All Questions
12,776 questions
25
votes
6
answers
15k
views
Does every distribution define a Radon measure?
On the one hand, Wikipedia suggests that every distribution defines a Radon measure:
http://en.wikipedia.org/wiki/Distribution_(mathematics)#Functions_as_distributions (revision from February 2010, ...
- 2,895
2
votes
1
answer
2k
views
Definition of a complex structure on a vector bundle
Given a Riemann surface $S$, e.g. $\mathbb{P}^1(\mathbb{C})$, with complex conjugation on the coordinates and a holomorphic vector bundle $E$ over $S$.
The complex conjugation $f$ is not holomorphic, ...
- 1,391
2
votes
1
answer
251
views
Help determining the asymptotic behavior of an integral involving rational functions.
Let $\phi:\mathbb{P}^1\to\mathbb{P}^1$ be a rational function of degree $d\geq2$. How can one prove, using the normalized spherical measure, that
$$\int_{\mathbb{P}^1(\mathbb{C})}|(\phi^n)'(z)|\ d\mu (...
user4324
5
votes
1
answer
495
views
On the failure of the infinite dimensional Brouwer Theorem
Let $K$ be the closed unit ball of some infinite dimensional Banach
space, and let $H$ be an autohomeomorphism of $K$, having fixed
points. Can $H/2$ be fixed point free ?
Also, let ${\mathcal{F}}$ :=...
- 4,060
5
votes
1
answer
1k
views
Orthogonal complements in Hilbert bundles
It's a standard fact that for a finite-dimensional vector bundle with an inner product, the othogonal complement of any subbundle is itself a locally trivial vector bundle.
What is known about the ...
- 8,803
0
votes
1
answer
1k
views
How to prove that rational functions satisfy a Lipschitz condition in the *chordal metric*?
How can one show that rational functions satisfy a Lipschitz condition in the chordal metric on the Riemann sphere?
user4324
1
vote
1
answer
359
views
the Cech-cohomology of the sheaf of germs of plurisubharmonic functions defined on a domain in C^n
we all know that if we consider the sheaf of germs of a holomorphic functions defined on a domain in C^n,we have too many beautiful theorems characterizing the geometry of the domain by consider the ...
- 399
28
votes
2
answers
3k
views
Restriction of a complex polynomial to the unit circle
I am pretty sure that the following statement is true. I would appreciate any references (or a proof if you know one).
Let $f(z)$ be a polynomial in one variable with complex coefficients. Then there ...
- 1,304
6
votes
1
answer
1k
views
Reference needed for: every idempotent in a C*-algebra is similar to a hermitian one
The result stated in the title is thoroughly standard - or that's the impression I got.
I seem to remember seeing it stated somewhere in a book I was reading in the library, and then reverse-...
- 25.8k
3
votes
0
answers
131
views
Slicing the fibres of a meromorphic function with the zero set of a section of an ample line bundle
I'm going through a proof of a vanishing theorem by Sommese ($H^{p,q}(X,L) = 0$ for $p+q > n+k$ if $L$ is $k$-ample) and have hit the following brick wall:
I've got a complex projective manifold $...
- 4,343
7
votes
2
answers
6k
views
Using Weierstrass’s Factorization Theorem
I am trying to factorize $\sin(x)\over x$ which by Taylor series expansion and using the roots is $$a \cdot \left(1 - \frac{x}{\pi} \right) \left(1 + \frac{x}{\pi} \right) \left(1 - \frac{x}{2\pi} \...
- 5,935
5
votes
0
answers
694
views
Has the Weierstass transform been used to give Hermite series representations of the Riemann zeta function?
The inverse of the Weierstrass transform
expands a function as a series of Hermite polynomials $H_{n}$. There are several ways to invert the Weierstrass transform which led me to the following ...
5
votes
1
answer
467
views
Info about Elton–Odell theorem
Hello everyone, could anyone please tell me where can I find information about the Elton–Odell theorem?
It states:
For any infinite dimensional Banach space $X$ there is a $q > 1$ so that $X$ ...
- 105
0
votes
1
answer
410
views
About vertex algebra, mode expansion
A vertex operator is a linear map associating every state to a operator-valued distributions (quantum field) on a algebra curve, which is also called operator-state correspondence.
Chose a local ...
- 787
4
votes
2
answers
2k
views
Convergence of Gaussian measures
Let $X$ be a separable Banach space with its Borel $\sigma$-algebra $\mathcal F$. Let $x_n \to x$ in $X$. Fix a Gaussian covariance operator $K$, and let $\mathbb P_n$ and $\mathbb P$ be Gaussian ...
- 8,512
1
vote
2
answers
3k
views
bounding roots of a polynomial with Rouche's Theorem
Suppose f(z) = z^n - k [ z^(n-1) + ... + z + 1 ] where n is a positive integer and k is a real constant such that nk<1.
I have shown that a root of this ...
- 27
17
votes
1
answer
2k
views
Which Fréchet manifolds have a smooth partition of unity?
A classical theorem is saying that every smooth, finite-dimensional manifold has a smooth partition of unity. My question is:
Which Fréchet manifolds have a smooth partition of unity?
How is the ...
- 4,519
15
votes
3
answers
2k
views
Disintegrations are measurable measures - when are they continuous?
This is a sequel to another question I have asked.
The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probability spaces; ...
- 8,512
4
votes
2
answers
1k
views
Can we extract information about how fast a function decay from its Laplace transform?
My question is whether we can extract information about how fast an integrable function converges to zero by looking at the asymptotics of its Laplace transform.
More concrete case, let $f:\mathbb{R} ...
- 1,839
6
votes
7
answers
8k
views
Existence of an extreme point of a compact convex set
The Krein-Milman theorem shows that a compact convex set in a Hausdorff locally convex topological vector space is the convex hull of its extreme points.
It seems this implies that a compact convex ...
- 169
3
votes
1
answer
1k
views
characterization of continuous functionals in weak-star topology
Reading Wojtaszczyk's Banach spaces for analysts, I'm trying to understand his proof that the space of all continuous linear functionals on $(X^\star,\sigma(X^\star, X))$ is $X$.
To show the $ \...
- 922
5
votes
1
answer
807
views
Self-adjoint extension of locally defined differential operators
The following is well known. Given a symmetric differential operator, like $\partial_x^2$, defined on smooth functions of compact support on $\mathbb{R}$, $C_0^\infty(\mathbb{R})$, one can count the ...
- 21.5k
5
votes
1
answer
403
views
Nonlinear Nuclear Operators ?
Is there a "right" definition of the nuclear
operator in the nonlinear framework ? Of course, such an operator
must be compact, while a linear operator should be "nonlinearly"
nuclear iff it is ...
- 4,060
10
votes
2
answers
629
views
What do the numbers G_4 and G_6 of a lattice actually measure?
If you have a lattice $L \subset \mathbb{C}$, you can compute the following numbers:
$
G_4(L) = \sum_{\omega \in L, \omega \neq 0} \frac{1}{\omega^4}, \quad G_6(L) = \sum_{\omega \in L, \omega \neq 0}...
- 1,821
4
votes
2
answers
340
views
Embeddings of Weighted Banach Spaces
Let be $d$ a positive integer, $\Omega=\mathbb{R}^{\mathbb{Z}^d}$ and fix $R\geq 2$. We define weighted Banach spaces
$$ \Omega_p:=\left\{ x\in \Omega\left| \left[\sum_{i\in\mathbb{Z}^d}\frac{|x_i|^...
- 2,044
3
votes
3
answers
584
views
Polynomials and L^p(R)
As someone who mostly does symbolic computation, I've always been puzzled by the fascination mathematicians seem to have with Lp(R) (for p<∞)? To be more precise, there are no non-trivial ...
- 11.8k
3
votes
1
answer
473
views
Is a function which is finitely multiple-valued in each variable separately, also finitely multiple-valued in all its variables jointly?
It is well known that under suitable conditions, a function which is:
a polynomial in each variable separately is a polynomial in all its variables jointly.
a rational function in each variable ...
- 371
0
votes
1
answer
288
views
The Quantum Operations On The Bipartite Systems
Given two distinct and noninteracting quantum mechanical
systems $\mathfrak{S}\_1$ and $\mathfrak{S}\_2$ with state spaces
$\mathcal H\_1$ and $\mathcal H\_2$, respectively, the state space
of the ...
- 1
6
votes
2
answers
3k
views
Dense inclusions of Banach spaces and their duals
This seems like a really simple question, but I'm struggling with it. Let $X$ be a separable Banach space, $H$ be a separable Hilbert space, and suppose $i : H \hookrightarrow X$ is a dense, ...
- 8,512
24
votes
1
answer
2k
views
How many ways are there to globalize Harish Chandra modules?
Suppose $G$ a reductive Lie group with finitely many connected components, and suppose in addition that the connected component $G^0$ of the identity can be expressed as a finite cover of a linear Lie ...
- 4,909
8
votes
3
answers
606
views
Compact Hausdorff and C^*-algebra "objects" in a category.
This is yet more on "algebraic objects in functional analysis".
Since Compact Hausdorff spaces are algebraic over Set, it seems to follow that one can find "Compact Hausdorff objects" in any suitable ...
- 26.8k
5
votes
3
answers
2k
views
When can a function be recovered from a distribution?
What properties does a distribution (in the generalized function sense) has to have in order to be a function. That is, when is $T(\varphi) = \int f \varphi$ for some $f$?
- 259
13
votes
0
answers
816
views
How hard is it to make a differential operator Hermitian?
Let $M$ be a closed finite-dimensional smooth manifold (over $\mathbb R$). Let $C^\infty(M) = C^\infty(M,\mathbb C)$ be the algebra of smooth complex-valued functions on $M$, with the natural complex ...
- 54.6k
4
votes
1
answer
822
views
What is the tensor product of $L^p(\bf R)$ with $L^q(\bf R)$?
I'm wondering: What is the tensor product of $L^p({\bf R})$ with $L^q({\bf R})$?
(For p=q=2, the answer clearly should be $L^2({\bf R}^2)$; for other values of $p$ and $q$, it is not at all obvious ...
- 41
0
votes
1
answer
198
views
An integral arising in statistics(2)
The integral I am interested in is:
$$t(x)=\int_{-K}^{K}\frac{\exp(ixy)}{1+y^{2q}}dy$$
$K<\infty$, q natural number
For q=1 one can use contour integration.
So for K>1 we have :
$$\pi/2-\...
- 267
7
votes
2
answers
684
views
Yet more on distortion
I would like to elaborate a little bit on my previous question which can be found
here.
Firstly, let me recall that a separable Banach space $(X, \| \cdot \|)$ is said to be
arbitrarily distortable ...
- 576
1
vote
1
answer
2k
views
spectra of sums and products in (Banach) algebras [was: Spectrum in Banach Algebra]
Let a,b be 2 elements in a Banach Algebra.Let Spec(x) denote the spectrum of an element x. If a,b commute with each other, then by Gelfand Transformation, we have Spec(a+b) is a subset of Spec(a)+Spec(...
- 11
8
votes
3
answers
2k
views
Definition of a von Neumann algebra
Is there a way to equip every C*-algebra A with a functorial topology such that
the canonical map A→A** is an isomorphism if and only if A is a von Neumann algebra?
Here A** denotes the dual of A* in ...
- 37.8k
7
votes
4
answers
946
views
On operator ranges in Hilbert & Banach spaces
Lemma 1 from Anderson & Trapp's Shorted Operators, II isLet $A$ and $B$ be bounded operators on the Hilbert space $\mathcal H$. The following statements are equivalent:
(1) ran($A$) $\subset$ ...
- 8,512
11
votes
1
answer
813
views
Approximation to divergent integral
Hi everyone,
I'm a physicist working on stochastic processes and I've come up against an integral that I'm not able to approximate using steepest descent (I don't have a large or small parameter), ...
- 111
12
votes
3
answers
1k
views
What's algebraic approach to QM good for?
The algebraic formulation of quantum mechanics (and related stuff, like quantum thermodynamics & dynamical systems etc.) via C*-algebras provides a viewpoint based mostly on abstract functional ...
- 3,323
6
votes
3
answers
3k
views
Zeros of the Weierstrass $\wp$-function
This question was prompted by the post here, and I asked this earlier, deleted it, and due to pressure exerted by Ilya Nikokoshev, I am asking it again. Apologies to Pavel Etingof.
Q1. Let $\Lambda$ ...
- 7,442
5
votes
2
answers
862
views
Hilbert $C^*$-modules and approximate units
Hi,
Given a $\sigma$-unital $C^*$-algebra $A$ and a full Hilbert $A$-module $E$, is it possible to find an approximate unit $ \{\epsilon_i\}, i\in I$ in $A$ such that each $\epsilon_i$ is of the ...
- 93
2
votes
2
answers
242
views
Simultaneous convergence of powers of unit complex numbers
Let $z_1,\ldots,z_n$ be complex numbers of modulus one. Does it exist an increasing sequence $k_j\in\mathbb{N}$ such that $\lim_{j\to\infty}z_i^{k_j}=1$ for all i?
- 971
8
votes
0
answers
605
views
convergence rate in Wiener's approximation theorem
Wiener has the following fantastic results about approximations using translation families:
Given a function $h: \mathbb{R} \to \mathbb{R}$, the set $\{\sum a_i h(\cdot - x_i): a_i, x_i \in \mathbb{...
- 1,839
5
votes
0
answers
537
views
Conditional probabilities in Banach spaces
This is the infinite-dimensional sequel to my question, Conditional probabilities are measurable functions - when are they continuous?.
Let $\Omega = \Omega_1 \times \Omega_2$ be a probability space ...
- 8,512
13
votes
1
answer
860
views
What does the incidence algebra of the lattices in C tell us about modular forms?
I have two different and probably unrelated questions that can both be superficially described by the title, so I hope you'll forgive me if I ask them together. They both fall under the category of ...
- 118k
4
votes
3
answers
2k
views
Algebraic Dual / Continuous Dual
Let $E$ be an infinite dimensional Banach space, let $E^{\ast}$ denote
its continuous (i.e., Banach space) dual, and let $E'$ be its algebraic
dual. Clearly, $E^{\ast}$ is a proper vector subspace of $...
- 4,060
0
votes
1
answer
412
views
An integral arising in statistics
The integral I need:
$$t(x)=\int_{-K}^{K}\frac{\exp(ixy)}{1+y^{2q}}dy$$
$K<\infty$, q natural number
For q=1 this integral is
$$\pi/2-\int_{Arc}\frac{\exp(ixy)}{1+y^{2}}dy $$
Where Arc ...
- 267
106
votes
6
answers
19k
views
Why does the Riemann zeta function have non-trivial zeros?
This is a very basic question of course, and exposes my serious ignorance of analytic number theory, but what I am looking for is a good intuitive explanation rather than a formal proof (though a ...
- 29k