All Questions
12,780 questions
5
votes
2
answers
385
views
on the approximation by holomorphic functions
Good evening,
I have a question on the approximation of holomorphic functions on a space of cartesian product type.
Question: Let $U,V$ be domains in $\mathbb{C}^n$ and $f\in \mathcal{O}(U\times V)$ ...
- 758
3
votes
2
answers
2k
views
Positive Fourier coefficients
Hi all,
Is there any general way to construct a smooth 2pi periodic function which vanishes on an interval and has positive Fourier coefficients?
And if that was too specific I can make this more ...
- 549
6
votes
1
answer
1k
views
On holomorphic branched coverings of a domain in the plane to the unit disk
This question is partly motivated by my answer to this question on math.stackexchange.
Let $\Omega$ be a bounded $n$-connected domain in the plane, bounded by $n$ pairwise disjoint Jordan curves.
...
- 2,154
1
vote
1
answer
767
views
Is the set of all probability measures weak* closed?
Let $(\Omega,\Sigma)$ be a measurable space. Denote by $ba(\Sigma)$ the set of all bounded and finitely additive measures on $(\Omega,\Sigma)$ (see http://en.wikipedia.org/wiki/Ba_space for a ...
- 101
0
votes
1
answer
286
views
Irreducible subspaces of separable Hilbert space
A question about definition: Let $\mathcal{H}$ be a separable Hilbert space over $\mathbb{C}$, with $B(\mathcal{H})$ the bounded linear operators on it. What does it mean to have an irreducible ...
- 103
-2
votes
1
answer
295
views
When does the adjoint operator map closed convex subsets to closed convex subset?
Let $T:X\rightarrow Y$ be a linear continuous map between Banach spaces $X$ and $Y$ and denote by $T':Y'\rightarrow X'$ the norm adjoint of $T$. Let $M\subseteq U'$ be a subset of
the unit sphere $U'$ ...
- 101
2
votes
2
answers
303
views
Analytic functions & convexity
It is known (Scwhartz-Pick) that holomorphic functions are distance non-increasing in the hyperbolic metric.Does it imply that they preserve convexity in the hyperbolic metric?
How about the ...
- 41
2
votes
2
answers
422
views
non-Identity operator on a separable Hilbert space
Suppose $\mathcal{H}$ is a separable Hilbert space over $\mathbb{C}$ (countable dimensions) with inner product $\langle,\rangle$. Let $A$ be a bounded linear operator on $\mathcal{H}$, i.e, in $B(\...
- 103
9
votes
3
answers
2k
views
Generalizations and relative applications of Fekete's subadditive lemma
Fekete's (subadditive) lemma takes its name from a 1923 paper by the Hungarian mathematician Michael Fekete [1]. A historical overview and references to (a couple of) generalizations and applications ...
- 10.5k
2
votes
0
answers
288
views
Limit of an inverse Mellin transform
In Edwards' very nice book ``Riemann's zeta function'' the following integral comes up in section 1.14. Suppose $\beta = \sigma + i\tau$ with $\sigma > 0$. Suppose $x > 1$. Fix some real number $...
- 21
11
votes
2
answers
2k
views
Spectrum of $L^\infty(X,\mu)$
Suppose that $(X,\Sigma,\mu)$ is a measured set with respect to $\sigma$-algebra $\Sigma$.
Suppose that $L^\infty(X,\mu)$ is the set of all $\mu$-equal bounded $\Sigma$-measurable functions on $X$. ...
2
votes
1
answer
213
views
Local convexity of C([a,b])
Let $C([a,b],\mathbb{R})$ denote the space of continuous functions from $[a,b]$ to the real numbers. For a function $f\in C([a,b],\mathbb{R})$ and $d\gt 0$, define
$$p_d(f) :=sup\{\lvert f(x)-f(y)\...
- 41
0
votes
1
answer
220
views
Frames and completeness
Let $H$ be a separable Hilbert space.
A sequence $\{f_{n}\}$ is a frame for a separable Hilbert space $H$ if there exists $A,B>0$ such that for all $f$ in $H$
$$
A\|f\|^2 \leq \sum |\langle f, ...
- 71
5
votes
2
answers
686
views
Bounded linear functionals and representations
Suppose that $A$ is a unital C$^*$-algebra and that $\varphi: A \to \mathbb{C}$ is a bounded linear functional. Then there exists a Hilbert space $H$, a representation $\pi: A \to B(H)$ and vectors $\...
- 153
1
vote
0
answers
125
views
Isomorphisms of group extensions arising from antisymmetric forms
Let $V,W$ be topological vector spaces and fix continuous antisymmetric bilinear forms $\omega_1:V\times V\to \mathbb{R}$, $\omega_2:W\times W\to\mathbb{R}$. Since $\omega_1$ is a 2-cocycle (in fact ...
- 1,411
18
votes
1
answer
996
views
Existance of certain almost invariant functions related to amenability and piece-wise transformations
We would like very much to know the answer to the following question:
Let $\|\cdot\|$ be any norm on $\mathbb{Z}^d$ and let $W(\mathbb{Z}^d)$ be the group of all bijections of $\mathbb{Z}^d$ such ...
1
vote
2
answers
316
views
Solutions of $\zeta(s) = 1$, $\zeta(\zeta(s)) = 1$ near a line and a circle, respectively?
Are solutions of $\zeta(s) = 1$ very near a line $\Re(s) = 54$ and solutions of $\zeta(\zeta(s)) = 1$ either on or very near a circle with center $\approx .00936$ tangent to $\Re(s) = 1$, known to ...
- 11
9
votes
1
answer
935
views
Question about an estimate in Hörmander's proof of Cartan's Theorem B
I have been working through the proof of Cartan's Theorem B that Hörmander gives in his book 'Introduction to Complex Analysis in Several Variables'. When I began, I skipped over some of the initial ...
- 19.3k
4
votes
0
answers
1k
views
The spectrum of a Markov Operator and Invariant Measures
Suppose I have a discrete-time Markov Chain (in an infinite dimensional state space $\Omega$) with Markov operator $P$, a linear operator on the space of bounded measurable functions on $\Omega$. (Or ...
- 509
1
vote
0
answers
465
views
Reference book for a stronger version of Goldstine theorem
Let me quote the proof of Goldstine theorem from wikipedia ( http://en.wikipedia.org/wiki/Goldstine_theorem ):
Given an $x^{**} \in B_{X^{**}}$, a tuple $(\phi_1, \dots, \phi_n)$ of linearly ...
- 311
5
votes
2
answers
356
views
$L^\infty$ properties of an infinite-dimensional Gaussian semigroup
Let $W$ be a separable Banach space and $\mu$ a Gaussian Borel measure on $W$ which is centered and non-degenerate. For $F : W \to \mathbb{R}$ bounded Borel and $t \ge 0$, let
$$P_t F(x) = \int_W F(x+...
- 29.8k
0
votes
1
answer
448
views
Uniform convergence of a series to exponent [closed]
I'm trying to prove that in the complex plane $\left(1+\frac{z}{n}\right)^n$ converges uniformly to $e^z$ in every closed disc $|z|\leq c$. I thought about showing the sequence as a logarithm of ...
- 3
1
vote
0
answers
119
views
Particular types of basis on a normed vector space of finite dimension
Is it true that on every normed vector space $V$ of dimension $n$ there exists a basis of norm $1$ vectors $v_i$, such that $\|\sum_{i=1}^n\epsilon_iv_i\|\geq 1$ for all possible combinations of $\...
- 31
1
vote
2
answers
581
views
Lower bound for the eigenvalue
For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined
$$
L_c(\psi)=(1-x^2)\frac{d^2\psi}{dx^2}...
- 71
0
votes
1
answer
2k
views
weak convergence in Sobolev spaces and pointwise convergence?
I encounter a problem when reading Struwe's book Variational Methods (4th ed). On page 38, it is assumed that
$\|u_m\|$ is a minimizing sequence for a functional $E$, i.e. $E(u_m)\rightharpoonup I$...
2
votes
0
answers
241
views
Plane Curve invariants via Contour Integrals
We learn in complex analysis class how to find the winding number of the contour $\Gamma$ around the origin.
\[ n = \frac{1}{2\pi i} \oint \frac{dz}{z} = \frac{1}{2\pi i} \oint d(\log z)
= \...
- 22.8k
2
votes
2
answers
2k
views
Understanding the inverse Laplace transform of a function with essential singularities
I need to do an inverse Laplace transform of a function with essential singularities for a specific problem. I find it is very similar to an equation J. Noolandi worked out in one of his papers in ...
- 23
8
votes
0
answers
1k
views
Strictly singular operators and their adjoints
This is a question I thought about a while back and figured I'd throw it out there to see if anyone has some insight that I am missing.
Let $X$ and $Y$ be infinite dimensional separable Banach ...
- 1,073
5
votes
2
answers
881
views
Fourier transform on locally compact quantum groups
I have read some articles on locally compact quantum groups and the Fourier Transform on them. I wonder why we define the Fourier transform as an operator valued functions from $L^1(\mathbb{G})$ to $L^...
- 71
0
votes
0
answers
150
views
$n$-th derivative of the prolate spheroidal function
For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined
$$
L_c(\psi)=(1-x^2)\frac{d^2\psi}{dx^2}...
- 71
8
votes
2
answers
583
views
Does every operator from a Hilbert space to $L^0$ factor through a canonical one?
Let $A:H\to L^0(S, \mu)$ be a continuous operator from a Hilbert space to the space of (equivalence classes of) measurable functions on a probability measure space $S$ with convergence in measure. Let'...
- 4,010
5
votes
1
answer
842
views
Hurwitz's automorphisms theorem with deformations
Hurwitz's automorphisms theorem bounds the order of the automorphism group of a negatively curved Riemann in terms of the genus.
Now suppose a finite group $G$ acts faithfully on a Riemann surface $...
- 17.6k
3
votes
1
answer
352
views
Integral Equation with "convolution"
I've got the following problem I'm working on which is related to some of my research:
Solve:
$f(x) = \int_{-\infty}^x G(x,y)f(y)f(x-y)dy$
for f, given $G$ which has whatever smoothness ...
- 31
11
votes
0
answers
601
views
High-dimensional geometry: Top-down Vs. Bottom-up
There are several ways to leverage one's intuition from low-dimensional geometry to understand high-dimensional phenomena. For example, one can get a clearer picture of the behaviour of high-...
- 1,666
1
vote
2
answers
425
views
( finite ) Blaschke product in higher dimensions ?
Hello, as we know, the (finite) Blaschke product $P$ in $\mathbb{C}$ or in $ \mathbb{R}^2 $ is defined by $\prod_{j=1}^{k} \frac{z-a_j}{1-\bar{a_j}z}, a_j \in \mathbb{D}$. I was wondering whether ...
- 1,471
1
vote
0
answers
121
views
showing convergence of a function recursion relation
I have obtained (formally) a perturbative solution
$$
H(y) = \sum_{n=0}^\infty \delta^n H_n(y)
$$
to the following integro-differential equation ($\delta$ is a small constant, $\nu$ is a L\'evy ...
- 351
5
votes
2
answers
2k
views
Double dual of C*-algebra
Suppose, that $A$ is a $C^{\ast}$-algebra. It follows, according to Sakai's book, that double dual of $A$ is also $C^{\ast}$-algebra. I'm not quite sure if I understand the proof correctly. Author ...
- 111
1
vote
1
answer
190
views
cardinality of discontinuity curves of BV function
If the function $f:R\to R$ is of BV class then it has at most countably many discontinuity points (since it can be represented as a sum of two monotonic function).
I am interested to know whether the ...
- 13
2
votes
2
answers
181
views
convergence of the coefficients of lacunary series
I just want to find some standard reference to the following result: let $(a_k)_k$ be the sequence of coefficients of a lacunary Fourier series which converges to an $L_1(T)$ function in the sense of ...
16
votes
1
answer
2k
views
Certain functional equations for the Riemann Zeta function?
Referring to this question I asked on math.SE.
I am posting a more generalized question here, for answers and further inquiry.
For the Riemann zeta function, we know of the standard functional ...
- 731
0
votes
1
answer
905
views
Hölder continuity of uniform limit of piecewise constant functions
Consider a piecewise constant function $v: [a,b] \rightarrow \mathbb{R}$ defined by a finite partition $a=t_0 < t_1 < t_2 < ... < t_s=b$ of the interval $[a,b]$, and constants $m_1,m_2,...,...
- 81
3
votes
0
answers
411
views
Connections between the "local parametrization theorem" and "Noether normalization theorem"
In the study of local theory for holomorphic varieties, the Local Parametrization Theorem states that in $\mathbb{C}^n$,for any irreducible germ of holomorphic variety $V$ at 0, there exist a ...
- 1,111
2
votes
1
answer
356
views
Analytic extension across the boundary.
Let $Q=[0,\infty)\times [0,\infty)\subset \mathbb C$ and $f: Q\times Q\to Q\times Q$ be a diffeomorphism.
such that $f$ is holomorphic in the interior of $Q\times Q$. Can we extend this map ...
- 541
0
votes
1
answer
446
views
About an integral transform or generalized Laurent series
We start with a little of context. I needed that a function from $\mathbb{R}^+$ to $\mathbb{R}$ could be represented in the following form, not necessarily uniquely:
$$
K(z)=\int_{-\infty}^{\infty}A(\...
- 157
1
vote
0
answers
153
views
The existence of the solution of the perturbed KdV Equation(semi-group operator)
Consider the perturbed KdV Equation$$u_t-6uu_x+u_{xxx}=\epsilon u,u(x,0)=f(x)$$where $f(x)=v(x,0)$，$v(x,t)$ is a soliton solution.$u$ satisfies the condition$u\to 0 $when $|x|\to\infty$
I want to use ...
- 61
1
vote
1
answer
4k
views
how to prove the range of a closed linear operator is closed ?
The closed range theorem tells us that given two banach spaces X,Y,and a closed densely defined linear operator T：$X \to Y$. We have the following equivalence $R(T)$ is closed in $Y \iff R(T^{*})$ is ...
- 1,644
1
vote
0
answers
271
views
When Pelczynski's property (V*) forces (V) in the dual space?
Recently, I have been asked if are there any sufficient conditions for a Banach space with Pelczynski property (V*) to have dual with property (V). Since it was a long standing open problem, I guess ...
2
votes
1
answer
1k
views
Are bounded functions L-1 compact?
Let $(X,\Sigma,\mu)$ be a finite measure space (i.e., $\mu(X) < \infty$). Let $\mathcal{F}$ be the set of $\mu$-measurable functions $f:X \to \mathbb{R}$ that are bounded in $[0,1]$, so that $0 \...
- 1,322
5
votes
3
answers
914
views
One-sided Cauchy principal value
What is the notion of a principal value of an integral when the singularity appears at one endpoint? Namely,
$ PV \int_a^b f(t) dt = ? $,
where the integral is convergent in the upper limit, but ...
1
vote
1
answer
201
views
Reference request for sums of Grothendieck spaces
I much appreciate your help with my previous question. Now, I'd like to ask about a reference. I need a fact asserting that if $p\in (1,\infty]$ (with emphasis on the case $p=\infty$) then the $\ell_p$...
- 125