All Questions
9,959 questions
2
votes
0
answers
114
views
non-closed weak graph limit of symmetric operators
Hi Everyone,
I was recently reading Reed & Simon's functional analysis textbook (the first volume), and it mentions casually on page 294 that weak graph limits of a sequence of symmetric ...
- 21
3
votes
2
answers
411
views
Convergence rate of an iterative process
I have the following iterative process
$$a_n=a_{n-1}(1-\phi(a_{n-1})),\quad 0< a_0<1,$$
where $\phi(x)$ is a continuous increasing function, $\phi(0)=0$, and if $x\in(0,1)$ then $0< \phi(x)&...
- 109k
2
votes
0
answers
176
views
Banach Algebras and the peripheral spectrum
Here is a little theorem that I'm trying to prove. I haven't seen it in literature before, but I think the applications will be quite useful, particularly in the context of Banach algebras.
Denote ...
- 25.8k
10
votes
1
answer
492
views
Which W*-algebras are the duals of C*-coalgebras?
A Banach algebra (assumed associative and unital) is precisely a monoid object in the monoidal category of Banach spaces, short linear maps, and the projective tensor product. A Banach coalgebra is ...
- 25.8k
1
vote
1
answer
2k
views
Basic questions about parabolic Holder space
Hi, I am interested in learning a bit more about this space. I have exhausted all the books available at my disposal, and none of them explain much of the basics for me. Here's a definition of this ...
CommunityBot
- 1
5
votes
0
answers
104
views
Regularity of simplices, part deux
This question is directly inspired by Pietro Majer's question and my answer to it.
One can define a simplex, and the dihedral angles thereof in an infinite dimensional Hilbert space (one has to take ...
CommunityBot
- 1
5
votes
2
answers
774
views
Can we calculate the inner product of a semicontinous function with the Dirac delta function?
Dear all,
It is clear that if $f:R\mapsto R$ is a continuous function, than $< f, \delta_x >=f(x)$. Now, if $f$ is only semicontinous, can we say that $< f, \delta_x >=f(x)$? I think this ...
- 3,322
1
vote
1
answer
3k
views
In Fourier Transforms: Positive Definite Functions, Bochner's Theorem, and Derivatives
I've been reading about Bochner's Theorem lately, but when I apply it to the derivative of a function, I seem to get a contradiction with the theorem.
"Bochner's theorem states that a
positive ...
CommunityBot
- 1
2
votes
0
answers
787
views
Regarding a proof in Bourbaki's Topological Vector Spaces
On Bourbaki's TVS Chapter IV pages 33-34, the last part of Proposition 2 can be formulated as follows:
Notations:
$K$ - The underlying field which is the real or complex number field;
$X$ - A ...
- 21
3
votes
1
answer
333
views
Stronger bound for a modified Lyapunov Equation
In regard to the stability analysis and control properties of the linear system $\dot{x}=Ax$.
Consider the solution $P$ of the continuous Lyapunov equation $AP+PA^T+Q=0$, where $A,Q,P
\in
{\mathbb{R}...
CommunityBot
- 1
2
votes
1
answer
4k
views
Gagliardo-Nirenberg inequality
I've read on another topic that general interpolation result from Gagliardo-Nirenberg inequality can be read as follow :
\begin{equation}
\|D^ju\|^1_{L^p} \leq C \|D^mu\|^a_{L^r} \|u\|^{1-a}_{L^q}
\...
- 16.2k
3
votes
1
answer
226
views
Analytic continuation of instantaneous eigenstates of a time-dependent hamiltonian
We are considering the instantaneous eigenstates of an analytically time-dependent hamiltonian and I would like to know how legitimate it is to extend them to the complex plane.
Specifically, our ...
- 54.2k
5
votes
1
answer
484
views
Complemented Subspaces and Riesz-Thorin interpolation
Riesz-Thorin interpolation may sometimes be applied to subspaces (of $\ell^p$ or $L^p$) when these are complemented and the spaces in the complementation comes from a common dense subspace. To be a ...
- 4,432
2
votes
2
answers
470
views
Linear coupled parabolic PDE system with Holder continuous coefficients
I am interested in proving existence/uniqueness to: find $u(x,t)$, $v(x,t)$ such that
$$u_t - a_1u_{xx} - a_2u_x - a_3u -a_4v = f$$
$$v_t - a_5u_{xx} - a_6u_x - a_7u - a_8v_{xx} - a_9v_x - a_{10}v = g$...
- 16.2k
4
votes
1
answer
1k
views
Convolution of a continuous function and a finitely additive measure
Let $f$ be a continuous function on $\mathbb R$ with compact support and $\mu$ a finitely additive measure which is in the dual space of $L^\infty(\mathbb R)$. Is the convolution
$f\ast \mu(x)=\int_{...
- 888
0
votes
1
answer
2k
views
Infinite linear span vs closed linear span
Hi,
Suppose we have a (real, separable) Banach space $V$ and a (linear) set $A\subseteq V$. I presume in general it might not be possible to write every element of the closed span of $A$ as an ...
- 888
3
votes
1
answer
949
views
ODE continuous dependence on parameters to PDE
I want to learn how to apply certain ODE theory to PDE. If we have a Banach space ODE $$x'(t) = f(t, x(t), p),$$ $$x(0) = x_0$$ where the equation is over same domain $t \in (a,b)$, then via the ...
- 3,322
1
vote
2
answers
318
views
Poisson modification of subharmonic function
Let $u\in C^2(\Omega)$ be such that $\Delta u \ge 0$ on $\Omega\supset \overline{B(a,r)}$. We consider the Poisson modification $U$ of $u$ for the ball $B(a,r),$ that is $U$ equals $u$ on $\Omega-B(...
- 60.6k
0
votes
1
answer
258
views
Convolution with an element in the dual space
We recall that if $f_1\in L^p(\mathbb R)$ and if $f_2\in L^q(\mathbb R)$ where
$1 \lt p \lt \infty$ and $\frac 1p+\frac1q=1$ then the function $f_1\ast f_2(x)=\int_{\mathbb R} f_1(x-y) f_2(y)dy$ is a ...
- 76
4
votes
1
answer
6k
views
Inverse of a function defined by an integral
Hi, I have a function defined by an integral as follows.
$$
z=f(w) = \int_0^w \frac{(\zeta-a_1)^{\alpha_1}(\zeta-a_2)^{\alpha_2}...}{(\zeta-b_1)^{\beta_1}(\zeta-b_2)^{\beta_2}...}\ d\zeta
$$
where $w$ ...
- 28.2k
0
votes
1
answer
193
views
Dissipative operator
Let $A$ a linear operator from his domaine $D(A)\subset H$ to $H$, with $H$ is a Hilbert space, such that $A$ is dissipative.
is it true that : if $\|y\|\leq \|z\|$ then $\|Ay\|\leq \|Az\|$?
Thank ...
CommunityBot
- 1
4
votes
1
answer
2k
views
Functional derivative of the square of an integral
I have the following functional
$$F(y)=\left[\int \frac{1}{y(x)+A}\cos(x)dx\right]^2$$
How do I find the functional derivative $dF$?
(I never encountered the square of an integral before when I did ...
5
votes
0
answers
240
views
Linear ODEs in a locally convex vector space
Let $X$ be a complete, locally convex, Hausdorff topological vector space over $\mathbb{C}$. Let $J \subset \mathbb{R}$ be an open interval. Consider the space $M = C^\infty(J,X)$ of smooth ...
- 235
3
votes
0
answers
269
views
Continuous selection given both upper and lower hemicontinuity
Suppose that $X,Y$ are compact metric spaces, and $g:X\rightarrow Y$ is a multivalued mapping that is both upper and lower hemicontinuous. Is there a single valued continuous selection of $g$? If it ...
- 203
-3
votes
1
answer
634
views
compactly supported harmonic functions [closed]
Do a significant class of compactly supported smooth functions u on Ω⊂Rn such that Δu≥0 exist?
Thanks!
- 4,899
0
votes
1
answer
156
views
Calculation of L2-dimension
For a group $G$, can we calculate $dim^{(2)}_{\mathcal{N}G}(\ell^2 G)$, where $\mathcal{N}G$ is the von Neumann algebra of $G$ and $\ell^2 G$ is the Hilbert space on $G$? I want to see whether this is ...
- 25.8k
9
votes
2
answers
1k
views
polynomials with minimal $L_\infty$ norm on multiple disjoint intervals
It is well-known that Chebyshev polynomials are the polynomials of minimal $L_\infty$ norm on [-1,1] with leading coefficient 1. But what if you want the minimal $L_\infty$ polynomial on two disjoint ...
- 91.8k
1
vote
1
answer
3k
views
How to show this Holder bound?
Define the seminorm on the space $S=[0,1]\times[0,T]$
$$\mid u\mid_{\alpha} = \sup\frac{|u(x, t) - u(y,s)|}{(|x-y|^2 + |t-s|)^{\frac{\alpha}{2}}}.$$
Define the norms on the same space
$$\lVert u \...
- 3,993
0
votes
0
answers
73
views
A constrained prolongement
Let $\Omega$ be a domain of $R^n$, let $\omega$ be open subset of $\Omega$ and let $\theta \in W^{2,\infty}(\omega).$
I am wondering about the existence of a function $\tilde{\theta} \in W^{2,\infty}...
- 25
8
votes
2
answers
1k
views
What is the simplest oscillatory integral for which sharp bounds are unknown?
I have either heard or read that sharp asymptotics and bounds for oscillatory integrals of the form
$ \int e^{i \lambda \Phi(x)} \psi(x) dx \quad \lambda \to \infty $
are unknown when the critical ...
- 156
0
votes
1
answer
320
views
Derivable functions & Sobolev spaces [closed]
Is a C^1-function in a bounded domain $\Omega\subset R^n;$ an element of the Sobolev space $W^{2,\infty}(\Omega)$ ?
- 4,899
3
votes
0
answers
395
views
Norm estimate for Moore-Penrose pseudo-inverse of $i^\ast T i$
Let $G$ and $H$ be Hilbert spaces, let $i : G \rightarrow H$ be an isometric inclusion (so $G$ is a subspace of $H$) and let $T : H \rightarrow H$ be a bounded linear operator with closed range.
That ...
- 5,327
7
votes
3
answers
2k
views
Why do we distinguish the continuous spectrum and the residual spectrum?
As we know, continuous spectrum and residual spectrum are two cases in the spectrum of an operator, which only appear in infinite dimension.
If $T$ is a operator from Banach space $X$ to $X$, $aI-T$ ...
- 23k
4
votes
2
answers
427
views
Choice of predictable (or jointly measurable) eigenvalues and eigenvectors of nuclear-operator-valued stochastic process
Let $q^{ij}$, $i,j\in\mathbb{N}$, be predictable real-valued stochastic processes. Let $(e^i)$, $i\in\mathbb{N}$ be an ONB of a separable Hilbert space $H$. Assume that $Q=\sum_{i,j=1}^\infty q^{ij}...
CommunityBot
- 1
3
votes
0
answers
302
views
Dense subalgebras of von Neumann algebras and increasing nets
[Question previously asked on Math.SE]
Let $N$ be a von Neumann algebra, and $A$ be a dense $∗$-subalgebra of $N$ (in the ultraweak topology) with $A''=N$. Is it true that:
For any $x∈N^+$, there ...
- 33
9
votes
1
answer
544
views
Question on Hilbert Manifolds
I have a very basic question on Hilbert manifolds.
Consider the Hilbert space
$$
\mathcal{H}:= L^2(S^1)
$$
with $S^1$ the unit circle.
On $\mathcal{H}$ let us introduce the equivalence relation
$$
...
CommunityBot
- 1
8
votes
2
answers
1k
views
Does infinite-dimensional Brownian motion live in hyperplanes?
I'll begin this question with the finite-dimensional case, as a
warmup.
Let me say a continuous path $\omega : [0,1] \to \mathbb{R}^d$ is
hyperplanar if there exists a nonzero $x \in \mathbb{R}^d$ ...
- 29.8k
4
votes
1
answer
2k
views
Composition of $C^{k, \alpha}$ function with $C^\infty$ function on a compact domain
(I asked this question on MSE but I did not receive an answer so I hope I can post here.)
Let $S$ be a compact set in $\mathbb{R}^2$ and let $C^{k, \alpha}(S)$ denote the usual Holder space with $k$ ...
- 3,322
4
votes
1
answer
4k
views
Weak compactness and weak sequential compactness in Banach spaces
If $E$ is a Banach space, $A$ is a subset of $E$ and is compact with the weak topology $\sigma(E,E')$, that is the most coarse topology which make every $f\in E'$ continuous, is it true that $A$ is ...
- 25
2
votes
1
answer
199
views
A Function with Exactly $k$ Minima in a Bounded Space
Is it possible to have a function with the following properties?
(i) The function maps a bounded $n$-dimensional space $\mathcal{X}$ (say $\left[0,1\right]^n$) onto a bounded interval $\mathcal{Y}$ (...
- 221
2
votes
1
answer
687
views
Solutions to Heat Equations with Obstacles!
Consider a closed Riemannian manifold $(M,g)$ and a positive function $\psi: M \to R$. Fix a point $p \in M$, I have been struggling to construct a solution to the heat equation, $\partial_t u = \...
- 3,322
2
votes
0
answers
202
views
Frames and reproducing kernels
Hello MathOverFlow
I have some questions about frames and reproducing kernels and here they are:
For a Hilbert space $H$ spanned by a frame $\lbrace f_n \rbrace$ there exists a reproducing kernel $K(...
- 21
0
votes
1
answer
338
views
Ultraweak closure inside a closed ball
Let $H$ be a Hilbert space, and $S\subseteq \mathcal{B}(H)$. We denote
$\bar S$ the ultraweak closure of $S$, and $B_r$ the closed ball of center 0
and radius $r>0$ of the normed space $\mathcal{...
- 10.1k
0
votes
1
answer
251
views
Schrodinger Operators with diverging Potential
Is it well known that if $ H = -\bar{\Delta} + V$ (which is defined over $ L^2( \mathbb{R} ^n $ ) and $ lim_{|x| \to \infty } = + \infty $, then $ H$ has compact resolvent?
Does someone know of any ...
- 5,005
0
votes
0
answers
143
views
description of a convex set of functions
Hi everyone,
I have a question about the characterization of a set of functions.
Let $\Phi$ a set containing all the functions $\phi(x): \mathbb{R}_+\rightarrow \mathbb{R}_{+}$ that satisfy the ...
- 96.4k
2
votes
1
answer
382
views
Function extension in a Sobolev space
Let $\Omega$ be a domain of $R^n$ and let $H^2(\Omega)$ be the usual Sobolev space.
Let $\emptyset\ne \omega_1\subset\omega_2$ be open subsets of $\Omega$, and let $\theta \in H^2(\omega_1)$.
I ...
- 34.7k
3
votes
1
answer
3k
views
Solving Functional Equation
Continue with my previous question “Regarding Kolmogorov's Superposition Theorem”, here are some further questions:
Question-1
Is it true: for any given $C^1$ continuous real function $f(x, y): \Re^2 ...
CommunityBot
- 1
6
votes
1
answer
363
views
von Neumann automorphisms: does convergence on a dense algebra imply $u$-convergence?
Let $M$ be a separable von Neumann algebra and let $A$ be a (von Neumann-)dense *-subalgebra.
Suppose that $\alpha,\alpha_1,\alpha_2,\dots$ are automorphisms of $M$, such that for every $a \in A$,
$$ \...
CommunityBot
- 1
0
votes
1
answer
156
views
Does homeomorphism preserves the family of cones?
Let me state my problem. Suppose we have a ball $B$ in standard $\mathbb{R}^3$, that is a $\varepsilon$-neighbourhood of $0$ point. Suppose we have a family of cones $X_C = \lbrace C > 0 \vert x^2 +...
- 15.4k
11
votes
1
answer
702
views
Kuiper's theorem via approximation
Kuiper's theorem says that the unitary group $U(H)$ of a separable infinite dimensional Hilbert space $H$ is contractible, if it is equipped with the norm topology.
Let's suppose, I do not know this ...
- 25.5k