All Questions
155 questions
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Number theory on Banach space $L^2(\mathbb R)$ meets linear independence?
Consider an orthonormal basis $(\varphi_k)$ of $L^2(\mathbb R)$ with Lebesgue measure.
I came along a nice number theoretic question in analysis:
Write $$f_k(x):=\int_{\left\lvert y \right\rvert \...
- 54.2k
4
votes
1
answer
860
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Lebesgue's integrability condition in several variables
The well known Lebesgue's condition of Riemann integrability says that a bounded function in one variable
$f\colon [a,b] \to \mathbb{R}$ is Riemann integrable if and only if it is continuous almost ...
- 41.1k
1
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1
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52
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Infinitely many independent functions that are only frequency localized?
A function $f \in L^2(\mathbb R^d)$ will be called $K$-frequency localized if the following inequality holds
$$\int_{\mathbb R^d} \lvert \widehat{f}(x) \rvert^2 x^2 \ dx \le K \int_{\mathbb R^d} \...
- 5,169
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1
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184
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Non-linear translation invariant functionals on $L^1$
I have recently come across a class of (possibly non-linear) operators $F$ defined on $L^1$ such that
$F \colon L^1(\mathbb R^d) \to \mathbb [0,+\infty]$;
$F(u(\cdot - z)) = F(u(\cdot))$ for every $...
- 4,361
1
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1
answer
131
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Convergence of $L^p$ of approximation
Let $f \in L^p(\mathbb R^n)$ be given. Consider a partition of rectangles $I_{ij}:=[x_i,x_{i+1}]\times [x_j,x_{j+1}]$ of $\mathbb R^2.$
Then, we may define the coefficients
$$\alpha_{ij}= \frac{1}{\...
- 128k
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votes
1
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113
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Verifying that a map to $L^2_{\text{loc}}$ is continuous
Let $M$ be a smooth manifold on which a Lie group $G$ acts properly, such that the orbit space $M/G$ is compact. Suppose $c:M\rightarrow [0,\infty)$ is a compactly supported smooth function with the ...
- 1,903
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1
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311
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Differentiation on $[0,1]$
EDIT:
Perhaps a more reasonable question after thinking about the answer I got would have been.
Is there a set $N$ of measure $1-\varepsilon$ and a disjoint partition of that set $N$ with finitely ...
- 536
1
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1
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165
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Integral function $z(x):=\int_{Y} f(x,y)d\mu(y)$ continuous?
Let $z(x):=\int_{Y} f(x,y)d\mu(y)$ for $x \in \mathbb R$ be an integral function where $\mu$ is a finite(!) Borel measure on $Y$ and $x \mapsto f(x,y)$ is continuous for every $y.$
Moreover, we know ...
- 128k
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1
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337
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Bochner measurable; continuous operator
It is well-known that if there is a function $f: \Omega \subset \mathbb R^n \rightarrow X$ with $\Omega$ open and $X$ is a Hilbert space, then continuity of $f$ implies also Bochner measurability of $...
- 66
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1
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209
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A particular measure of noncompactness?
I am working on an article based mainly on the notion of Measure of non-compactness, to study a particular type of fixed point theorems.
Let $\mathcal M $ to be the family of all nonempty bounded
...
- 128k
1
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1
answer
285
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Recover norm from integral
I am given the following expression where $f \in L^2(\mathbb{R}^2, \mathbb{R}^{2 \times 2})$
$$\int_{\mathbb{R}} \int_{\mathbb{R}} \langle g(x), f(x,y) h(y)\rangle dx dy.$$
The functions $g$ and $h$ ...
- 39k
1
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1
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Function is almost everywhere 1 w.r.t. sequence of regular Borel probability measures
Let $\epsilon>0$ be given. Let $Y$ be a compact, Hausdorff space and let $U\subseteq Y$ be an open subset. Assume that $(\mu_n)_{n\in\mathbb{N}}$ is a sequence of regular Borel probability measures ...
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58
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Absolute continuity of DOS measure for Schrödinger operators
Kotani theory gives roughly that for ergodic operators there is a certain equivalence between absolutely continuous spectrum and an absolutely continuous density of states measure.
I would like to ...
- 21
1
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1
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178
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Bochner integrability within a subspace
Let $(H,||\cdot||_H)$ be a Banach space and $K$ a (not necessarily closed) subspace. Suppose that $K$ is a Banach space under another norm $||\cdot||_K$, which satisfies
$$||x||_H\leq ||x||_K$$
for ...
- 29.8k
10
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2
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3k
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Absolute continuity on $R^{n}$
I know the definition of absolute continuity if there is a function $f:(a,b)\rightarrow R$.
I wonder what is an analogy of this concept if we have a function $f:A\rightarrow R$, where $A\subset R^{n}$ ...
- 4,713
1
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1
answer
201
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Existence of a certain norm on space of measurable functions
Suppose $X$ is a measure space with measure $\mu$. Given a strictly increasing continuous (or sufficiently nice) function $\phi:[0, \infty)\to [0, \infty)$ with $\phi(0)=0$. Is it true that we can ...
- 17k
6
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1
answer
2k
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About weak convergence of probability measure
Suppose $\mu_j$ is a sequence of measures on $\mathbb{R}$. By the definition of weak convergence of measures, $\mu_j$ weak converges to $\mu$ means that for any bounded continuous function $f$, there ...
- 29.8k
7
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549
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Counter-example to the completeness of the Wasserstein metric
$\newcommand{\P}{\mathcal{P}}$
Let $(E,d)$ be a complete metric space, let $\P(E)$ be the set of all probability measures on $(E,\mathcal{B}(E))$. Let $W_d$ be the $1$-Wasserstein (Kantorovich) ...
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1
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1
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139
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Compactly supported functions and projections
Let $\Omega$ be an open subset of $\mathbb{R}^n$ and take a family of continuous compactly supported functions $f_n$ on $\Omega$ normalized to one (in the $L^2$ sense).
Then, these functions span a ...
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1
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1k
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Pointwise convergence implies uniform convergence?
Let $K$ be an integral kernel of a bounded operator $S:L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n) $ defined like
$$(Sf)(x)= \int_{\mathbb{R}^n}K(x,y)f(y)dy.$$
Assume that $K\in C^{\text{bounded}...
- 24.2k
9
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1
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636
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Is there a characterization of the Hausdorff measures?
It is known that there is a unique measure on the Borel $\sigma$-algebra of $\mathbb{R}^n$ such that the measure of the rectangle $\prod_i [a_i,b_i[$ is $\prod_i (b_i-a_i)$. This is the Lebesgue ...
- 702
9
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1
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570
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Elements of $L^p$ and nice representatives of equivalence classes
Considering $L^p$ $( 1 \leq p < \infty)$ as a normed vector space, each element of $L^p$ is actually an Equivalent class. Take $[f] \in L^p $ as an Equivalent class, What is the Nicest possible ...
- 369
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1
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1k
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For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?
I am trying to characterize all measures on $\mathbb{R}$ such that
$$
\sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty,
$$
where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...
CommunityBot
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18
votes
1
answer
3k
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How bad can the second derivative of a convex function be?
One can easily construct an example of a measurable function $f:(a,b)\to \mathbb{R}$ which satisfies the following property:
$$\label{p}\tag{P}
f\notin L^1(I),\ \mbox{for each interval}\ I\subset (a,...
11
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3
answers
3k
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Dual space of $L^2(\mathbb{R},L^1(0,1))$?
I was wondering what the dual space of $L^2(\mathbb{R},L^1(0,1))$ is? (equipped with Lebesgue measures)
Formally, one would suspect that it is just $L^2(\mathbb{R},L^{\infty}(0,1))$. But this may be a ...
CommunityBot
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1
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91
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Design measure, which cannot be factorized as a product of measures
Let $\mathcal{S}_x$ and $\mathcal{S}_y$ be a finite discrete sets, such that
$$
0 < |\mathcal{S}_x| < \infty, \qquad 0 < |\mathcal{S}_y| < \infty, \qquad \mathcal{S}_x \cap \mathcal{S}_y =...
- 24.8k
1
vote
1
answer
287
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Interpolation between $L^1$ and $L^2$ spaces
I was wondering whether the following interpolation between $L^1$ and $L^2$ spaces is true:
Let $f \in \mathbb{R}^n$ be such that
$$ \alpha_1:= \int_{\mathbb{R}} \left\lVert f(x_1,\cdot,....\cdot) \...
- 7,249
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1
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Alternative proof of a theorem of Riesz
My question is not research level, but I have not received any feedback on Mathstack; so I am posting it here. I am aware of the traditional proof of the Riesz Theorem that relates linear functionals ...
- 3,816
1
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1
answer
114
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Reference request: regularity of functionals on the space of probability measures
Let $\mathcal M=\mathcal M(\mathbb R^d)$ be the space of finite measures on $\mathbb R^d$, and $\mathcal P=\mathcal P(\mathbb R^d)\subset\mathcal M$ be the space of probability measures. Let $F:\...
CommunityBot
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4
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1
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283
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Absolutely continuity in variation of constant formula
We are talking here about the initial value problem on some Hilbert space $H$
$$y'(t)=Ay(t)+f(t), \\ y(0)=y_0 \in D(A).$$(Problem 1.13 in the reference)
Then $y(t)=e^{At}y_0 + \int_0^t e^{A(t-s)}f(s) ...
- 6,045
3
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1
answer
133
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Restrictions on spectral measure
Given any Borel measure $\mu$ on $\mathbb{R}$, define a map that sends any $f\in C_c(\mathbb{R})$ to $$T_\mu(f)(y)=\int \langle\exp(-i x \lambda),f(x)\rangle\exp(iy\lambda)d\mu(\lambda).$$
Here $\...
- 24.2k
3
votes
1
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355
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convex function with distributional Hessian $D^2 f \leqslant \lambda$, $\lambda$-concave?
Let $f:R^n \to R$ be convex (may not $C^1$), $$[D^2f]=[D^2f]_{ac}+[D^2f]_s=[h_{ij}] L^n+[D^2f]_s$$ is the Lebesgue decomposition of the Hessian matrix. Where $[h_{ij}]$ is the density w.r.t the ...
CommunityBot
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4
votes
1
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393
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How can I show that "almost all function" have property P?
The following is cross-posted from
https://math.stackexchange.com/questions/1391293/is-almost-all-function-a-well-defined-concept
since I didn't (yet) get an answer there.
(I hope that's okay?)
...
CommunityBot
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2
votes
1
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140
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interpret of Picone inequality for non-regular functions
Assume $\Omega \subset \mathbb{R}^N$, $ N>4 $ is open set.
There is a well-known picone identity that says
Let $u,v \in C^2(\Omega)$ satisfy $v>0$ and $-\Delta v \geq 0$ in $\Omega$. The ...
- 3,584
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3
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dual space of a subspace of the space of bounded measures
Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Equipped with the weak convergence, the dual space of $\mathcal{M}$ is $\mathcal{C}_b(\mathbb{R})$ consisting of continuous ...
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0
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63
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The union of weighted compact supported continuous function
Let $\Omega\subset \mathbb R^N$ be open. Given a weight function $v\geq 1$ such that $v\in L^1_{\text{loc}}(\Omega)$ and $l.s.c$. Also supposethere exists a Lipschitz continuous sequence $v_n$ such ...
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2
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785
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Is taking the product of signed measures weakly continuous?
For a Polish space $X$, let $C_b(X)$ denote the real Banach space of bounded continuous real-valued functions on $X$. Let $M(X)$ denote the space of all finite signed Borel measures on $X$, equipped ...
CommunityBot
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4
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1
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209
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Simultaneous approximation of arbitrary functions in Hölder space and in $L^2(\mu)$ by a smooth function and its derivative
Let $\mu$ be a probability measure on the circle $S^1=\mathbb{R}/\mathbb{Z}$ which is singular with respect to the Lebesgue measure $\lambda$. Consider the functions spaces $L^2(\mu)$ on the one hand, ...
- 14.4k
7
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1
answer
306
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An indicator of a planar subset as an element of a tensor product
Denote $I=(0, 1)$, and let $\mu$ be the Lebesgue measure on $I$. Does there exist a function $f$ on $I\times I$ viewed as an element of the space $L^\infty(\mu\times\mu)$ such that
$$
f^2=f
$$
(that ...
CommunityBot
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0
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260
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Generating the sigma algebras on the set of probability measures
I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and $\triangle\left(X,\...
- 11
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0
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448
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Lebesgue point and regularity of functions
A known theorem says that for $f \in L_{loc}^1(\mathbb{R}^d)$, almost every point is a Lebesgue point.
I know too a theorem saying that for $f \in W_{loc}^{1,p}(\mathbb{R}^d)$ , every point is a ...
- 21
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3
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1k
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A natural center of a convex weakly compact set in Banach space
Question: Let $S$ be a convex weakly compact set in Banach space $H$. Propose a natural way to define the unique center $O \in S$.
Motivation: A lot! For example, in game theory $S$ can be a set of ...
- 423
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0
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451
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Why does it seem that $rca=rba$? [closed]
The following paradox has got me stumped. I'm hoping someone can point out the error.
Take a locally compact metric space $X$ and define the $C_b(X)$ and $C_0(X)$ as the spaces of continuous real-...
- 459
2
votes
1
answer
577
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When is the bound in Riesz-Thorin Interpolation Theorem attained?
Let me recall the statement of Riesz-Thorin theorem (see also http://en.wikipedia.org/wiki/Riesz%E2%80%93Thorin_theorem).
Theorem (Riesz-Thorin): Let $(X,\mu)$ and $(Y,\nu)$ be $\sigma$-finite ...
- 13k
0
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0
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405
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Dual of the space of vector valued Borel measures
What is the dual of the space of all vector valued Borel measures?
- 21
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2
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957
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Do semi-continuous functions generate bounded Borel measurable functions as a $C^*$-algebra?
This question is related to Question 2 of my previous posting.
Question. Let $\mu$ be a Radon measure on a compact Hausdorff space $\Omega$ and $L^{\infty}(\Omega,\mu)$ the set of essentially bounded ...
CommunityBot
- 1
4
votes
1
answer
260
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Weak continuity of Lebesgue decomposition
Let $X$ be a space with its $\sigma$-algebra $\mathcal{B}$; we are given a finite measure $\mu$ and a sequence of finite measures $\nu_n$ such that, for every bounded continuous function $f:X\to\...
- 190
3
votes
0
answers
860
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decreasing rearrangements: why the asymmetry of measure-preserving maps?
Ryff proved in 1970 that the decreasing rearrangement $f^*$ of a, say, continuous function $f:[0,1]\to\mathbb{R}$ admits a measure preserving map $\phi$ such that $f=f^*\circ\phi$. In general it is ...
CommunityBot
- 1
4
votes
1
answer
370
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Norms for complex measures
I'm searching for a state of the art introduction to norms on the space of complex measures (on $\mathbb R^n $, for example, or some compact subset thereof). I'd be interested in inequalities of the ...
CommunityBot
- 1
2
votes
1
answer
469
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If two probability distributions have the same weak limit and one of them satisfies Large Deviation Principle, what can we say about the other?
If the probability distribution function of two sequences of random variables have the same weak limit and one of the sequences satisfies a Large deviation principle, then does it imply that the other ...
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