All Questions
12,777 questions
2
votes
1
answer
570
views
Is a polynomial positive on the sphere a sum of squares of spherical harmonic polynomials?
Let $p\in {\mathbb{R}}[x_1,\ldots, x_n]$ be a homogenous polynomial of even degree $2d$. If $p$ is positive on the unit sphere $S\subset {\mathbb{R}}^n$, then does there exist some $m>0$ and ...
user2529
5
votes
1
answer
578
views
Infimum over all vector-valued L^2 spaces
Suppose I have a Banach space $E$ (which may be finite dimensional if you wish), a Hilbert space $H$ and a tensor $\tau \in H\otimes E$ in the algebraic tensor product. There are lots of ways to ...
- 18.7k
3
votes
1
answer
1k
views
Borel-Cantelli lemma for general measure spaces (those with infinite measure)
The Borel-Cantelli lemma is often stated for a probability space or spaces with finite measure.
But it seems to me that it still holds if the space $X$ is of infinite measure. I seem to be able to ...
- 31
1
vote
1
answer
754
views
Addition of essentially self-adjoint operators
Say $A, B : H \supset D \to H$ are essentially self-adjoint operators on the dense common domain $D$. $H$ is some Hilbert space. Does it hold that $A + B$ is also essentially self-adjoint? If not, can ...
- 403
19
votes
4
answers
4k
views
Ramanujan's eccentric Integral formula
The wikipedia page on Srinivasa Ramanujan gives a very strange formula:
Ramanujan: If $0 < a < b + \frac{1}{2}$ then, $$\int\limits_{0}^{\infty} \frac{ 1 + x^{2}/(b+1)^{2}}{ 1 + x^{2}/a^{2}}...
- 4,795
59
votes
7
answers
29k
views
Learning roadmap for harmonic analysis
In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas ...
4
votes
1
answer
875
views
equality in noncommutative Hölder inequality
Let $1\leq p,q,r\leq \infty$ such that $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$. Let $S_p$ denote the Schatten space. For any $x\in S_p$ and any $y\in S_q$ we have
$$
||xy||_{S_r} \leq ||x||_{S_p}||y||_{...
- 1,222
7
votes
0
answers
496
views
planar mappings that preserve elliptic measure
Let $D_1$ and $D_2$ be two bounded simply connected Jordan domains in $\mathbb{R}^2$. By Carathéodory's Theorem there exists a homeomorphism $f:\bar{D}_1 \to \bar{D}_2$ such that the restriction $f:...
- 551
5
votes
1
answer
512
views
$C_0$-semigroups applications
My graduation thesis was about stability theorems for $C_0$-semigroups (see the Wikipedia article for the definitions: http://en.wikipedia.org/wiki/C0-semigroup). I would like to know if there is ...
- 2,222
0
votes
1
answer
901
views
Schwartz space inequality
Let $g$ be a function in the Schwartz space $\mathscr S (\mathbb R)$. Show that for any $l \ge 0$, we have $\sup_x |x|^l |g(x-y)|\le A_l (1+|y|)^l$ by considering separately the cases $|x|\le 2|y|$ ...
4
votes
0
answers
442
views
Why is Mellin-inverse of Gamma periodic?
Specific Case
The periodicity is obvious from computation:
$$\cal{M}^{-1}\{\Gamma\}(x) := \frac{1}{2\pi i}\int_{c}\Gamma(s)x^{-s}d s=e^{-x}$$
However, is there a way to see directly from the integral ...
- 1,243
7
votes
1
answer
1k
views
Geometric interpretation of the argument of the Fubini-Study bilinear form on projective space?
Let $s_0$ and $s_1$ be the holomorphic sections of the tautological bundle $O(1)$ over the complex projective line ${\mathbb{CP}}^1$ which correspond to the functions $1$ and $\frac{x_1}{x_0}$ in the ...
user2529
1
vote
0
answers
369
views
Infinite internal direct sums of subspaces
Given a compact Hausdorff space $K$ such that $C(K)$ is of density $\omega_1$. Suppose that every copy of $c_0(\omega_1)$ in $C(K)$ is complemented. Let $\{Y_\alpha\colon\alpha<\omega_1\}$ be a ...
0
votes
2
answers
444
views
Sobolev space: probably simple ode....
I am trying to solve for $y(x)$ in terms of $f(x)$ in a convenient space (eg. $\dot{H}^2(\mathbb{T})$-zero mean). Here is the ode:
$y(x)+y(x)y'(x)=f(x)$.
I think a contraction mapping argument will ...
- 9
2
votes
0
answers
140
views
WLD Banach spaces
Does anyone know of an example of a weakly Lindeloff determined (WLD) Banach space which does not contain c_0 and is not weak Asplund? I believe the example of a WLD, non-weak Asplund space by Argyros ...
- 21
11
votes
4
answers
1k
views
Norm continuous infinite dimenisonal representation of a Lie group
Given a Lie group G and an infinite dimensional Hilbert space $\mathcal{H}$. In the literature I have only encountered the two following notions of a representation $\pi$ of G on $\mathcal{H}$ :
1) $\...
- 403
44
votes
1
answer
4k
views
Example of a compact set that isn't the spectrum of an operator
This question is somewhat ill-posed (due to the word easy) and is triggered by idle curiosity:
Is there an easy example of a (separable, infinite-dimensional) Banach space $X$ and a nonempty ...
- 5,743
8
votes
1
answer
1k
views
Is there a regular Dirichlet form with no associated Feller process?
I'm reading Dirichlet Forms and Symmetric Markov Processes by M. Fukushima, Y. Oshima, and M. Takeda (hereafter, [FOT]). In Chapter 7, where they discuss the construction of a Markov process ...
- 29.8k
1
vote
1
answer
977
views
Fourier transform of distributions with non-standard test functions
This might be a quite simple question for function analysis standards, but it has some obstacles. I'll try to improve the readability a bit by not using the full tex code. A short motivation:
Given a ...
- 278
26
votes
3
answers
4k
views
Riemann mapping theorem for homeomorphisms
How do you prove to any two simply-connected domains in the plane are homeomorphic without using the Riemann mapping theorem? An elementary proof would be nice.
- 1,159
2
votes
3
answers
759
views
How to prove/disprove that quasiconformal maps send measure-zero sets to measure-zero sets
$Qn#1 $
: Let $f:U\to V$ be a $K$ quasiconformal homeomorphism ( NOT diffeomorphism ) of plane open subsets of $C$. By my definition of quasiconformality, I mean 1)$f$ is continuous, 2)the weak ...
- 1,471
2
votes
0
answers
604
views
Morera type theorems
In Stein and Shakarchi, Complex Analysis, Princeton lectures in Analysis, Chapter 2, Problem 2 an interesting question is posed. The problem section in each chapter contains more complicated problems, ...
- 2,222
5
votes
3
answers
3k
views
What is the $L^p$-norm of the (uncentered) Hardy-Littlewood maximal function?
The (uncentered) Hardy-Littlewood maximal function $M(f)$ of (a locally integrable) function $f$ on $\mathbb{R}^{n}$ is defined by the rule $M(f)(x)=\sup_{\delta>0,\left|y-x\right|<\delta} \text{...
- 1,993
3
votes
0
answers
496
views
When checking if a harmonic function is continuous on its boundary, is a dense subset enough?
Let $U$ be an open connected subset of $\mathbb{C}$ and let $u:U\rightarrow \mathbb{R}$ be harmonic and bounded on $U$.
Let $f:\partial_\infty U \rightarrow \mathbb{R}$ be a continuous function, ...
- 1,601
10
votes
2
answers
2k
views
Characterize where the Dirichlet Problem for the Laplacian is always solvable
Conway's 1978 textbook Functions of One Complex Variable I gives an unsatisfying characterization of the regions for which the Dirichlet Problem can always be solved, and then comments no cleaner ...
- 1,601
2
votes
0
answers
366
views
Are affine continuous functions on Bauer sub-simplices of the probability measures given by integration over continuous functions?
Let $X$ be a compact (non-metrizable) Hausdorff space and $\mathcal{P}(X)$ the set of Radon probability measures with
weak-$*$ topology (weak topology induced by the continuous functions).
Consider a ...
- 888
21
votes
4
answers
2k
views
A question on the integral of Hilbert valued functions
This questions stems from an attempt to recast in a form suitable for teaching some standard computations which are usually proved by handwaving, without much care about the details. My hope is that ...
- 9,007
7
votes
2
answers
523
views
Are these two definitions of nef-ness equivalent for Moishezon manifolds?
Recently, I have been learning about nef line bundles. I know that when $X$ is projective or Moishezon, a line bundle $L$ over $X$ is said to be nef iff $$L.C=\int_{C}c_{1}(L)\ge 0$$ for every curve $...
- 247
9
votes
2
answers
477
views
An extension of Gaussian Isoperimetry
The Gaussian isoperimetric inequality (Tsirelson,Sudakov, Borell) states that among all sets of given Gaussian measure in the n-dimensional Euclidean space, half-spaces have the minimal Gaussian ...
- 949
0
votes
1
answer
307
views
hesse matrix under diffeomorphism
Let $u : U \rightarrow \mathbb{R}$, where $U \subset \mathbb{C}^{n}$ be a strictly plurisubharmonic smooth function and consider its complex hesse matrix $Hess^{\mathbb{C}}(u)$. Furthermore consider a ...
- 3
1
vote
1
answer
312
views
Invertibility of frame/sampling operator on Bargmann-Fock spaces
Let $F_\alpha ^p (\mathbb{C}^n)$ for $1 < p < \infty$ and $\alpha > 0$ be the Bargmann-Fock space defined as the Banach space of entire functions $f$ such that $f(\cdot) e^{- \frac{\alpha}{2} ...
7
votes
1
answer
592
views
topologies on U(H)
There are many topologies on the algebra $B(H)$ of bounded operators on Hilbert space:
the weak, strong, ultraweak (also called σ-weak), ultrastrong (also called σ-strong), and some more......
- 43.2k
1
vote
1
answer
318
views
Does the Bergman kernel always arise as the Jacobian of a biholomorphism?
Let $0 \in D$ be a bounded domain. Is it true that we can always find a injective holomorphic map $F: D \to \mathbb{C}^n$ such that $JF=K(z,0)$? Here, $K$ denotes the Bergman kernel of $D$, and $JF$ ...
- 1,159
2
votes
1
answer
196
views
Coefficients of lacunary series on quasiconformally transformed unit disk
Say I have a lacunary $q$ series $s(q)=\sum_{n=0}^{\infty} a_{n}q^{n}$ , and I have a quasiconformal transformation $\xi$ which preserves the boundary of the unit disk in $\mathbb{C}$ such that if $|q|...
- 975
0
votes
1
answer
622
views
products in the category of banach spaces
Let $\{X_{\alpha} \}_{\alpha \in A}$ be a collection of Banach spaces. It is easy to show that $ P = \{(x_{\alpha}) : {\rm sup}_{\alpha} \|x_{\alpha} \| < \infty \} $ with $\| (x_{\alpha} ) \| = {\...
- 3,830
0
votes
2
answers
273
views
Do extracted weak $H^{1,2}$-limits and $C^0$-limits coincide?
Let $I$ be a bounded interval and consider a sequence $(u_k)$ in $H^{1,2}(I)$ (usual Sobolev space). Suppose furthermore, that the sequence $(u_k)$ is bounded in $H^{1,2}(I)$. Then, by Rellich, we can ...
- 2,935
2
votes
1
answer
447
views
flatness of coherent analytic sheaf
I meet a problem like this : given a short exact sequence $0\rightarrow E_1\rightarrow E_2\rightarrow E_3\rightarrow 0$ , where $E_i,i=1,2,3$ are coherent sheaves over a compact complex manifold $X$ . ...
- 399
4
votes
2
answers
1k
views
Reference for Neumann-Laplacian
Let $\Omega\subset R^d$ be a bounded, smooth domain. Consider $A=-\Delta$ subject to homogeneous Neumann boundary conditions in $L^p$-spaces. Does anybody know a good reference book on basic results ...
- 225
7
votes
2
answers
657
views
Subspaces isomorphic to $C[0, \omega_1]$
Let $\omega_1$ be smallest uncountable ordinal. I am trying to understand the possible "large" subspaces of $C[0,\omega_1]$, namely those which are isomorphic to the whole space. Therefore I have the ...
- 11.3k
2
votes
0
answers
139
views
Question on Bergman minimal domains
Let $D \subseteq \mathbb{C}^n$ be a bounded domain and let $t \in D$. We say that $D$ is a minimal domain with center $t$ if for each biholomorphism $F:D \to D' \subseteq \mathbb{C}^n$ such that $JF(t)...
- 1,159
7
votes
1
answer
423
views
Best constant in comparison between Rademacher and gaussian averages?
Let $(g_k)$ be a sequence of independent standard gaussians variables on a fixed probability space $\Omega$. Let $(\epsilon_k)$ be a sequence of independent rademacher variables.
What is the best ...
- 1,222
8
votes
3
answers
2k
views
Evaluating the integral $\int_0^\infty \frac{\psi(x)-x}{x^2}dx.$
Let $\psi(x)=\sum_{n\leq x} \Lambda(n)$ be the weighted prime counting function. I am trying to evaluate the integral $$\kappa:=\int_{1}^{\infty}\frac{\psi(x)-x}{x^{2}}dx$$ in several different ways. ...
- 11.4k
19
votes
4
answers
5k
views
Explicit extension of Lipschitz function (Kirszbraun theorem)
Kirszbraun theorem states that if $U$ is a subset of some Hilbert space $H_1$, and $H_2$ is another Hilbert space, and $f : U \to H_2$ is a Lipschitz-continuous map, then $f$ can be extended to a ...
- 1,839
3
votes
4
answers
2k
views
Suitable references for the the Stone-von Neumann Theorem
Hi all,
I am working on a mathematical physics project now and I need to understand the Stone-von Neumann Theorem properly. Wikipedia says that it is any one of a number of different formulations of ...
- 1,719
0
votes
1
answer
318
views
Complement of a hypersurface in a projective space
When does the complement of a hypersurface in a projective space admit a nonconstant holomorphic function?
- 103
5
votes
1
answer
878
views
Short five lemma in Banach spaces
Denote by $\mathbf{Ban}$ the category of Banach spaces and bounded linear maps and by $\mathbf{Banc}$ the subcategory of Banach spaces and linear contractions. The isomorphisms of $\mathbf{Ban}$ are ...
- 1,848
3
votes
2
answers
1k
views
who can give me a example of coherent sheaf
What are examples of coherent sheaves $\mathfrak{F}$ on a compact complex $n$-fold with $\dim \operatorname{Supp} \mathfrak{F}=p$ , where $0\leq p \leq n$ ? And how can they be described in local ...
- 399
1
vote
1
answer
410
views
how to prove the following fact in sheaf cohomology ?
Let $X$ be a compact complex n-fold . Then for every coherent sheaf $\mathfrak{F}$ on $X$ , and every holomorphic line bundle $L$ on $X$ , then the dimension of $H^0 (X,\mathfrak{F}\otimes\mathcal{O}...
- 399
12
votes
4
answers
4k
views
Locally constant functions with compact support = smooth ?
Hello,
I have a trivial question, but I hope that you don't mind helping. I often get confused with basic definitions.
Let F be a p-adic field. Then (from what I understand) $C_c^{\infty}(F)$ is the ...
- 231
2
votes
1
answer
680
views
spectra of sums in (Banach) algebras
A similar question was already asked in question titled "Spectra of sums and products in (Banach) algebras [was: Spectrum in Banach Algebra]".
Answer there led me to the following question.
If for ...
- 179