Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

learn more… | top users | synonyms (1)

0
votes
0answers
19 views

inclusion maps interpolation spaces

I apologize in advance if the question below is a nonsense or a trivial one but I have no intuition as for the interpolation theory. Let $\overline{A}:=(A_0,A_1)$ and $\overline{B}:=(B_0,B_1)$ are ...
3
votes
2answers
178 views

Example of measure of non-compactness

I can't understand the following example of measure of non-compactness, which was given in this article. Definition: A non-negative function $\phi$ defined on the bounded subsets of $X$ will be ...
4
votes
0answers
54 views

Are Sobolev trace spaces equal from both sides of the boundary?

Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure. Assume $\partial\Omega=\partial\Omega'$. Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...
-4
votes
0answers
37 views

Linear Independence for functions defined by integration [on hold]

Given that the set of functions $$f_i(x,y), \quad i=1,\dots,n$$ are linearly independent for $(x,y) \in [0,1]^2$. Is the set of functions, $g_1,\ldots,g_n$, defined by $$ g_i(x) = \int_{y\in [0,1] } ...
0
votes
0answers
116 views

Radius of convergence of Taylor expansion of $z \mapsto (1 - z \cdot a)^{-1}$ [migrated]

Let $A$ be a Banach $\mathbb{C}$-algebra with norm $\text{N}(-)$ and let $a \in A$. Where can I find a reference to/can somebody supply a proof of the following posited equality?$$\max_{z \in ...
4
votes
0answers
80 views

Compensated compactness for system of conservation laws?

As far as I knew, the method of compensated compactness can be used only for one-dimensional scalar and $2\times 2$ systems of conservation laws, i.e. $u_t+f(u)_x=0$. But if I understood correctly ...
11
votes
2answers
278 views

Quantum Hamiltonian for an Inverse Cube Force Law

If you have a nonrelativistic quantum particle in $\mathbb{R}^3$ in an attractive inverse cube force, its Hamiltonian is $$ H = -\nabla^2 - \frac{c}{r^2} $$ where I'm keeping things simple by ...
4
votes
1answer
122 views

Kernel of the natural map between group $C^*$-algebras

Let $\Gamma$ be a discrete group. We can form two $C^*$-algebras: the universal (or full) and reduced, to be denoted by $C^*_u(\Gamma)$ and $C^*_r(\Gamma)$ (respectively). Both of them are completions ...
5
votes
0answers
131 views

Tensorization of Orlicz norm?

Associated with a convex function $\phi:[0,\infty)\mapsto[0,\infty)$ satisfying $\lim_{x\to 0} \frac{\phi(x)}{x} = 0, \lim_{x\to\infty}\frac{\phi(x)}{x} = \infty,$ the Orlicz norm of a random variable ...
1
vote
0answers
61 views

Comparing Dirichlet energy and area of a Surface-immersion

Let $(F,g)$ be a closed Surface, $(M,h)$ a Riemannian 3-Manifold and $f: F \to M$ a smooth immersion. Denote by $f^*(h)$ the pullback metric on $TF$ induced by $f$ and let $dV_g$ and $dV_{f^*(h)}$ be ...
0
votes
0answers
57 views

Singular integral equation

Investigating a control problem for heat equation I stacked on solution of this integral equation which seems to be singular: $$ \int_0^1\mathcal{K}(\tau)u(\tau)d\tau=\mathcal{M}, $$ in which ...
2
votes
1answer
124 views

Hardy space, Lebesgue space for $p<1$,

We denote $\mathcal D'(\mathbb R^n)$ the space of distributions, and $\mathcal D(\mathbb R^n)$ the space of smooth, compactly supported functions. Let $\rho\in \mathcal D'(\mathbb R^n)$ such that ...
0
votes
0answers
41 views

An eigenvalue convergence question [closed]

The following question is from Eberhard Zeidler's book in Nonlinear Functional Analysis volume 1, question 1.5b page 40. Let $(X,(\cdot | \cdot))$ be a Hilbert space, take the operator: ...
0
votes
1answer
116 views

On the Riesz representation theorem II

I have a follow-up question to On the Riesz representation theorem . Let $V$ be a subspace of a Hilbert space, and let $V^\times$ be the space of all antilinear functionals on $V$, equipped with the ...
4
votes
0answers
44 views

Determinant of quotient of unbounded operators

I have been trying to prove this for a while but failed so far. Let $A$ and $B$ are two positive, self-adjoint operators with compact resolvent on a Hilbert space $H$ defined on the same dense ...
0
votes
0answers
66 views

A question in compact set

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
0
votes
0answers
74 views

When do block sequences yield disjoint subspaces?

Let $X$ be a Banach space having a (unconditional, normalized) Schauder basis $(e_n)_n$. Suppose that $Y$ and $Z$ are (closed) block subspaces of $X$ having normalized block bases (with respect to ...
3
votes
1answer
44 views

Restriction of a semigroup to a form domain

Say, we have a Hilbert space $H$ with a semibounded self-adjoint operator $A:D(A)\to H$ generating a strongly continuous semigroup $T(t):H\to H$. Is it possible to restrict $T(t)$ to a form domain of ...
1
vote
0answers
78 views

Fredholm subvector spaces of $B(\mathcal{H})$

Let $\mathcal{H}$ be a separable Hilbert space with orthonormal base $\{e_i\}\;\;i\in \mathbb{N}$. Definition: We say a subvector space $W\subset B(\mathcal{H})$ is a Fredholm subspace if ...
4
votes
1answer
279 views

On the Riesz representation theorem

Let $V$ be a vector space with inner product $(\phi,\psi)$ antilinear in the second argument - not necessarily a Hilbert space. Let $\Phi$ be an antilinear functional on $V$. What are the precise ...
4
votes
0answers
156 views

An equivalent relation on the space of polynomials in one complex variable

Let $P(z)$ be a polynomial with complex variable $z$. We consider the following distribution for the roots of $P(z)=0$: the distribution is a triple $(n_{1},n_{2},n_{3})$ where these integers are ...
2
votes
1answer
125 views

Isometry Group of real Hilbert space?

Does the isometry group of a real separable infinite-dimensional Hilbert space have two connected components? Or, conversely, is the there even a Kuiper's theorem in the real case? How does the ...
1
vote
0answers
69 views

Construct a PDE solution from a net of approximations

Consider $P$ a linear partial differential operator in $\Bbb R ^n$. Consider some boundary condition given in the generic form $C(u) = 0$, that guarantees a unique solution (if any) of $Pu = 0$. Let ...
0
votes
0answers
24 views

Absolute continuity and the Luzin N-Property for functions of two variables

It is a well known fact that absolutely continuous functions of one real variable have the so-called Luzin N-property. That is, if $E\subset\operatorname{Domain}(f)$ has zero measure, then $f(E)$ has ...
4
votes
2answers
249 views

Vector Fields in a Riemannian Manifold

Suppose $(M,g)$ is a Riemannian manifold. Is there a way to classify manifolds where there exists a vector field that commutes with the laplace beltrami operator? Thanks
10
votes
5answers
449 views

Identities and inequalities in analysis and probability

Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of ...
1
vote
0answers
59 views

Weighted Sobolev spaces over open/closed intervals

I am escalating this question from SE, as I have been unable to obtain any guidence in relation to a possible solution. Some context, I am working with weighted Sobolev spaces of the form ...
-2
votes
1answer
161 views

No Hilbert space can have countable Hamel basis without using Baire's Category theorem [closed]

I want to prove that no Hilbert space can have countable Hamel basis just using the fact that any finite dimensional subspace is closed (more specifically without using Baire's theorem). I saw a paper ...
0
votes
0answers
48 views

Existence of a solution to $xf(x) = \int_0^1 k(x,y) yf(y) dy$

Crosspost - I asked very similar question on math.sx. Let $X = (0,1)\times (0,1)$ and $k\colon X \to \mathbb{R}$ be a Lebesgue measurable non-negative function such that $$ \int_0^1 k(x,y) dy = ...
2
votes
1answer
109 views

Completion of $C_0^{\infty}(\mathbb{R}^N)$ with norm $\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $

I have a question that I could not find it any where. Is the completion of $C_0^{\infty}(\mathbb{R}^N)$ with the respect to norm $$\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, ...
0
votes
0answers
26 views

Wiener amalgam space $W(\mathcal{F}L^{2}, L^{1}) \subset L^{1}$?

(I have asked this question on SE but could not get any answer and hope this is o.k for MO) Let $X=\mathcal{F}L^{p}=\{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{p}(\mathbb R)\},$ and $\|f\|_{X}= ...
3
votes
1answer
47 views

Under what hypothesis on the domain is the X-ray transform/John transform operator bounded?

I asked this question on math stackexchange, without any reply yet. Link:http://math.stackexchange.com/questions/1401580/under-what-hypothesis-is-the-x-ray-transform-john-transform-operator-bounded ...
0
votes
0answers
51 views

$L^2$-boundedness of integral operator

Let $a:{\bf R}^d\to M^{d\times d}$ semi-definite matrix consisted of smooth functions i.e. $$ \langle a(x) \xi,\xi \rangle=\sum\limits_{k,j=1}^d a_{kj}(x)\xi_j \xi_k \geq 0, \ \ x\in {\bf R}^d, \ \ ...
1
vote
0answers
171 views

Density of subspace with nonlocal/Wentzell boundary condition

Given the space $F$ defined by: $$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$ I want to prove that the subspace $E$ of $F$ defined by ...
0
votes
1answer
82 views

Is there relation between vector valued RKHS and interpolation space?

Vector valued RKHS which is covered extensively in the book "Pick Interpolation and Hilbert function spaces" . In a different context interpolation space is defined in the wikipedia link: ...
0
votes
0answers
46 views

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 ...
1
vote
1answer
96 views

Positive linear functionals on group algebras generated by one element

Let $g$ be a group element, let $G$ be the group generated by $g$, and let $\mathbb C G$ be the group algebra on $G$. If we define $\tau:G\to\mathbb C$ as \begin{align*}\tag{1} ...
4
votes
0answers
196 views

Area defined with $\pm$ closedness

Denote $B_n\subset\Bbb R^n$ to be unit ball at origin. Denote $S\subset B_n$ to region of type $\mathsf I$ if it satisfies $$s\in S\iff\forall t\in S, s+t\in S\mbox{ or }s-t\in S$$ I am convinced ...
3
votes
1answer
204 views

Does there exist a norm on continuous real-valued function space?

I know the space of continuous real-valued function on closed set can be given a norm by integral. How about the continuous funcion on the real line? It may be non-integrable, like f(x)=x^2. So, does ...
1
vote
1answer
92 views

Extension by zero in Sobolev spaces

Let $\Omega$ be an open bounded set of $R^n$, and let $\omega$ be an open subset of $\Omega$ s.t $\overline{\omega} \subset \Omega.$ For $f\in H_0^1(\omega)$, it is known that the extension of $f$ to ...
1
vote
0answers
74 views

Can we have extension of Mercer theorem to interpolation? [closed]

This question is related to Mercer theorem, Reproducible kernel Hilbert space(RKHS) and interpolation. The wikipedia links are https://en.wikipedia.org/wiki/Mercer%27s_theorem and ...
2
votes
2answers
105 views

The convolution between weighted $L^1$ space and normal $L^1$ space

Let $\omega\in L^1_{\text{loc}}(\mathbb R^N$) be given. We assume that $\omega\geq 1$, l.s.c, and satisfies, for a constant $C>0$, $$ \frac{1}{|B(x,r)|}\int_{B(x,r)}\omega(y)dy\leq C\omega(x) $$ ...
2
votes
1answer
72 views

Is there any parameter space of Cramér–Rao_bound

It is known that Cramér–Rao_bound is the lower bound of variance of a parameter. A useful link is https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound There is also a term called ...
0
votes
0answers
50 views

Second order differentiability of subharmonic function almost everywhere?

The following general definition of subharmonic function comes from the classical text book [elliptic partial differential equations of second order] by Gilbarg and Trudinger. We call a function $u$ ...
15
votes
1answer
317 views

The Gelfand duality for pro-$C^*$-algebras

The Gelfand duality says that $$X\to C(X)$$ is a contravariant equivalence between the category of compact Hausdorff spaces and continuous maps and the category of commutative unital $C^*$-algebras ...
0
votes
1answer
117 views

How can I show that “almost all function” have property P?

The following is cross-posted from http://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?) ...
4
votes
2answers
199 views

Well-complemented copies of $\ell_p^n$

This must be surely known but I couldn't locate this problem in the literature. It popped out in a priori unrelated approximation problem but if true, would help me greatly. Let $p\in (1,\infty)$. ...
5
votes
0answers
172 views

Version of Stone Weierstrass for functions not vanishing at infinity

I am trying to see what is known about uniform density of function spaces in $C(\mathbb{R}^n)$ or $C_b(\mathbb{R}^n)$ (bounded continuous functions on $\mathbb{R}^n$). By uniform density, I mean ...
0
votes
1answer
178 views

Is this set of function belongs to $L^\infty$?

Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $u\in SBV\cap L^\infty(\Omega)$ be given. We write $$ Du = \nabla u\lfloor \mathcal L^N + (u^+-u^-)\otimes \nu_u\mathcal ...
4
votes
1answer
171 views

Is the space of vectorial functions that are Dunford integrable complete?

Let $X$ a Banach Space and $(\Omega, \Sigma, \mu)$ a measure space. A function $F:\Omega\rightarrow X$ is Dunford integrable if $x^\ast\circ F$ is $\mu$-integrable for every $x^\ast\in X^\ast$. The ...