All Questions
12,777 questions
50
votes
4
answers
6k
views
The maximum of a polynomial on the unit circle
Encouraged by the progress made in a recently posted MO problem, here is a "conceptually related" problem originating from a 2003 joint paper of Sergei Konyagin and myself.
Suppose we are given $n$ ...
- 23k
4
votes
1
answer
246
views
Are all continuous linear operators on the space of entire functions "simple"?
Let $\langle \operatorname{Ent},+,\cdot \rangle$ be the (complex) vector space of entire functions.
For all members $n$ of $\{1,2,3,...\}$, define $||\cdot ||_n : \operatorname{Ent} \to \mathbb{R}$ ...
user5810
2
votes
1
answer
265
views
Multiple ergodic averages with varying number of terms
Hi. I've been stuck on the following question for some time.
Consider a sequence of functions $\left( f_n \right)$ from an ergodic space $\left( \mathsf{X}, \mathsf{S}, \mu \right)$ to $\left[ 0,1 \...
1
vote
1
answer
496
views
Can be this operator extended to an unbounded self-adjoint operator ?
Consider an enumeration $\{q_1,q_2,\ldots\}$ of $\mathbb{Q}\cap [1,\infty)$ and a orthogonal Schauder basis $\{e_1,e_2,\ldots\}$ of $\ell^2(\mathbb{N})$. Define
$Ae_{2k-1}=e_{2k-1}$ and $Ae_{2k}=...
- 2,044
0
votes
1
answer
365
views
Integral in a σ−convex set.
Having had no (proper) answer to this question, I formulate the remaining case as a new question as follows. With $I=[0,1]$, let $E$ be a separable (real) Banach space, and let $\gamma:I\to E$ be ...
- 3,584
5
votes
2
answers
1k
views
Sobolev imbedding on Riemannian manifolds
Let $(M, g)$ be a non-compact smooth Riemannian manifold of dimension $n \ge 2$, and $G$ a subgroup of the isometry group of $(M,g)$, say with $G$ contained in the component of the identy.
Let $W^{1,...
- 397
3
votes
1
answer
556
views
Convergence of mountain pass solutions of $-\Delta u+u=u|u|^{p-2}$
Consider the following equation in $\mathbb{R}^N, N \ge 3$:
$$
(E) \quad -\Delta u +u=|u|^{p-2}u,
$$
where $2 < p < 2^{*} =2N/(N-2)$.
Denote by $J: H^1(\mathbb{R}^N) \to \mathbb{R}$ the ...
- 397
1
vote
1
answer
580
views
Relation between partially computable function and complex function
Given a partially computable function, is there an analytic complex function which is equal to it at every point of it's domain? Or under what condition does a partially computable function correspond ...
- 3,094
2
votes
2
answers
2k
views
Does the Euler product formula diverge for any zero of the Riemann zeta function?
Simple question (but not for me):
Does the Euler product formula diverge for any zero of the Riemann zeta function?
The reason why I ask this is that I heard we should not use the Euler product ...
- 255
1
vote
1
answer
235
views
Existence of a special holomorphic function
How can you prove the existence of a nonzero function from the subset $U= \{z| 0 \leq Re z \leq 1\}$ of $\mathbb C$ to $\mathbb C$ which is holomorphic on the interior of $U$ and vanishes on the right ...
- 1,282
21
votes
4
answers
3k
views
Is the Euler product formula always divergent for 0<Re(s)<1?
It is known that the Euler product formula converges for $\Re(s)>1$
(and there it represents the Riemann zeta function).
My question: Is the Euler product always divergent for
$0 < \Re(s) < ...
- 255
2
votes
1
answer
431
views
Sobolev imbedding
It is known that, for $n \ge 3, 2 < p< 2^*$, the imbedding $H^1(\mathbb{R}^n) \hookrightarrow L^p(\mathbb{R}^n)$ is not compact. Let $G=O(n_1) \times O(n_2)\times\cdots\times O(n_k)$, with
$n_1+...
- 397
40
votes
4
answers
4k
views
Polynomials on the Unit Circle
I asked this question in math.stackexchange but I didn't have much luck. It might be more appropiate for this forum. Let $z_1,z_2,…,z_n$ be i.i.d random points on the unit circle ($|z_i|=1$) with ...
- 3,626
5
votes
1
answer
2k
views
Proof of a concentration compactness lemma
Hi I'm stuck with the proof of a concentration-compactness lemma.
We have the following equation in $\mathbb{R}^N, N \ge 3$:
$$
-\Delta u +u=|u|^{p-2}u,
$$
where $2 < p < 2^{*}$.
The functional ...
- 397
9
votes
3
answers
2k
views
Trace theorem for $C^{k,1}$ domains
What are the best results on (Sobolev space) trace theorems for $C^{k,1}$ domains?
For $k=0$, e.g., when the domain is Lipschitz, from e.g. the works of Martin Costabel and Zhonghai Ding, it is known ...
- 3,322
23
votes
1
answer
2k
views
Which Fréchet spaces have a dual that is a Fréchet space?
I've read the claim that Fréchet spaces that are not Banach spaces never have a dual that is a Fréchet space, but have not been able to find a proof of this statement. Is it trivial or does someone ...
- 1,544
8
votes
1
answer
1k
views
Ring of continuous functions, reference request.
I am looking for a reference for the following facts in functional analysis and topology. (If these "facts" are not true, I suppose I'm looking for the closest approximation which is true.)
Let $X$ ...
- 13.3k
42
votes
7
answers
5k
views
Is there an integration free proof (or heuristic) that once differentiable implies twice differentiable for complex functions?
The title pretty much says it all. I am revisiting complex analysis for the first time since I "learned" some as an undergraduate. I am trying to wrap my head around why it should be the case that a ...
17
votes
2
answers
5k
views
Positive-Definite Functions and Fourier Transforms
Bochner's theorem states that a positive definite function is the Fourier transform of a finite Borel measure. As well, an easy converse of this is that a Fourier transform must be positive definite.
...
- 4,952
4
votes
6
answers
925
views
Quasiconformal harmonic extension of a quasi-symmetric map on $S^1$
Hello ,we know that for given $h:S^1\to S^1$, we can solve the Dirichlet problem on $\bar{D} $ with the boundary value $h$ and in fact this extension, which is the complex harmonic extension $H=E(h) $ ...
- 1,471
11
votes
1
answer
642
views
Random walk origin return monotinicity
Consider a Markov chain on $\mathbb{Z}^d$ with transition kernel $P$ for adjacent vertices (non-diagonal). Essentially this is a $d$ dimensional random walk with the probability of a transition ...
- 4,952
14
votes
3
answers
778
views
Is analytic Quillen-Suslin simple?
This question is motivated by a sentence on the Wikipedia entry for Quillen-Suslin theorem. This theorem states that every algebraic vector bundle on affine space is trivial. The analogous result is ...
- 156k
15
votes
1
answer
2k
views
Holomorphic line bundles on a punctured disc
Is every holomorphic line bundle on the - say - punctured unit disc $\dot{\Delta} \subseteq \mathbf{C}$ trivial? Griffiths-Harris (p. 39) prove that $H^{p,q}_{\overline{\partial}}(\Delta) = 0$ (for $q ...
- 1,704
3
votes
0
answers
385
views
Off-diagonal asymptotic expansion of the Bergman kernel on hyperbolic Riemann surfaces
Let $X$ be a compact Riemann surface of genus at least 2. Let $K$ denote the canonical line bundle, and $E$ be any vector bundle. Let $P^{(m)}$ be the projection map from the space $L^2(X,K^mE)$ of ...
user2529
5
votes
2
answers
491
views
Is independence meaningful for commutative $C^*$-algebras?
I don't know very much about spectral theory so probably the answer to my question has a basic reference which I would appreciate.
Let's say I have two self-adjoint operators on a Hilbert space and ...
- 2,243
12
votes
1
answer
2k
views
Hardy spaces: analysis <---> martingales
Let $H^p$ be the Hardy space of analytic functions on the open unit disk $\mathbb{D}$: $f \in H^p$ if $f$ is analytic on $\mathbb{D}$ and $\sup_{r < 1} \int_0^{2\pi} |f(re^{i\theta})|^p d\theta <...
- 943
3
votes
1
answer
371
views
If $X$ fails to be holomorphic, what is $\mathcal{L}_X \bar\partial - \bar\partial \mathcal{L}_X$?
I present a simplified version of the problem, and if that's easy to answer then someone can try answering the more complicated version.
Simplified version
Suppose $X$ is a tangent vector field on a ...
- 33
0
votes
2
answers
796
views
Extending Continuous Sublinear maps on dense subsets of a Banach space
Suppose X' and Y are Banach spaces and X is a linear subspace dense in X'. Let T be a continuous map of X to Y satisfying:
(1) ||T(x+y)|| is less than or equal to ||T(x)||+||T(y)||.
Please prove ...
- 11
18
votes
2
answers
961
views
How to prove that $e^{zw}$ can not be written as a rational expression in functions in $z$ and in $w$?
Let $K$ be the field of fractions of
$\mathbb{C}[[z]]\otimes_{\mathbb{C}}\mathbb{C}[[w]]\subset \mathbb{C}[[z, w]].$ Given
a formal power series in $t, f\in \mathbb{C}[[t]],$ is there any simple ...
- 689
1
vote
3
answers
492
views
Cos[Im[zetazero(n)]Log(prime)] spans a countable dense set in [-1,1]?
It is known that cos(N) spans a countable dense set in [-1,1].
(N: any natural number)
As far as I know generally, for any continuous function f defined in [a,b],
f is Riemann integrable where its ...
- 255
0
votes
2
answers
390
views
Given: $\phi(s) = \sum_p \frac{\log p}{p^s}$ Why is: $\lim_{e \rightarrow 0} e \phi(1+e)$ = 1 ?
Problem:
Given: $\phi(s) = \sum_p \frac{\log p}{p^s}$
(summation is only over primes)
Why is: $\lim_{e \rightarrow 0} e \phi(1+e)$ = 1 ?
Context: http://mathdl.maa.org/images/upload_library/22/...
- 663
5
votes
2
answers
833
views
Conformal structure determined by principal curvatures
On a smooth surface of positive Gauss curvature that is embedded in Euclidean 3 space the principal curvatures seem to define a smooth field of ellipses whose major and minor axes are lined up with ...
- 439
14
votes
3
answers
2k
views
Meromorphic 1-form and Picard's theorem
Let $D$ be the open unit disk in the complex plane and $U_1,U_2,\,\ldots\,,U_n$ be an open cover of the puntured disk $D^*= D\setminus\{0\}$. Suppose on each open set $U_j$ there is an injective ...
- 229
1
vote
0
answers
180
views
Generalized vector bundles with singularities on Riemann surfaces
Let $X$ be a Riemann surface of genus $g \geq 2$ or in other words a complex curve.
Let $P_1, \ldots, P_m$ be points in $X$ and $E \rightarrow X$ surjective map such that is is a complex $n$-...
- 121
7
votes
3
answers
4k
views
Measures on infinite dimensional Banach spaces
Does there exist a Borel measure or any valid measure on an infinite dimensional Banach space such that a bounded open set in this space has a positive measure ?
8
votes
1
answer
2k
views
Limits of Nilpotent and Quasi-nilpotent Operators in a $\mathrm{II}_1$-factor
A bounded operator $A$ in a Hilbert space is called nilpotent if there exists $n$ such that $A^{n}=0$. An operator is called quasi-nilpotent iff
$$
\limsup_{n\to\infty}{ \|A^{n}\|^{1/n}}=0.
$$
Every ...
- 3,626
6
votes
1
answer
404
views
Unique preduals up to (nonisometric) isomorphism?
It's well known that there are Banach spaces which has a unique isometric predual-- for example, any von Neumann algebra. As other questions on here (for example, Isomorphisms of Banach Spaces ) ...
- 18.7k
9
votes
1
answer
943
views
Removable sets for harmonic functions and Hardy spaces of general domains
Let $\Omega$ be a domain of the complex plane. The Hardy space $H^p(\Omega)$ is defined, for $1 \leq p<\infty$, as the class of functions $f$ that are holomorphic on $\Omega$ such that $|f|^p$ has ...
- 2,154
3
votes
0
answers
637
views
Fixed point theorem for convex, closed multivalued mapping
There is well-known fixed point theorem theorem for multivalued l.s.c. maps, based on Michael selection theorem:
Suppose, that $X$ is compact, convex and metrizable in locally convex Hausdorff ...
- 696
4
votes
1
answer
1k
views
Sobolev-Slobodeckij spaces for p=infinity
For $1\leq p<\infty$ an approach to define fractional Sobolev spaces is by Sobolev-Slobodeckij spaces a generalisation of Hölder continuity. For example letting $U\subset\mathbb{R}^n$ then,
$
\...
- 3,217
3
votes
1
answer
535
views
Uniqueness of analytic continuation on a domain of C^n.
Hi. I have been struggling with this question for a while now.
Given a domain $\Omega \subset \mathbb{C}^n$, $\Omega^\prime = \Omega \cap \mathbb{R}^n$, and an analytic function $f: \Omega \...
11
votes
1
answer
676
views
Analysis and finitely generated groups
Dear all, this is perhaps a bit a vague question, but some references would already be very helpfull.
So let $G$ be a finitely generated group and choose some finite set of generators. This allows to ...
- 8,098
12
votes
3
answers
3k
views
Infinitesimal generators of stochastic processes
What's the $L^1$ analogue of Stone's theorem saying that any strongly continuous 1-parameter unitary groups has a unique self-adjoint generator?
More precisely: let $X$ be a measure space ($\sigma$-...
- 22.3k
2
votes
1
answer
465
views
Complex version of Farkas' lemma
It is well known result of linear optimization theory that, if a finite set $S$ of linear inequalities in real variables $x_1,x_2, \ldots ,x_n$ implies a linear inequality $i$ in $x_1,x_2, \ldots ,...
- 3,595
3
votes
2
answers
1k
views
Sum of two closed operators closable
I found this question on another forum, and after processing it a bit, I didn't find a good answer. The question is:
Is the sum of two closed operators closable? If not, give an example of two ...
- 2,222
9
votes
2
answers
928
views
Property (T) for pseudogroups
Let $H$ be a Hilbert space, $S(H)$ be the inverse semigroup (pseudogroup) of linear maps between (closed) subspaces of $H$ preserving the dot product (the operation is composition of partial maps). ...
user6976
2
votes
0
answers
296
views
Is the complex harmonic extension of a $\mathcal{C^r}$ map from $S^1$ to $\mathbb{C}$ is smooth upto the boundary ?
Suppose we have a map $ h : S^1\to \mathbb{C} $ that we know is a $\mathcal{C^r} $ map ( in the sense of a map between 1-manifold ( or in the sense of a $2\pi$ periodic map from $\mathbb{R}\to \mathbb{...
- 1,471
6
votes
0
answers
1k
views
Computing the Chern class for a flat line bundle using the holonomy group?
Let $X$ be a Riemann surface of genus $g \geq 2$. I would like to consider flat line bundles on $X$. Flat line bundles can be identified with representations $\pi_1(X) \rightarrow U(1)$. From Chern-...
- 121
3
votes
2
answers
642
views
Localization of Laplacian eigenfunction on the unit square?
Let A be the unit square, $\{u_k\}$ is the set of all L2-normalized Laplacian eigenfunctions with Dirichlet boundary condition. Is it true that for any open subset V, $C_V = \inf\limits_k \int\...
5
votes
3
answers
987
views
Boundedness of Laplacian eigenfunctions
Let $A$ be a bounded domain in $\mathbb R^d$, $d>1$, and $\{u_k\}$ is the set of all $L^2$-normalized Laplacian eigenfunctions on $A$ with Dirichlet boundary condition (i.e., $\|u_k\|_2 = 1$).
Is ...