All Questions
12,776 questions
1
vote
0
answers
283
views
Density of Dolean exponentials in L2 and Wiener Measure
Assume that W is the classical Wiener space C([0,1],R) note $\mu$ the Wiener measure, and denote by $\mu_s$ the image of $\mu$ under the maping $T: W ->W$ such that$ T(w)= \sqrt(s) w$ . Denote by $...
- 19
8
votes
1
answer
612
views
Is the set of exponentials open?
Let $A$ be a $C^*$-algebra or some norm-closed algebra of operators on a Hilbert space.
In the old paper
Hille, E. On Roots and Logarithms of Elements of a Complex Banach Algebra, Math. Annalen, ...
- 25.5k
-1
votes
1
answer
311
views
A differential equation
let $g(s)$ be real-valued function defined on $[0,T]$ such that $g(T)=0$ and suppose that $g$ is a "nice function"
Assume that $0<\gamma<1$, $v$ is a positive number, and
$$\frac{dg}{ds}+(v\...
- 1
1
vote
1
answer
706
views
Plancherel-Polya Type Inequality for non-compactly Fourier-supported Functions??
Hi!
The Plancerel-Polya inequality can be stated as follows:
Let $0 < p\le \infty$ and $ \nu \in \mathbb{Z}$. Suppose that $g$ is a (smooth) function satisfying $\mbox{supp }\hat g \subset \...
- 979
2
votes
1
answer
1k
views
Hilbert Schmidt operators
I don't know much about the theory of Hilbert spaces but a research project has me working with them a little bit. In particular requiring an operator to be Hilbert-Schmidt is a recurring condition. ...
- 3,968
7
votes
3
answers
1k
views
Nice orthonormal basis for L^2(Cantor set)
Let X be the Cantor set, which we view as the space $2^\mathbb{N}$ (the set of all infinite binary sequences), equipped with the product topology. We can construct a Borel probability measure $\mu$ on ...
- 81
4
votes
2
answers
707
views
Selecting basic sequences
Suppose $(x_\alpha)_\alpha$ is an uncountable, linearly independent family of norm one vectors in a Banach space. Can one always select a basic sequence (or at least a minimal system) from this family?...
- 355
1
vote
2
answers
541
views
Weierstrass Theorem [closed]
Hi--
Where can I find a proof of this theorem: For each $r \in \mathbb{Z}_{+}$,
there exists a complex entire function $f(z)$ such that $f(r) \neq 0$ but
$f(r+1)=f(r+2)=\cdots =0$,
i.e. $f(z) \in ...
- 4,795
18
votes
5
answers
2k
views
What is the spectrum of the ring of entire functions?
Let $\mathcal{O}(\mathbb{C})$ be the ring of entire functions, that is, those functions $f : \mathbb{C} \to \mathbb{C}$ which are holomorphic for all $z \in \mathbb{C}.$ For each $z_0 \in \mathbb{C}$.
...
- 3,268
5
votes
1
answer
666
views
Question regarding divergence
Let $E$ be a closed and convex set of distributions on a finite set $A$. Let $P',Q'\notin E$ and let $P^{\star},Q^{\star}$ be their respective estimates in $E$ with respect to the KL-divergence, i.e.,...
- 779
1
vote
1
answer
635
views
Closed range for a continuous linear transformation
I have a Banach space $B$ and a continuous linear transformation $F:B \rightarrow B\times B$. One of the induced transformations $F(1):B \rightarrow B$ and $F(2):B \rightarrow B$ into the factors of ...
- 549
11
votes
0
answers
528
views
Adeles of Holomorphic Functions
In number theory, an adele is a restricted product of elements of the completion at each prime. For function fields, we take (a kind of) product of the completion at each point, and at non-singular ...
- 15.4k
0
votes
1
answer
652
views
Fiberwise torsion free and generically null sheaf for flat morphism
Hi.
Has some one an example of sheaf $A$ on flat morphism $f:X\rightarrow S$ of reduced complex spaces with fibers of constant positive dimension (or locally noetherian excellent schemes without ...
- 435
17
votes
3
answers
2k
views
What is a reasonable finitary analogue of the statement that harmonic functions are smooth?
In my answer to this question on MU, I suggested that the OP think about the difference between real-differentiable and complex-differentiable functions by using a sort of finitary analogue. One way ...
- 118k
3
votes
1
answer
575
views
Kahler forms on Cohen Macaulay spaces
Hi.
Can anyone answer the two following questions:
For $n$-dimensional $X$ Cohen-Macaulay complex space, is it true that the sheaf of top degree homolorphic forms $\Omega^{n}_{X}$ has no ...
- 435
6
votes
1
answer
978
views
Simply-connected domain around a curve
In a current project with a colleague, we have come across the following reasonably classical-sounding geometric question. While not vital to our work, it would be interesting if anyone has seen this ...
- 6,548
7
votes
1
answer
737
views
Question about projections on a Hilbert space
Sorry for the vague title, I can't think of a better one that isn't overly long.
Suppose that $S$ is a commuting set of projection operators on a Hilbert space. I'll introduce the following notation: ...
- 391
7
votes
2
answers
790
views
Question about von Neumann algebra generated by a complete algebra of projections
Hi all, sorry if this is a dumb question, I don't know much about von Neumann algebras except the definition and a few relevant facts I've managed to prove by myself so I expect the answer will turn ...
- 391
1
vote
1
answer
717
views
Double dual space of a C* algebra A
We know that $A$ embeds into $A$** (the double dual space of $A$ ). Is the following true? If $\Psi$ is in $A$** and weak* continuous, is there an element $a \in A$ such that $ \Psi$ is the ...
- 831
7
votes
0
answers
189
views
When is the Locus of Equi-modular points of two monic polynomials with integer coefficients contained in the unit disk?
If $\lambda_{1}(z)$ and $\lambda_{2}(z)$ are two monic polynomials (relatively prime) with integer coefficients and $$\Gamma:=\lbrace z \rm{\ s.t.\ } |\lambda_{1}(z)|=|\lambda_{2}(z)|\rbrace,$$ when ...
- 103
8
votes
1
answer
920
views
Looking for references talking about category of topological vector spaces
It's known that category of topological vector spaces is not abelian but quasi-abelian or exact category. I am looking for the references playing with this category(category theory). All the related ...
- 5,515
5
votes
2
answers
579
views
Improved versions of discontinuous functions
Given a set X (such as the set of points in an interval), the space ℝX of all real-valued functions on X is not usually the function space we work with -- it is "too large" in some sense. Thus, ...
- 8,903
21
votes
0
answers
876
views
Are the eigenvalues of the Laplacian of a generic Kähler metric simple?
It is a theorem of Uhlenbeck that for a generic Riemannian metric, the Laplacian acting on functions has simple eigenvalues, i.e., all the eigenspaces are 1-dimensional. (Here "generic" means the set ...
- 6,247
7
votes
3
answers
1k
views
If *Y* is weakly dense in *X*, is the unit ball in *Y* necessarily dense in the unit ball in *X*?
Let X be a normed space and denote by X* the space of all bounded linear functionals on X. Take a linear subspace G ≤ X* which separates the elements of X, i.e., for each x ∈ X, there is an f &...
- 73
3
votes
1
answer
1k
views
Cyl(E) = Borel(E) for E non-reflexive Grothendieck Banach space
This is sort of a follow-up to Borel(X) = \sigma(X') for X non-separable
PROBLEM: Given a Banach space $E$ over $\mathbb{K} \in \{\mathbb{C}, \mathbb{R}\}$ that has the Grothendieck property. ...
- 1,338
17
votes
1
answer
847
views
Irrational Numbers and the Riemann Surface of a Multi-Valued Function
Suppose a meromorphic function $f(z)$ has two poles, with residues $1$ and $\gamma$, respectively. Then the topology of the Riemann surface of the anti-derivative of $f(z)$ depends on whether or not $\...
- 15.4k
2
votes
1
answer
2k
views
Cesaro convergence implies weak convergence of a subsequence
Suppose a bounded sequence $(x_n)$ converges to $x$ in the Cesaro sense (i.e., $\frac{1}{n}(x_1 + x_2 + \dots + x_n)\rightarrow x$) in a separable Hilbert space $H$. How to prove that some subsequence ...
- 5,019
1
vote
2
answers
3k
views
Weak-* compactness in L^1
Hey I'm really stuck on what I think is an interesting 'paradox'. Consider the sequence of functions $f_n = 1_{[n,n+1]}$ (indicator functions of the interval $[n,n+1]$.
These are uniformly bounded ...
- 19
3
votes
1
answer
615
views
When is a fixed point of f^n a fixed point of f?
Let $E$ be a Banach space and $f:E\to E$ be a continuous map. By $f^n$ we denote the $n$-th iterate of $f$, i.e. $f^n:=\underbrace{f\circ f\circ\cdots \circ f}_{\text{n times}}$. Let $x_0$ denote a ...
- 41
13
votes
4
answers
1k
views
"Simple" Kahler manifolds
I have some lecture notes from Demailly on Kahler geometry where he talks about "variétés Kahleriennes simples", which are defined as Kahler manifolds $X$ such that for very generic points $x_0$ in $X$...
- 4,343
11
votes
1
answer
603
views
Reference for a particular Radon transform on non-positively curved spaces
Let me first recall that the classical Radon transform takes a (smooth compactly supported, say) function $f$ defined on $\mathbb{R}^n$ as an input, and gives as output the map $H\mapsto \int_H f$ for ...
- 14.4k
14
votes
1
answer
1k
views
Any further applications of Freudenthal's 1936 Spectral Theorem?
Seemingly completely forgotten, back in 1936, the Dutch mathematician Freudenthal, quite well known at the time, proved his so called Spectral Theorem, see chapter 6 in Luxemburg & Zaanen : Riesz ...
5
votes
2
answers
1k
views
Applications of minmax theorem(s)
Intro We suppose $X$ and $Y$ are nonempty sets and f: $X\times Y \rightarrow \mathbb{R}$. A minimax theorem is a theorem that asserts that, under certain conditions,
$$ \inf_Y \sup_X f = \sup_X \...
6
votes
1
answer
444
views
When does a matrix define a convolution operator on a hypergroup?
Let $H$ be a discrete hypergroup. Suppose I have a matrix $A=(A_{x,y})$ indexed over $H$ with nonnegative entries which defines a bounded operator on $\ell^2(H)$. When does there exist $f\in\ell^1(H)$ ...
- 5,425
39
votes
3
answers
6k
views
On linear independence of exponentials
Problem.
Let $\{\lambda_n\}_{n\in\mathbb N}$ be a sequence of complex numbers . Let's call a family of exponential functions $\{\exp (\lambda_n s)\}_{n\in\mathbb N}$ $F$-independent (where $F$ is ...
- 22.3k
24
votes
3
answers
9k
views
Functions of several complex variables: book recommendations?
Can anyone recommend a good comprehensive introduction to functions of several complex variables that a) is fairly up to date, b) isn't a geometry or an algebra book only, but takes multiple ...
18
votes
1
answer
3k
views
Let a function f have all moments zero. What conditions force f to be identically zero?
Throughout, let $f$ be a Lebesgue measurable function (or continuous if you wish, but this is probably no easier). (Questions with distributions etc. are possible also but I want to keep things simple ...
- 1,990
0
votes
1
answer
6k
views
Is there an Isomorphism between R and C under addition? [duplicate]
Possible Duplicate:
AC in group isomorphism between R and R^2
Somewhere, I recall being told that there is an isomorphism between $\mathbb{R}$ and $\mathbb{C}$ under addition. However, despite a ...
- 5,759
7
votes
1
answer
577
views
Does a crossed product R⋊_α F_n of the hyperfinite factor of type II_1 and a free group have the QWEP?
Let $\mathcal{R}$ be the hyperfinite factor of type $\rm{II}_1$ and let $\mathbb{F}_n$ be a free group with $n$ generators. Let $\alpha$ be an action of $\mathbb{F}_n$ on $\mathcal{R}$.
Does the von ...
- 1,222
18
votes
1
answer
5k
views
Unbounded linear operator defined on $l^2$
Let $l^2$ be a Hilbert space of infinite sequences $(z_0, z_1, \cdots)$ with finite $\sum_{i=0}^{\infty} |z_i|^2$.
Are there any simple example of unbounded linear opearator $T: l^2 \to l^2$ with $D(...
- 2,821
15
votes
1
answer
2k
views
Dirichlet series expansion of an analytic function
Let $F(s)=\sum_{n\geq 1}\frac{a_n}{n^s}$ be a Dirichlet series with (finite) abscissa of absolute convergence $\sigma_a$. It can be shown that $\forall \sigma >\sigma_a:$
$$\lim_{T\to\infty}\frac{1}...
- 7,127
3
votes
1
answer
1k
views
Adjoint/transpose of wavelet transform
I'm using a wavelet transform in Matlab, so I think of it as a black-box. I'll represent it here as $W(x)$. There's a reconstruction function as well, which I'll write as $W^\dagger(y)$. I can ...
- 170
0
votes
0
answers
520
views
Motivation of proof of Riemann-Roch for elliptic curve and generalizations
Given a lattice $L \subseteq \mathbb{C}$, Alain Robert defines a theta function as a meromorphic function such that $\theta(z+\omega)=a(\omega) e^{\pi h(\omega)(z+\frac{\omega}{2})} \theta(z)$ for all ...
- 15.4k
10
votes
2
answers
960
views
Stone-Weierstrass for cones
A version of the Stone-Weierstrass Theorem asserts: If A is a linear subspace of C(K), the set of continuous functions on a compact space, and if A is a subalgebra that contains the constant functions ...
- 101
4
votes
1
answer
228
views
When can closedness of the range of an operator be checked on a positive cone?
Let $T:X\to Y$ be an operator between Banach spaces $X$ and $Y$. Assume that $X$ has a positive cone $C\subset X$, which generates $X$: every element of $X$ can be written as a difference of elements ...
user2412
2
votes
2
answers
874
views
Dimension of the space of harmonic functions on the unit ball
Is the dimension of the space of $H^2(B)$ harmonic functions on unit ball $B\subset\mathbb{R}^d$ countably or uncountably infinite?
- 21
5
votes
2
answers
3k
views
Uniform convergence of difference quotient
Let $\phi\in C^\infty_c(\mathbb R)$ be a smooth function with compact support.
For $h>0$ define the difference quotient $\phi_h\in C^\infty_c(\mathbb R)$ by $\phi_h(t)=\dfrac{\phi(t+h)-\phi(t)}{h}$...
- 3,184
4
votes
3
answers
3k
views
Distributional derivative of non continuously differentiable functions
Hello,
let $f$ be a continuously differentiable function on $R^n$. Then its classical derivative and its distributional derivative coincide.
It is known (cf. Rudin, Functional Analysis, Sect. 6.13) ...
- 5,327
0
votes
1
answer
2k
views
pure dimensional and embedded components
Hi.
Let $X$ be a pure $n$-dimensional complex subspace of manifold $Z$. It is true that $X$ has no embedded components? (perhaps that is obvious with Weierstrass preparation theorem or Noether ...
- 435
6
votes
1
answer
581
views
A puzzling question on real interpolation
Suppose an operator $T$ is bounded on $L^2$ and also bounded from $L^{1}$ to $L^{1}$-weak. Then by Marcinkewicz interpolation one gets that $T$ is bounded on every $L^{p}$ for p between 1 and 2. ...
- 9,007