All Questions
9,959 questions
2
votes
2
answers
539
views
Fourier transform of $e^{it|\xi|^{\alpha}}$
Consider the fourier transform of $e^{it|\xi|^{2\alpha}}$ ($\alpha>0$)in $\mathbb{R}^n$,let $K_{\alpha}=\mathcal{F}(e^{it|\xi|^{\alpha}})$,so $K$ is a tempered distribution.Now I want to know if ...
CommunityBot
- 1
6
votes
2
answers
1k
views
Commuting Linear Operators In Hilbert Spaces
Let $V$ be a finite dimensional vector space over the complex field $\mathbb C$. Let $L:V\rightarrow V$ be a linear operator. Using the matrix of $L$ and the Jordan canonical form it is easy to find ...
- 5,654
1
vote
1
answer
426
views
Continuous embedding of Hardy space in Lebesgue space
I would like to have a reference to the following statement which I think is true:
$$h^1 \hookrightarrow L^1.$$
The closest I came to this is in D. Goldberg's paper, "A local version of real Hardy ...
CommunityBot
- 1
7
votes
2
answers
2k
views
Arbitrary union of meagre open sets
Let $X$ be a topological space. A subset $M$ of $X$ is called meagre (or of first category) if it is covered by the union of a countable family of closed subsets of $X$ with empty interior.
Can you ...
- 4,586
6
votes
0
answers
324
views
Ricci-flat metrics on Cotangent bundles in adapted complex structure
greetings,
Let $(M,g)$ be a compact Riemannian manifold. On some neighbourhood $X$ of the zero section in the cotangent bundle $T^{*}L$ we have a complex structure $J$ and a Kähler form $\omega$ s.t. ...
- 13
8
votes
1
answer
668
views
Hahn-Banach theorem with real extended valued function
Hello to everyone,
My problem is the following: I have this version of the Hahn-Banach theorem:
Let V be a vector space and let $p:V\rightarrow \mathbb{R}$ be any
convex function. Let $W$ be a vector ...
- 83
4
votes
1
answer
426
views
Smooth functions tangent to the leaves of a foliation
Given two smooth manifolds $M$ and $N$, it is known that if $M$ is compact, then $C^\infty(M,N)$ is a Fréchet manifold whose tangent space at $f \in C^\infty(M,N)$ is the space
$$T_f C^\infty(M,N) = \...
- 5,343
2
votes
1
answer
240
views
BM-distances between $B(\ell_p^n)$ and $\ell^{n^2}_p$
Let $\ell_p^n$ be the $n$-dimensional real or complex $\ell_p$-space and let $\mathscr{B}(\ell_p^n)$ be the space of matrices on $\ell_p^n$ endowed with the operator norm. I am looking for any ...
- 31.5k
3
votes
2
answers
1k
views
Infinite dimensional manifold
In Hamiltonian mechanics, one essentially work with $\mathbb{R}^{2n}$. However, this is only a local description of our configuration manifold $M$. More precisely, the mechanical system is regarded as ...
- 25.3k
5
votes
1
answer
762
views
Schwartz Kernel theorem for tempred functions
Let $T(R)$ denote the space of tempered functions on the line,
i.e. the smooth functions that give Schwartz function after a
multiplication by any Schwartz function, equipped with the natural
nuclear ...
- 25.3k
2
votes
1
answer
253
views
A question on Schwartz distributions
I have a question on the tempered distributions, namely, continous functionals on Schwartz class endowed with the weak* topology. Is is a Barreled space, say, a space whose convex, balanced, ...
- 16.4k
7
votes
0
answers
299
views
Generalized Skorokhod spaces
Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without ...
- 8,512
2
votes
0
answers
82
views
Description of the norm of certain interpolation space
Dear all,
I suspect that there should be some detailed description of the norm (or of the unit ball) of the following complex interpolation space (for any $0< \theta < 1$): $$\Big(B(\ell_1^n, \...
- 769
1
vote
1
answer
357
views
semi group of contractions
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, and let $B$ is a monotone linear operator such that $D(A)\subset D(B)$.
...
CommunityBot
- 1
11
votes
1
answer
633
views
Inequivalent complete norms and the axiom of choice
Hi,
I've been wondering about the following :
Is it possible, without the axiom of choice, to have two inequivalent complete norms on a vector space?
All the examples of inequivalent complete norms ...
2
votes
1
answer
400
views
Existence of a measure under certain condition
Hi everyone,
my problem seems quite simple: I have a set $\Gamma$ along with a nice $\sigma$-algebra $\mathscr{B}$. Then I have a vector space of bounded measurable functions $A \subset \mathscr{B}_{\...
- 586
2
votes
1
answer
224
views
Subalgebras of $B(E)$
Let me fix an infinite-dimensional (complex) Banach space $E$. There is a cute result of Bracic and Kuzma which says that every maximal abelian subalgebra of $\mathscr{B}(E)$, the algebra of bounded ...
- 2,378
6
votes
2
answers
605
views
$\beta\mathbb{N}$ vs $\beta\mathbb{Z}$
Just started learning the Stone-Cech compactification of discrete groups this week. My motivation comes from a question on $\beta\mathbb{Z}$. Surprisingly, I realized there are muchhhh more literature ...
- 18.7k
10
votes
0
answers
508
views
Tensorial decomposition of $B(H)$
Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
- 3,937
11
votes
0
answers
758
views
A basic question on Stone-Cech compactification of $\mathbb{Z}$
Can the identity isomorphism on the additive group $\mathbb{Z}$ be extended to a non-identity semigroup isomorphism on $\beta\mathbb{Z}$, and still preserves $\beta\mathbb{Z}\setminus\mathbb{Z}$? ...
- 895
-2
votes
1
answer
1k
views
holomorphic extension of a function [closed]
hi,
I have the following question: let $U \subset \mathbb{C}^{n}$ be some open set containing zero. let $\tilde{U} = U \cap \mathbb{R}^{n}$. assume we have a real-valued analytic function $f : \tilde{...
- 91.8k
0
votes
1
answer
1k
views
A continuous linear functional on $L^\infty(R)$ that vanishes on $C(R)$.
Intuitively, I want to construct the functional F in this way:
$$F(f)=\lim_{x\rightarrow 0+}f(x)-\lim_{x\rightarrow 0-}f(x)$$
for $f\in L^\infty$. I know this is not well defined so I'd like to find ...
- 91.8k
0
votes
1
answer
488
views
Discrete Sobolev space of $R^n$ valued maps
Can some one tell me the reference or any idea how to take the Discrete Sobolev space work defined for a scalar valued map to the space of maps which are vector valued.Let's say
$f:\Omega \...
- 3,388
15
votes
2
answers
3k
views
What do we actually know about logarithmic energy ?
In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by
$$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course it is not well ...
- 802
0
votes
1
answer
795
views
Can we construct a Hilbert space where the operator following differencial operator is symmetric?
I'd like to know if one can define a pertinent Hilbert space where the operator
$$A_p v := -\frac{1}{2} v" + (vF + v\int_\mathbb{R} Sp + p\int_\mathbb{R} Sv )'$$ is symmetric. Here, $p$ satisfies ...
CommunityBot
- 1
1
vote
2
answers
437
views
Analytical predicate for integers over complex numbers
A complex number $z$ is an integer if and only if $\sin(\pi z)=0$.
It follows that a complex number $z$ is an integer if and only $\sin^2(\pi z) = 0$. So for a real analytic function $f$ and any real ...
- 114k
8
votes
2
answers
472
views
Notion of generalized function/distribution for functional derivatives?
Is there any work on defining something analogous to a generalized function for functions whose domain is a Hilbert or Banach space? Is there an extension of the notion of Frechet/Hadamard/Gateaux ...
- 784
4
votes
2
answers
473
views
Galois cover via C star algebras
Hello to all, here's my question, I hope it's not too trivial. I haven't found reference for it so far.
We know that abelian C star algebras are the same as locally compact spaces.
Now what is the ...
CommunityBot
- 1
1
vote
1
answer
111
views
Log-nonexpansive functions: terminology and references
During my recent work in the optimization of positive valued functions, the following class of functions proved to be exceptionally important.
(Defn.). Let $h: (0,\infty) \to (0,\infty)$ be ...
- 12.7k
1
vote
1
answer
340
views
Reflexive Besov spaces
I don't know whether the Besov space $B^1_{1,1}$ on a one dimensional torus is reflexive or not? Can someone help me please?
- 39.1k
10
votes
1
answer
929
views
Non-probabilistic proof of the Johnson–Lindenstrauss lemma
The Johnson–Lindenstrauss lemma states that a small set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are ...
- 2,378
16
votes
3
answers
791
views
Random products of projections: bounds on convergence rate?
The von Neumann-Halperin [vN,H] theorem shows that iterating a fixed product of projection operators converges to the projector onto the intersection subspace of the individual projectors. A good ...
- 201
3
votes
1
answer
189
views
Sufficient (and concrete) condition for a function to satisfy some measure theoretic property
I'm interested in the following property, for a positive and locally bounded function $\omega:\mathbb{R}\to\mathbb{R}^d$, $d\ge 1$: there exists a countable sequence of open and pairwise disjoint sets ...
- 3,270
0
votes
1
answer
382
views
Double duals characteristic [closed]
Recall that (for $1\le p<\infty$), $\ell^p = \{\{a_n\}_{n=1}^\infty:\sum\limits_{i=1}^\infty|a_i|\lt\infty\}$, with norm $||\{a_n\}||=(\sum\limits_{i=1}^\infty|a_i|^p)^{\frac{1}{p}}$.
It is well ...
- 32.5k
2
votes
1
answer
202
views
tensor product with projective topology
There are two Banach spaces $X,Y$. These spaces have unconditional Schauder bases $\{e_i\}$ and $\{f_i\}$ respectively.
Is this right that $e_i\otimes f_j$ is the unconditional Schauder basis in $X\...
CommunityBot
- 1
6
votes
1
answer
751
views
left- and right- Folner sets
Given an amenable group, it is a standard trick to turn a left-invariant mean ( i.e. a continuous positive normalised linear functional $m:\ell_\infty(G) \to \mathbb{R}$ such that $\forall g \in G, m \...
- 17k
1
vote
1
answer
1k
views
Almost analytic continuation
Let $f\in S^{\alpha}$ for some $\alpha \in \mathbb{R}$(which means that f is smooth and satisfies $|D^{\beta}f|\leq C(1+|x|)^{\alpha-\beta}$),a function $\tilde{f}$ on $\mathbb{C}$ is called an almost ...
- 21
0
votes
1
answer
142
views
A special Integral Kernel
Does there exist either one / general class of non-negative definite , symmetric Integral Kernel map satisfying the following properties ??
$f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$
$K:L^2(\...
- 16.2k
0
votes
0
answers
186
views
Properties of Eigenfunctions of a Kernel
I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple and provide me some references.
I've and Kernel function $K(x,y)$
$f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$...
- 17
0
votes
0
answers
255
views
Convergence of a function in a metric space to its metric.
Given a metric space $(\mathbb{A},d)$ in $\mathbb{R^n}$ with a metric $d$ being the Euclidean metric:
If $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent sequence where $A$ ...
- 101
2
votes
1
answer
261
views
Is C^{k+1}(X) compactly contained in C^{k}(X) for a closed manifold X?
Hi all,
I apologize if this question is too low level for mathoverflow. I'm happy to move it to math.stackexchange if so.
Let $X$ be a closed manifold, let $k$ be a nonnegative integer and let $C^k(...
- 2,409
2
votes
1
answer
403
views
The set of Upper semi-continuous functions as a ring.
I should recall that the surgenfery topology on the real numbers is denoted by $\mathbb{R}_l$, and has the set
{$[a , b): a,b \in \mathbb{R} $} as it's base.
If $X$ is a topological space, an upper ...
- 149k
4
votes
1
answer
243
views
When is Prim(A) of an infinite discrete group hausdorff ?
Does anyone know, if the following result has been proved ?
Let G be an infinite discrete group. A = L1(G) it's algebra and Prim(A) the set of prime ideals with spectral topology.
The result is :
...
- 21
7
votes
1
answer
1k
views
Helmholtz-Decomposition on compact Riemannian manifolds
For smooth domains $\Omega$ in $\mathbb{R}^n$ it is known that one can decompose vector fields in $L^p(\Omega)^n$, $1 < p <\infty $ into a "gradient"- and a "divergence-free"-part such that
$L^...
CommunityBot
- 1
0
votes
0
answers
164
views
Can we separate Toeplitz matrices for negative and positive eigenvalues?
Consider a Toeplitz matrix T which has both positive and negative eigenvalues. Can we prove that there exist two Toeplitz matrix T1 and T2 such that T1+T2=T and T1 has only one positive Eigenvalues ...
- 9
1
vote
2
answers
659
views
A function which belongs on a concrete Besov Space
Please, anyone of you know a simple example of a function which belongs to the Besov Space with $p=q=\infty$ and $s=0$ (over $\mathbb{R}$ or $I\subset\mathbb{R}$ where $I$ is a closed interval). I ...
- 16.2k
9
votes
0
answers
397
views
Does the algebra of bounded variation functions have a "noncommutative geometric" meaning and generalization?
According to Gelfand-Naimark theory, $C^*$-algebras of continuous functions $\mathcal{C}^0(X,\mathbb{C})$ on a compact Hausdorff topological space completely capture its topology. Furthermore, every ...
- 23.3k
3
votes
1
answer
678
views
Is this kernel space of finite dimension ?
Assume that $P \in \Psi^{m}(X)$ (X is a $C^{\infty}$ manifold)is properly supported and has a real principal part p which is homogeneous of degree m.I'm interested in the existence theorem(at least ...
- 16.4k
2
votes
0
answers
259
views
Common eigenvector
I have little experience with functional analysis beyond an undergraduate basic course, and I'm dealing with the following problem:
let $V$ be an infinite-dimensional locally convex (but not normed!) ...
- 6,962
26
votes
3
answers
7k
views
Dual of bounded uniformly continuous functions
Let $(X,d)$ be a metric space, and let $C_u(X)$ be the Banach space of bounded uniformly continuous functions on $X$ (with the uniform norm). How can I characterize its dual space $C_u(X)^*$?
I ...
- 2,409