space?
Let $V$ be an arbitrary vector space over some field $\mathbb{K}$ (UPD: of characteristic 0), $V^*=\mathrm{Hom}(V,\mathbb{K})$ its linear dual. Let $\mathrm{Sym}_\mathbb{K}(V^*)$ be the free ...
- 480
31
votes
4
answers
8k
views
Counterexamples to differentiation under integral sign?
I'm exploring differentiation under the integral sign (I want to be much faster and more assured in doing this common task). So one thing I'm interested in is good counterexamples, where both ...
- 831
-3
votes
0
answers
66
views
Exercise generalizing (related to) Hölder's inequality
I came across this exercise and feel absolutely stuck:
Let $p, q, r \in (1, \infty]$ be such that $1/p + 1/q = 1 + 1/r$. Suppose that $F : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ satisfies ...
- 12.9k
3
votes
0
answers
67
views
Effective action of unbounded operators on subspaces outside their domains of definition
Consider a densely defined, self-adjoint operator
$$
H: \mathcal{D} \rightarrow \mathscr{H}.
$$
Assume for simplicity that $H$ is nonnegative.
We want to effectively restrict this operator $H$ to a ...
- 133
2
votes
0
answers
104
views
Uncertainty principle: minimize $\int_{-\infty}^\infty |t| |\widehat{f}(t)|^2 dt$ for $f$ of compact support
This is a question of uncertainty-principle type stemming from Eigenvalue of a convolution and a restriction?
Let $f:\mathbb{R}\to \mathbb{R}$ be even, absolutely continuous and supported in $[-\frac{...
- 20.2k
0
votes
0
answers
64
views
$L_1$ norm of $f\in L^1(\mathbb{R}^n)$ compactly supported and its change of variable
Let $M\in\mathbb{R}^{n\times n}$ be an invertible matrix, denote its induced linear map on $\mathbb{R}^n$ also by $M$, and let $f\in L^1(\mathbb{R}^n)$ be compactly supported.
I am wondering if we can ...
- 91
3
votes
1
answer
353
views
Sequential separability on $C_p(X)$
Definition. Let $E$ be a topological space. Suppose that $E$ contains a sequence $\{x_n\}$ such that for every $x\in E$, there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ with $x=\lim x_{n_k}$. ...
- 4,713
4
votes
1
answer
273
views
Eigenvalue of a convolution and a restriction?
Let $\epsilon>0$ be small. Let $\eta(t) = \frac{2\epsilon}{\epsilon^2+(2\pi t)^2}$ (the Fourier transform of $x\mapsto e^{-\epsilon |x|}$). Let $V$ be the space of integrable, bounded functions $f:\...
- 20.2k
3
votes
0
answers
82
views
+50
Tight upper bound for $m[Q^k - Q^{k+1}]$ for completely positive linear maps
Let $m: \mathcal{L}(\mathbb{R}^{d \times d}) \to \mathbb{R}$ be the function
$$
m[H] = \frac{\lambda_{\max}(H[\mathbf{I}])}{\lambda_{\max}(H)},
$$
where $\lambda_{\max}$ denotes the largest eigenvalue....
- 673
0
votes
1
answer
115
views
Bounding $(\int_{S^1}\left|\partial_r u(r\omega)\right|^2 d\omega)^{1/2}$ with $(\iint \frac{|u(x)-u(y)|^2}{|x-y|^{2+2s}} dxdy)^{1/2} $?
The following inequality is trivially true
$$\left(\int_{S^1}\left|\frac{\partial u}{\partial r}(r\omega)\right|^2 d\omega\right)^{1/2} \le \left(\int_{S^1}\left|\nabla u(r\omega)\right|^2 d\omega\...
CommunityBot
- 1
0
votes
1
answer
546
views
Orlicz–Sobolev spaces
Let $A$ be an N-function and suppose that
$$\int^{+\infty}_1\frac{A^{-1}(\tau)}{\tau^{1+\frac{1}{n}}}d\tau=+\infty. $$
We denote by $\widehat{A}$ an N-function equal to $A$ near infinity and $\widehat{...
CommunityBot
- 1
2
votes
1
answer
431
views
Density of $w^*$-support points
I am looking for a simple proof of the following theorem — wasn't able to come up with one myself. Should be a use of the Bishop–Phelps theorem, in some way:
Let $X$ be a Banach space, $D \subset X^*$ ...
CommunityBot
- 1
3
votes
0
answers
90
views
About BMO space on smooth open bounded domain
Let $\Omega$ be any open domain in $\Bbb R^d$.
Define the $\text{BMO}(\Omega)$ space as
$$ \text{BMO}(\Omega)= \big\{u\in L^1_{loc}(\Omega)\,\,:\,\, |u|_{\text{BMO}(\Omega)} <\infty \big\},
$$
...
- 1,101
2
votes
0
answers
224
views
A deceptively simple regularity problem for functions on the plane
By various meanderings and toying with simpler problems, my current research has lead me to the following quite straightforward question, which I am wholly unable to answer:
Consider a twice ...
- 989
2
votes
0
answers
77
views
Function that is (essentially) a self-convolution but not a multiple of a self-convolution
Call a function $F:\mathbb{R}\to C$ nice if it is of the form $F = f\ast \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$. (Of course nice functions are precisely those whose Fourier transform is ...
- 20.2k
1
vote
1
answer
294
views
Regarding subspace generated by the polynomial multiples of outer functions
Let $\mathbb{D}$ and $\mathbb{T}$ denote the open unit disk and unit circle in $\mathbb{C}$ respectively. We write $Hol(\mathbb{D})$ for the space of all holomorphic functions on $\mathbb{D}.$ The ...
CommunityBot
- 1
12
votes
1
answer
394
views
Is $X\times X$ homeomorphic to $X$ for a space of probability measures?
Let $\mathcal M_1(S)$ be the (compact, metrizable) space of probability Borel measures on the circle $S=\{z\in\mathbb C: |z|=1\}$ with its weak $*$ topology, so $\mu_n\to\mu$ if and only if
$$
\int_S ...
- 4,461
0
votes
0
answers
46
views
Amenability of locally convex algebras
Let $A$ be an amenable Banach algebra, and let $A_w$ denote $A$ with the weak topology. Clearly, $A_w$ is a Hausdorff locally convex algebra (l.c.a.).
Q0: Is $A_w$ amenable as a l.c.a. in the sense ...
- 2,605
0
votes
1
answer
157
views
Weak convergence of $f(x,e^{itx})$
This is the desired result (what I want to prove):
$$f(x,e^{itx})\overset{t\to\infty}{\rightharpoonup}\frac{1}{2\pi i}\oint_{|z|=1}\frac{1}{z}f(x,z)dz \tag{1}$$
Given that $f\in C([a,b]\times\{e^{i\...
- 233
4
votes
1
answer
195
views
Asymptotic spectrum of a complex Sturm-Liouville differential operator
Let $\varepsilon > 0$ and consider the (complex) Sturm–Liouville differential operator on $[0,1]$ given by
$$
\mathcal{L}_\varepsilon f(x) = \varepsilon^2 f''(x) + i V(x) f(x),
$$
with Neumann ...
CommunityBot
- 1
1
vote
0
answers
85
views
Density of a subset of Schwartz space in the fractional Sobolev space
It is known that the Schwartz space $\mathcal{S}(\mathbb{R}^N)$ is dense in the fractional Sobolev space $H^s(\mathbb{R}^N)$, (where $0<s<1$), as $C_{c}^{\infty}(\mathbb{R}^N) \subset \mathcal{S}...
4
votes
2
answers
871
views
Decay of eigenfunctions for Laplacian
Consider the discrete second derivative with Dirichlet boundary conditions on $\mathbb C^n$.
Its eigendecomposition is fully known:
see wikipedia
It seems like the largest eigenvalue $\lambda_1$ is ...
CommunityBot
- 1
3
votes
0
answers
61
views
On the relative growth rates of occupancy times in ergodic theory
Let $(X, \mathcal{F}, \mu)$ be a general measure space, and let $T: X \to X$ be a measure-preserving transformation on $X$. Assume that $T$ is ergodic and satisfies the property that, for any set $A \...
- 105
6
votes
0
answers
105
views
Dependence on Urysohn's Lemma in Cartan's Construction of Haar Measure
This question was posted by someone else on stackexchange three months ago, but no one has answered as of yet:
Cartan's 1940 paper, Sur la mesure de Haar, claims to provide a proof of the existence ...
- 85
4
votes
0
answers
90
views
Hölder stability of the PDE $\partial_t u (t, x) = \operatorname{div} \{ a (t, x) \nabla u(t, x) \}$
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\eps}{\...
- 835
1
vote
1
answer
151
views
Is smoothness preserved under an isometric isomorphism?
Let $(X, \|.\|_1)$ is isometrically isomorphic to $(X, \|.\|_2)$ and $\|.\|_2\leq \|.\|_1$. Assume that $x_0$ is a smooth point of $(X, \|.\|_1)$ and $\|x_0\|_2=1$. According to the definition of a ...
- 128k
2
votes
0
answers
157
views
About the 7.3.5. Corollary of the book "Measure Theory" by V.I. Bogachev
According to the 7.3.5. Corollary of the book "Measure Theory" by V.I. Bogachev we have the following result:
Let $(X,\tau)$ be a completely regular space and let $\Gamma$ be a
family of ...
- 627
2
votes
0
answers
62
views
Bessel spaces and Triebel Lizorkin
It is known that bessel potential spaces $H^{s,p}$ coincide with Triebel-Lizorkin spaces $F^{s}_{p,2}$ for $s\in \mathbb{R}$ and $1<p<\infty$. Im wondering what can be said por $p=1$ and $p=\...
1
vote
0
answers
58
views
duality of sobolev spaces. Representation of elements in the dual
I'm trying to understand $(W_0^{1,p} (Ω))^*=W_0^{-1,p^*} (Ω)$, and what a proper representation of its elements is. I understand the basics such as: if $f∈L^{p^*} (Ω)⇒f∈W_0^{-1,p^*}(Ω)$ and the ...
1
vote
1
answer
102
views
Every tight $\tau$-additive finite measure is Radon
According to the 7.2.2 Theorem of the book "Measure Theory" written by V.I. Bogachev, every tight $\tau$-additive finite measure is Radon. The proof says: "The restrictions of a $\tau$-...
- 3,768
0
votes
0
answers
42
views
questions on stochastic kernels and pushforward operator
Let $f:X \rightarrow \Delta (Y)$ and $g:X \rightarrow \Delta (X)$ be two kernels. For any bounded measurable function $h_Y:Y \rightarrow \mathbb{R},$ define $F(h_Y):X \rightarrow \mathbb{R}$ such that ...
- 1
2
votes
1
answer
236
views
Self-adjointness of generator and semigroup of an SDE
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\eps}{\...
- 6,696
0
votes
0
answers
90
views
How to show a point is a weak* -weak continuous for the identity map on $X_1^*$ or on $X_1^{**}$?
I am trying to understand the Remark 3.2 mentioned in the paper titled as "On Weak*
-Extreme Points in Banach Spaces" written by S. Dutta and T. S. S. R. K. Rao (http://library.isical.ac.in:...
- 113
-3
votes
2
answers
195
views
Which self homeomorphisms preserve measure on a torus, apart from affine? [closed]
Which self homeomorphisms preserve measure on a torus, apart from affines? Affine is the composition of rotation and automorphism. Measure is the Lebesgue measure.
- 17.9k
6
votes
1
answer
956
views
a claim for a proof of the invariant subspace problem [closed]
Recently four mathematicians claimed to have proven the invariant subspace problem, which is the problem that states
Does every bounded operator on a separable Hilbert space have a non-trivial ...
- 12.5k
1
vote
1
answer
379
views
Creating an inverse system which "stratifies density"
Setting:
Let $X'$ be a dense subset of an infinite-dimensional Fréchet space $X$ and suppose that $(X_n')_{n \in \mathbb{N}}$ is a nested sequence of non-empty subsets of $X'$ satisfying
$$
\bigcup_{n ...
CommunityBot
- 1
29
votes
3
answers
3k
views
Is there a probability theory developed in intuitionistic logic?
Since Boole it is known that probability theory is closely related to logic.
According to the axioms of Kolmogorov, probability theory is formulated with a (normalized)
probability measure $\mbox{...
CommunityBot
- 1
0
votes
1
answer
142
views
Why are the homeomorphisms from the unit circle to the unit circle preserving measure affine? [closed]
Why are the homeomorphisms from the unit circle to the unit circle preserving measure affine? The affine is composition of rotation and continue automorphism.
- 128k
18
votes
1
answer
2k
views
Function of two sets intersection
Let $U$ be the set of all nonempty subsets of $[0,1]$ that are a union of finitely many closed intervals (where an "interval" that is a single point does not count as an interval). Does ...
CommunityBot
- 1
2
votes
1
answer
605
views
Strong convergence of differential quotient in $L^2(0,T;V^*)$
I have got a problem regarding the weak differentiability of Bochner-integrable functions. Let $(V,H,V^*)$ be a Gelfand-triple and
\begin{align*}
w \in W(0,T) := \{w\in L^2(0,T;V) ~\vert~ \exists w' \...
CommunityBot
- 1
0
votes
1
answer
124
views
Holomorphic functions of certain blow up at origin
Suppose that $D=\{z\in \mathbb C\,:\, |z|\leq 1\}$ and let $f$ be holomorphic on $D\setminus\{0\}$ such that $|f(z)|\leq e^{\frac{1}{|z|}}$ for all $0<|z|\leq 1$ and assume additionally that $\lim\...
- 6,476
0
votes
0
answers
71
views
Fourier decay implies what kind of regularity
We consider a function $f:\mathbb R^2 \to \mathbb C$ that is compactly supported and bounded. In addition, we know that
$$\lim_{\vert x\vert \to \infty} \vert x \vert^2 \vert \hat{f}(x)\vert =0,$$
...
1
vote
0
answers
146
views
integral over the unit sphere of $\Bbb C^n$
Please, is there a way to calculate this integral
$$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$
where $ z $ is a fixed point in the complex unit ball ...
- 511
0
votes
1
answer
239
views
Are ALL linear functionals on $C[0,1]$ generated by measures? [closed]
Consider derivative of the convolution of a given function $f(\cdot)$ with a fixed $C^\infty$ function $s(\cdot)$, evaluated say at $1/2$. Is there a measure which generates the functional so defined?
- 2,183
2
votes
0
answers
63
views
Lipschitz retraction constant of $B^+$ into $S^+$ in $L^2([0,1])$
In Hilbert space modeled by $L^2([0,1])$ we can define a set $B^+=\{x\in B(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ and $S^+=\{x\in S(0,1): x(t)\geq0 \quad \forall t\in [0,1] \}$ where where $B(...
