All Questions
10,448 questions
3
votes
0
answers
116
views
On a functional equation of Mahler?
Recently, I was trying to introduce the concept of natural boundaries to a fellow math student, and what greater way to do this than using an example? In particular, I tried to use as an illustration, ...
- 131
3
votes
2
answers
228
views
Sobolev extension problems of $W^1_\infty(\Omega)$
Recently I have read the paper Whitney's problem on extendability of functions and an intrinsic metric written by Nahum Zobin and published by Advances in Mathematics in 1998. I am confused about one ...
- 69
2
votes
0
answers
55
views
Distance between a Hölder function and a Sobolev ball
Let $\Omega$ denote $[0, 1]^n$ and let $\|\cdot\|_{k, p}$ and $|\cdot|_{m, \alpha}$ denote norms of Sobolev space $W^{k,p}(\Omega)$ and Holder space $C^{m, \alpha}(\Omega)$, respectively.
My question ...
- 380
1
vote
1
answer
76
views
Proving that a function $f(x,y)$, that is unbounded in every direction, is uniformly bounded below by $1$ outside some disc of large enough radius
I have a smooth function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the curve $(ta,tb)$, then $$\lim_{t\to\infty}f(ta,...
- 765
1
vote
1
answer
115
views
Block-diagonal embedding of $U(n)$ into $U(mn)$
What is known about the subgroup $U(n)\subset U(mn)$ for $m,n\in\mathbb{N}$ given by the diagonal embedding
$$ \alpha\mapsto \text{diag}(\alpha,\cdots, \alpha),$$
for $\alpha$ appearing $m$ times?
For ...
2
votes
1
answer
318
views
Understanding the Schrodinger flow——Symplectic Banach manifold
This question was posted on https://math.stackexchange.com/questions/4925369/understanding-the-schrodinger-flow-symplectic-banach-manifold but recieve nothing. I really want to know the something ...
4
votes
1
answer
187
views
Topology on the space of compactly supported functions
Let $X$ be a locally compact Hausdorff topological space, and let $C_c(X)$ be the space of compactly supported $\mathbf{R}$-valued continuous functions. It is a well-known fact that this space is not ...
- 43
7
votes
1
answer
271
views
Existence of a linear map resulting in the determinant being an elementary symmetric polynomial
Let $1 \leq k \leq n$ be fixed integers. Let $\mathcal{M}_n^{\mathrm{H}}$ be the set of $n \times n$ complex Hermitian matrices (if it makes it easier to answer this question, you may instead use the ...
- 6,351
3
votes
1
answer
169
views
Copy of $\ell_\infty$ inside $\ell_\infty(\Gamma)$ containing given subspace
To complete a proof I need to know if the following is true:
Given a non-empty set $\Gamma$ and a separable subspace $Y$ of $\ell_\infty(\Gamma)$, there exists a subspace $A$ of $\ell_\infty(\Gamma)$ ...
- 271
8
votes
0
answers
177
views
Understanding spaces of negative regularity
I apologize if this question is too basic for this site, but I posted it on mathSE and did not get any responses (link can be found here) so I'm crossposting it here.
Let $C^k(\mathbb{R}^n$) be the ...
- 721
2
votes
0
answers
95
views
On analytic functions on the complement of a curve without jump across the curve almost everywhere
Question. Suppose $f$ is an analytic function on $\mathbb C\setminus\mathbb R$ and assume that the boundary values of $f$ from above and below the real axis (denoted $f_\pm$ respectively) exist almost ...
7
votes
1
answer
299
views
Intermediate spaces of test functions between $\mathcal{S}$ and $\mathcal{D}$?
On $\mathbb{R}^n$, let $\mathcal{S}(\mathbb{R}^n)$ be the Schwartz space and $\mathcal{D}(\mathbb{R}^n)$ be the space of smooth, compactly supported functions.
According to p.145 of the book by Reed &...
- 3,477
26
votes
2
answers
6k
views
Understanding a simplifying assumption in proof of the invariant subspace problem
In a recent preprint On the invariant subspace problem in Hilbert spaces Per H. Enflo claims to have solved the invariant subspace problem, showing that every bounded linear operator on a separable ...
- 423
3
votes
0
answers
58
views
Infinitesimal generators of random evolutions
Consider two state spaces $X$ and $Y$ and infinitesimal generators of Markov processes $(A_y)_{y\in Y}$ and $B$, on $X$ and $Y$ respectively. We assume that $A_y$ share the same domain $D(A)$, and ...
- 31
0
votes
1
answer
123
views
Proving a Fourier transform inequality for functions with mixed variable bounded support
I'm working on a problem involving the Fourier transform and have encountered an inequality that I am unsure how to prove. I would greatly appreciate any help or guidance you can provide.
Let $\gamma\...
- 131
2
votes
1
answer
258
views
Differential equation involving square root
I am absolutely not familiar with differential equations. However, I am facing the following differential equation:
\begin{equation}
a(x)y^{\prime}(x)+b(x)y(x)=c(x)\sqrt{y^{2}(x)+d(x)}
\end{equation}
...
- 163
1
vote
0
answers
51
views
Compact embeddings RKHSs into Sobolev Spaces
Let $\mathcal{H}$ be an RKHS over an open domain $\Omega \subseteq \mathbb{R}^d$. Are there conditions under which $\mathcal{H}$ can be compactly embedded in a Sobolev space $W^{s,p}(\Omega)$ for ...
8
votes
1
answer
537
views
Reference request: Expository paper on the use of functional analysis in differential and integral equations
Some textbooks on functional analysis do not hint that a major raison d'être of the subject is its use in the study of differential and integral equations. The reader could go all the way through ...
3
votes
1
answer
199
views
Can gradient zero implies that a function is constant with Hörmander vector fields
Let $X=(X_1,\cdots,X_m)$ be a system of Hörmander vector fields defined on $\mathbb{R}^n$. The Sobolev space $W_{X}^{1,p}(\Omega)$ is defined by
$$W_{X}^{1,p}(\Omega):=\{u\in L^p(\Omega)|X_iu\in L^p(\...
- 561
1
vote
0
answers
34
views
Discrepancy between probability measures, tested against bounded functions of bounded variance
When studying some concentration inequalities, it became relevant to consider the following discrepancy between two probability measures $\pi$ and $\nu$ (treating $\sigma \in \left( 0, \frac{1}{2} \...
- 801
5
votes
1
answer
375
views
Looking for a counterexample: Conditioning increases regularity?
Let $p(x,y,z)$ be a joint density (over $\mathbb{R}^3$) under no smoothness or regularity assumptions, besides its existence. I am looking for a (counter)example where $p(y|x)$ is less regular than $p(...
- 51
1
vote
1
answer
132
views
Deriving a specific bound for functions in Hardy Space
Reading some article a while ago I read the following: (here $H^2$ represents the Hardy space)
Let $f\in H^2$ be such that $f(0)=1$, and let $0<\lvert\lambda\rvert<1$, then $$\lVert f(\lambda z)\...
0
votes
0
answers
66
views
Equality between operators, on dense subspace, from a quadratic form point of view
Let $L \ge 1$ and consider a finite box $\Omega = [0,L]^{d} \subset \mathbb{R}^{d}$. The set of functions:
$$\psi_{p}(x) = \frac{1}{L^{d/2}}e^{i\langle p,x\rangle} \quad p\in \frac{2\pi}{L}\mathbb{Z}^{...
- 1,305
3
votes
0
answers
69
views
Perturbation of one-parameter groups of unitary operators
Let $H$ be a Hilbert space and let $h$ be a fixed, densely defined, possibly unbounded, self-adjoint operator on $H$. Letting $B(H)$ denote the space of all bounded operators on $H$, it is well ...
- 2,263
2
votes
0
answers
94
views
Nemytskij operator for Lebesgue variable UNBOUNDED exponent spaces
Let $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function (i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is measurable for all $t\in\mathbb{R}$), where $\Omega\...
- 1,759
0
votes
1
answer
150
views
When are infimal convolutions contractions?
Let $X$ be a separable Fréchet space and $\varphi,\psi:X\to \mathbb{R}$ be a lower semi-continuous and convex function with $\psi$ bounded below and coercive. Consider the infimal convolution
$$
\...
- 364
6
votes
1
answer
528
views
A functional equation
I am working on some physics problem and got stuck with the following equation: Let $a$ be a very small positive number. Is there a bounded function $F$, $0 \leq F \leq 1$, such that for all $x \in \...
- 63
6
votes
1
answer
292
views
Analytic maps on Banach spaces: analyticity upgrade
Consider the following problem.
Let $E,F,G$ be real or complex Banach spaces, such that $F\subset G$ with continuous embedding. Let $U\subset E$ an open set and
$$ f:U\to G $$
an analytic map, such ...
- 1,075
3
votes
1
answer
228
views
Is compact-open topology stable with respect to injective limits?
Let $X$ be a locally convex space, and $\{Y_i;\ i\in I\}$ a covariant system of locally convex spaces over a partially ordered set $I$, i.e. a system of linear continuous mappings is given $\sigma^j_i:...
- 7,426
2
votes
0
answers
83
views
What is Lipschitz constant of the radial renormalization $(X,\|\cdot\|_a) \rightarrow (X,\|\cdot\|_b)$ on a normed vector space $X$
Suppose that $X$ is a vector space with two norms $\|\cdot\|_a$ and $\|\cdot\|_b$. The mapping
$$
f(x) = \frac{\|x\|_{a}}{\|x\|_{b}} x, \qquad \forall x \in X,
$$
with $f(0)=0$
is a radial and maps ...
- 5,327
3
votes
1
answer
287
views
Unitary transformations of Vandermonde matrices forms a smooth manifold?
The space of all Vandermonde matrices $V$ with $r$ variables and degree $n$ (as below) forms an embedded submanifold of $\mathbb{R}^{(n+1) \times r}$ when $x_{i} \in \mathbb{R}$. It is naturally a ...
- 275
3
votes
0
answers
148
views
Embeddings of Bochner-Sobolev spaces with second time derivative
NOTE: I also asked this question here in MSE.
In the weak theory of evolution PDEs, the Bochner-Sobolev spaces are frequently used. For $a,b \in \mathbb{R}$ and $X,Y$ banach spaces, we define these ...
- 91
3
votes
1
answer
182
views
Let $\phi : A \to B$ be a surjective $*$-homomorphism of star algebras, is there any good notion of "normal bundle of $B$ in $A$"?
Let $\phi : A \to B$ be a surjective $*$-homomorphism of star algebras (maybe more restricted kind of star algebra), is there any good notion of "normal bundle of $B$ in $A$"? By a "...
- 31
1
vote
1
answer
127
views
approximating differentiable functions with double trigonometric polynomials
Let $Q = [0,1]^2$. For sake of notation, let
$$
f^{(i,j)}(x,\xi) = \frac{\partial^{i+j}}{\partial x^i \partial \xi^j}f(x,\xi).
$$
Fix some non-negative integer $k$. Moreover let $f\in C^k(Q)$ if
$$
\|...
- 171
0
votes
1
answer
151
views
Super-reflexivity is separately determined
I've found this result that states that super-reflexivity is separably determined, i.e., if every separable subspace $Y\subset X$ of a Banach space is superreflexive then $X$ itself is superreflexive. ...
- 145
-2
votes
1
answer
241
views
Does a group representation being transitive on a basis imply irreducibility?
Let $G$ be an infinite discrete group and $\pi$ a representation of $G$ on the Hilbert space $H$.
Suppose that the group representation is transitive on an orthonormal basis $B = \{e_j\}_{j=1}^{\infty}...
0
votes
0
answers
95
views
Orthogonalization of symmetric non-degenerate bilinear forms
It is well-known that given a field $k$ with characteristic different from $2$, every symmetric non-degenerate bilinear form $B$ over a finite-dimensional space can be orthogonalized. This means that ...
- 1,485
0
votes
0
answers
34
views
Locally compact groupoid with range map restricted to isotropy groupoid is open
Suppose the action groupoid 𝐺=𝐻⋉𝑋, where 𝐻 is a locally compact group and 𝑋
a locally compact space is such that isotropy subgroups of H are isomorphic to each other.
Can this be an example of a ...
4
votes
3
answers
482
views
Does the uniform boundedness principle holds for multilinear maps as well?
This question has been motivated by weak* completeness of distributions.
According to the answer in the above post, any barrelled locally convex topological vector space $E$ satisfies the uniform ...
- 3,477
2
votes
1
answer
216
views
Forming real positive semidefinite matrices from complex matrices
I have asked this question on the Mathematics Stack Exchange: https://math.stackexchange.com/questions/4924554/forming-real-symmetric-positive-semidefinite-matrices-from-complex-matrices.
Let $Q \in \...
- 31
2
votes
0
answers
137
views
Why a function induced by the infimum of the arclength of curves is Lipschitz?
Recently I have read a paper "Weighted Trudinger-type Inequalities" written by Stephen M. Buckley and Julann O'Shea and published by Indiana University Mathematics Journal in 1999, MR1722194,...
- 69
1
vote
0
answers
88
views
Schauder estimate for $f \in L^\infty$
I was reading an article where at some point the author uses the following estimate:
Let $u$ be a solution of
$$\Delta u = f \quad \text{in } B_1$$
for $f \in L^\infty$. Then $u \in C^{1,1 - \...
- 452
6
votes
1
answer
252
views
Poisson kernel for the orthogonal groups
For the complex ball $|z|^2\le 1$ in $\mathbb{C}^n$, there is a Poisson kernel proportional to $|x-z|^{-2n}$. This is generalized to the unitary group $U(N)$ so that in the complex matrix ball $Z^\...
- 1,549
0
votes
0
answers
56
views
What is the maximum of $ \frac{\sin(n(x+a))}{\sin(x+a)} + \frac{\sin(n(x-a))}{\sin(x-a)}$?
I have asked this here. Due to inactivity and no satisfying answers, I am asking here. Hope that's okay.
We know the global maxima of the function $\frac{\sin(nx)}{\sin(x)}$
is $n$ (thanks to this ...
- 213
0
votes
0
answers
85
views
Measurable selection for the mean value theorem
When we use the mean value theorem we come across the problem of measurability of the argument. The problem is somehow like that:
Let $f:\Omega\times [0,1]\to\mathbb{R}$ be a Caratheodory function (i....
- 1,759
2
votes
0
answers
102
views
Existence of unique-up-to-shift solution of a Volterra equation
Let $\Delta=\{(t,s):\ 0<s\leq t\leq1\}$, and suppose $k:\Delta\to\mathbb R$ and $f:(0,1]\to\mathbb R$ are continuous. Further assume that for every $t\in(0,1]$, the function $k(t,\cdot):(0,t]\to\...
- 537
6
votes
1
answer
342
views
If $t \to \lVert f(\cdot,t) \rVert_{L^2_x}^2$ is absolutely continuous, can we interchange the spatial integral and time derivative? (from MSE)
I originally posted this question on MSE. But it seems more nontrivial than expected, so I guess MO is a more appropriate place to ask.
I repeat the question for the sake of completeness:
Let $f(x,t) ...
- 3,477
7
votes
2
answers
2k
views
Method of characteristics for higher order PDEs in more than two variables
I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
- 8,998
2
votes
2
answers
151
views
Upper bound $\int_{\mathbb{R}^d \times \mathbb{R}^d} |fx)-f(y)| (1+|y|) \ell (x) p_t (x-y) \, \mathrm d x \, \mathrm d y$ in $t$
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
We fix $\alpha \in (0, 1)$ and $c>0$. Let $f : \bR^d \to \bR$ and $\ell : \bR^d \to \bR_+$ be measurable such that $\ell$ ...
- 825
6
votes
2
answers
449
views
Asymptotic behavior of the "Cauchy square" series
$\renewcommand{\ge}{\geqslant}\renewcommand{\le}{\leqslant}$
$\newcommand{\pa}[1]{\left( #1 \right)}$
Let us take $\alpha > 0$, $x_1 := \alpha$ and for any $n \ge \mathbb{N}$,
\begin{align*}
\boxed{...
- 61