All Questions
10,934 questions
0
votes
0
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38
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Are measures singular with respect to all representing measures in $\mathbb{D}^n$ always concentrated on null-sets? Will it also be a Henkin measure?
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(\...
- 63
1
vote
0
answers
176
views
Unitary operators in $\mathcal{U}(L^2(X,\mu))$ not coming from $\operatorname{Aut}(X,\mu)$
$\DeclareMathOperator\Aut{Aut}$Let $(X,\mu)$ be a measured space with probability measure $\mu$ and $\Aut(X,\mu)$ denote the group of measure preserving bijections of $(X,\mu)$. It is easy to see ...
- 105
0
votes
0
answers
107
views
Does the sequence of bounded symmetric square integrable holomorphic functions have a convergent subsequence?
Let $f$ be a bounded holomorphic function on $\mathbb D^2$ and $s : \mathbb C^2 \longrightarrow \mathbb C^2$ be the symmetrization map given by $s(z) = (z_1 + z_2, z_1 z_2),$ for $z = (z_1, z_2) \in \...
- 41
2
votes
0
answers
111
views
Everywhere-defined unbounded operators between Banach spaces
In this post, it is said that there are no constructive examples of everywhere-defined unbounded operators between Banach spaces; every example furnished must use the axiom of choice. This seems like ...
- 151
1
vote
1
answer
146
views
Form of a hereditary subalgebra of $C^*$-algebra $C_0(X)$
I would like to show that:
"every hereditary subalgebra $U$ of a $C^*$-algebra $C_0(X)$ for a locally compact Hausdorff Space $X$ has the form $J_E := \{f \in C_0(X) : f|_E=0 \}$ for a closed ...
- 11
3
votes
2
answers
250
views
Existence of a positive measurable set with disjoint preimage under iterated transformation
Let $(X,\mathcal B,\mu)$ be a atomless probability measure space and $T:X\to X$ be a non-singular transformation such that $\mu\left({x\in X: T^n(x)=x}\right)=0$ for every $n\ge 1$. Let $A\in \mathcal ...
- 105
2
votes
0
answers
71
views
How to naturally define an output space with certain properties
Consider the following regression problem $v=A(u) + \varepsilon$
for some operator $A:\mathcal{U} \rightarrow \mathcal{V}$ and some function spaces $\mathcal{U},\mathcal{V}$, mapping from $\mathcal{X}$...
- 51
2
votes
1
answer
136
views
Is there a scalar product which makes orthonormal the family of complex functions $ (f_n)_{ n \geq 1 } $?
Let $ (f_n)_{ n \geq 1 } $ be a family of complex functions defined as follow,
$ \forall n \geq 1 $,
$$ f_n (z) = \dfrac{1}{n^{z}} $$
I would like to ask you if it is possible to construct a ( non-...
- 595
8
votes
1
answer
390
views
Order bounded version of monotone complete $C^*$-algebras
Let $A$ be a $C^*$-algebra with self-adjoint part $A_{\operatorname{sa}}$. Then $A$ is called monotone complete if every increasing norm bounded net in $A_{\operatorname{sa}}$ has a supremum (with ...
- 12.6k
6
votes
1
answer
403
views
Do acyclic amenable groups exist?
Is there an example of a nontrivial discrete amenable group with vanishing integral homology?
To put the question in contrapositive. Given arbitrary acyclic group $Q$, is there some reason for the ...
- 4,600
2
votes
1
answer
176
views
Does $i\partial_tu = \Delta^2 u$ exhibit more or less dispersion than $i\partial_t u= \Delta u$?
Consider the initial-value problems in $d=1$
$$\begin{cases} i\partial_tu = \Delta^2 u \\
u(x,0)=u_0
\end{cases}$$
and
$$\begin{cases} i\partial_t u= \Delta u \\
u(x,0)=u_0,
\end{cases}$$
Solutions to ...
- 860
11
votes
3
answers
1k
views
"Simple" integral equation
Let $H(z)$ be a continuous solution of the problem
$$
H(z)=\frac1{1-z}\int_z^1 \frac{2\zeta}{1+\zeta} H(\zeta^2)\,d\zeta,\ \ \ z\in[0,1);\ \ \ H(1)=1.
$$
Is it true that $H(0)=1-\ln2$? The question ...
- 283
7
votes
3
answers
909
views
Using the Stone-Weierstrass theorem to solve an integral limit
The following question was posted on math stack exchange here but it got no answers
Let $c\in (1, +\infty)$ and $f \colon [0, c] \to \mathbb{R}$ be a continuous and monotonically increasing function ...
- 331
5
votes
0
answers
180
views
A variant of Schwarz's theorem on generators of smooth $G$-invariant functions
Let $V$ be a finite dimensional euclidean space and let $G\subset O(V)$ be a finite (or compact) group, let $\mathbb{R}[V]$ be the algebra of polynomial functions on $V$. If $E\subset \mathcal{C}^\...
- 4,123
2
votes
1
answer
155
views
Tempered distributions at non-coinciding points and density of Schwartz functions
In the previous question, I find that situation is much less favorable than expected…. So I add more details to focus on the specific case I have in mind.
Let us consider the Schwartz space $\mathcal{...
- 3,477
1
vote
1
answer
121
views
Question on complemented subspaces of a product space
Assume that we have closed subspaces $Y_1$ and $Y_2$ of Banach spaces $X_1$ and $X_2$, respectively. If the product $Y_1\times Y_2$ is complemented in $X_1\times X_2$, does it follow that $Y_i$ is ...
- 19
1
vote
0
answers
66
views
The derivative of semigroup in the weak sense imply strong sense
Suppose $X$ is a Banach space, and $T(t)$ $t\ge0$ is a strongly continuous semigroup with generator $A$. Assume $\frac{T(t)-I}{t}x$ weakly converges to $y\in X$ when $t\to 0$, then I need to prove $x\...
- 775
4
votes
2
answers
257
views
Simple proof that exactness implies strong mixing
Let $f$ be a continuous map defined on a compact metric space $X$. Suppose that $f$ preserves the Borel probability measure $\mu$ and that, for every positive-measure set $A\subseteq X$, we have $$\...
- 63
4
votes
2
answers
360
views
Functions with asymmetrically decreasing Fourier transform?
$\def\ii{{\rm i}}\def\bbR{\mathbb R}\def\bbC{\mathbb C}\def\bbNo{\mathbb N_0}\def\Fou{\mathscr F}$Specifically, I would like to have a compactly supported continuous function $f=u+\ii\,v:\bbR\to\bbC$ ...
- 3,584
4
votes
1
answer
222
views
estimate a singular integral using a dyadic decomposition
Let $0<\alpha_{j}<1$, $j=1,\dots,d+1$. I am trying to estimate the following singular integral:
$$I(y_{1},\dots,y_{d},z):=\int_{\substack{ x\in[0,1]^{d}\\
1/2<|x|<1}}
\frac{d x_{1} \dots d ...
- 852
1
vote
1
answer
120
views
Characterization of an integral operator with a Bessel kernel
I am considering the following integral operator: $$K(\sigma)(\theta)=\int_0^{2\pi} \sigma(\theta') J_0(|e^{i\theta}-e^{i\theta'}|)\,d\theta',$$ where $J_0$ is the Bessel function of order $0.$
I am ...
- 141
4
votes
1
answer
205
views
Existence of an $\alpha$-Hölder continuous function whose graph has positive Hausdorff measure of maximal dimension
It is standard that if $f:[0,1] \rightarrow \mathbb{R}$ is $\alpha$-Hölder continuous, then its graph has Hausdorff dimension at most $2-\alpha$. My naive expectation was that "most" graphs ...
- 45
2
votes
0
answers
187
views
How can the maximal ideal space of the Fourier Stieltjes algebra be non-separable?
I have been asking a fair few (probably elementary) questions about abstract harmonic analysis lately. By means of explanation, I am just feeling around the subject at the moment and trying to build ...
- 1,955
2
votes
0
answers
184
views
Example of space which is weak Hahn-Banach smooth but not Hahn-Banach smooth
A Banach space $X$ is said to be Hahn-Banach smooth if every linear functional on $X$ has a unique norm-preserving extension over $X^{**}$. Weak Hahn-Banach smoothness is what if the above condition ...
- 521
2
votes
0
answers
88
views
Dependence and $L^2$ projections of functions
tl;dr: Is it possible that the best approximation to a nonnegative function of three variables with a bivariate function is no better than the best univariate function?
Let $w$ be a density on $\...
- 21
1
vote
2
answers
162
views
Proof that the closed convex hull of a weakly convergent sequence has empty interior using properties of $c_0$?
Let $X$ be an infinite-dimensional Banach space, and $x_n\to 0$ weakly in $X.$ Let $K$ be the closed convex hull of $\{x_n\}.$
I remember a proof that $K$ has empty interior as follows: define a map $...
- 1,755
3
votes
1
answer
208
views
Density beyond Stone–Weierstrass
$\DeclareMathOperator\tr{tr}$I need density assertions for spaces of polynomials which are not (that I know of) algebras. One goes like this: Fix $n\in\mathbb N$ let $S$ denote the set of self-adjoint ...
- 460
1
vote
1
answer
232
views
When is this operator positive semi-definite?
I have the following operator
$$\Phi(\chi_A)=\int \text{d}\eta\, \text{d}\zeta\,\chi_A(\eta,\zeta)\,e^{i(\eta \hat{P}+\zeta\hat{Q})}.$$
With $\chi_A$ the indicator function associated to a set $A\...
7
votes
1
answer
401
views
High dimensional Lusin conjecture
Lusin, in 1913, while considering the properties of Hilbert's transform, conjectured that every function in $L^2[-\pi, \pi]$ has an a.e. convergent Fourier series. Kolmogorov, in 1923, gave an example ...
- 879
2
votes
1
answer
103
views
Sufficient conditions for the space of Radon measure to be a Banach space
Let $\mathcal{X}$ be a Hausdorff space and consider the space of Radon measures with bounded total variation $M(\mathcal{X})$ on $\mathcal{X}$.
Usually, the additional assumptions on $\mathcal{X}$ are ...
- 171
3
votes
2
answers
392
views
Monotonicity of matrix conjugation
Let $A$ and $B$ be positive-definite matrices such that $A \le B.$
By matrix monotonicity of the root, this also implies that $A^{\alpha} \le B^{\alpha}$ for $\alpha \in [0,1].$
I am now curious under ...
-2
votes
1
answer
118
views
Mismatch between equivalent definitions of the Bohr compactification of the reals
I feel I'm overlooking something very silly.
The Bohr compactification of $\mathbb R$ has two equivalent definitions.
The set of (possibly discontinuous) homomorphisms $\mathbb R \to \mathbb T$ under ...
- 1,955
5
votes
1
answer
561
views
interiors of positive cones in ordered Banach spaces
I have a couple of questions about ordered Banach spaces and interiors of their positive cones. I would appreciate your insights and any recommended references.
I want to know several examples of ...
- 79
15
votes
4
answers
1k
views
Steinhaus theorem and Hausdorff dimension
Assume for simplicity that sets $A_i\subset\mathbb{R}$ are compact. If $A_1$ and $A_2$ have positive length, then $A_1+A_2$ contains an interval. That is a variant of the classical Steinhaus theorem ...
- 28k
8
votes
2
answers
675
views
Question on whether, "An entire function, nowhere zero, has an entire logarithm," holds for matrix-valued entire functions as well
It is known that an entire function that is nowhere zero must be the exponential of another entire function.
Does this hold for matrix-valued functions as well? That is, given a matrix-valued entire ...
- 286
1
vote
0
answers
63
views
Is there any other norms besides cut norm defined on graphon?
Let $\mathcal{W}$ denote the space of all bounded symmetric measurable functions
$W : [0, 1]^2 \rightarrow \mathbb{R}.$ For any $W\in\mathcal{W}$ we say it is a kernel and define its cut norm $\lVert ...
- 349
0
votes
0
answers
97
views
Heine-Borel property for (probability) measures on $\mathcal{S}'$?
For Euclidean space $\mathbb{R}^n$, it is a well-known fact that the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ is a Montel space. Moreover, the space of tempered distributions $\mathcal{S}'(\mathbb{R}...
- 3,477
2
votes
0
answers
261
views
When is an unbounded averaging operator on $\mathbb{R}\to \mathbb{R}$ closed?
Let $\{a_n\}_{n=1}^\infty$, $a_n\in \mathbb{R}$. Consider the following linear operator $A$ on functions $f:\mathbb{R}\to \mathbb{R}$:
$$(Af)(x) = \sum_{n=1}^\infty a_n f(x+n)+ \sum_{n=1}^\infty a_n f(...
- 20.2k
6
votes
2
answers
2k
views
Weak convergence + convergence of the norm implies strong convergence in Orlicz spaces
It is known [1, proposition 3.32] and a classical trick in PDEs that, in any uniformly convex Banach space $X$, weak convergence $x_n\rightharpoonup x$ together with convergence of the norm $\|x_n\|_X\...
- 5,380
2
votes
1
answer
104
views
Strong convergence of a sequence in $L^2((0,T); H^{s,2}(\Omega)) \cap C([0,T];H^{-s,2}(\Omega))$, $0<s<1$
Let $u_n$ be a sequence with $u_n \in L^2((0,T);H^{1,2}(\Omega))$ and $\frac{\partial u_n}{\partial t} \in L^2((0,T);H^{1,2}(\Omega)^*)$. Then, how could one get a subsequence of $u_n$ that strongly ...
- 101
5
votes
2
answers
1k
views
How can I prove this special version of the Poincaré inequality?
I am reading a paper, Is $L^2$ Physics-Informed Loss Always Suitable for Training Physics-Informed Neural Network?, which uses an "application" of Poincaré's inequality. The form I know and ...
3
votes
1
answer
172
views
1-1 map on the $\{0,1\}^k$
Let integer $k>0$ and let $\{0,1\}^k$ denote the set of all $1\times k$-dim vectors whose every coordinate is eithor 0 or 1, for example, $(0,1,1,0,\dots,1,0,0,1)$. For any such vector $\alpha$, ...
- 349
2
votes
1
answer
197
views
Prékopa-Leindler style inequality?
Does anyone know a simple proof of the following Prékopa-Leindler style inequality:
If we have $f_1,f_2,g_1,g_2$ strictly positive functions on $\mathbb{R}$ such that, for any $x_1,x_2 \in \mathbb{R}$,...
- 125
4
votes
3
answers
869
views
Can these identities for the Euler-Mascheroni constant be proven?
I stumbled upon these 4 limit/integral identities involving Euler's constant aka gamma (~0.5772). They appear to be valid based on inspection but I have no idea how to prove them. In addition, I have ...
- 194
0
votes
1
answer
317
views
A variation of the Riesz Lemma
Given a normed space $X$, a closed proper subspace $Y$ and $\alpha\in (0,1)$, the Riesz Lemma states that there is $x\in X$ such that $\|x\|=1$ and $d(x,Y)>\alpha$. Observe that also $d(-x,Y)=d(x,Y)...
- 153
2
votes
1
answer
98
views
Locally compact groupoid with a Haar system such that the range map restricted to isotropy groupoid is open
Can somebody provide an example of a locally compact groupoid $G$ with a Haar system such that the range map restricted to isotropy groupoid of $G$ is open?
I could not find any specific example for ...
0
votes
1
answer
127
views
Why $\int_{S^{n-1}} |\hat{f}(w)|^2d\sigma(w) < \infty$?
Let $f\in L^p(\Bbb R^n)$ and $S^{n-1}$ be the Unit sphere.
Why
$\int_{S^{n-1}} |\hat{f}(w)|^2d\sigma(w)<\infty$ when $1<p<2$.
$\hat{f}$ is the Fourier transform fora function f.
- 9
4
votes
0
answers
256
views
Singularity of singularities and second microlocalization: a question that come from the stabilization of damped wave equation
In the paper [2], the Authors introduce a tool called second microlocalization, which is difficult for me. Although I have searched a lot of papers on the internet, nevertheless the material that I ...
1
vote
0
answers
215
views
Computing a closed form representation for a Fourier series summation
I want to compute a closed form representation for the below given summation expession.
$$g_{\lambda}(\boldsymbol{x}) = \sum\limits_{\boldsymbol{l}\in\mathbb{Z}^m} \frac{1}{1+\lambda\|\boldsymbol{l}\|...
- 698
1
vote
0
answers
205
views
Uniqueness for Volterra equation with initially (linearly) unbounded kernel
Letting $D:=\{(x,y):\ 0\leq x\leq y\leq1 \text{ and } y>0\}$, I have a continuous function $k:D\to[0,\infty)$ that satisfies some properties that I list below. I'm interested in continuous and ...
- 537