All Questions
12,777 questions
7
votes
1
answer
2k
views
A good reference for the wave front set
Hello,
I am wondering whether anyone know some good references for the theory of wave front set, microlocal analysis? I have some basic knowledge of distribution theory at the level of the Rudin's ...
- 1,649
0
votes
1
answer
222
views
Bounding near the boundary for a Sobolev function.
Let $f: \Omega\rightarrow \mathbb{R}$ where $\Omega\subset\mathbb{R}^d$ is bounded with lipschitz smooth boundary. Further suppose that $f\in\mathcal{H}^{\tau}(\Omega)$, $\tau>\frac{d}{2}$ (i.e. $f$...
- 3,217
44
votes
10
answers
47k
views
Is square of Delta function defined somewhere?
I am wondering whether anyone knows if the square of Dirac Delta function is defined somewhere.
In the beginning, this question might look strange. But by restricting the space of the test functions, ...
0
votes
1
answer
234
views
A property of "Schwartz" quadratic forms
Consider $K(x, y)$, $f(x)$ Schwartz functions and $g(y)$ a tempered distribution. Suppose $$K(x, y) = K(y, x)$$ Define
$$h(t) = \int f(x - t) K(x, y) g(y - t) dx dy$$
It appears to me $h(t)$ is a ...
- 1,368
3
votes
1
answer
418
views
Conjugate Groups of (quasi) Fuchsian Groups
I apologize in advance if this question is so trivial or too low level.
Let $\Gamma$ be a Fuchsian group. Let $\mathcal{F}$ be the set of pairs $(\mu,f)$, where $\mu \in L^\infty(\mathbb{C})$ such ...
- 245
6
votes
1
answer
1k
views
Holomorphic functions in almost-complex geometry
Maximum principle implies that every holomorphic function on a compact complex manifold is constant. Is this still true if the manifold is only almost complex?
11
votes
1
answer
546
views
How manifold-like is Aut(C^n) in the holomorphic category?
This question is similar to, but not the same as this one. Take the space of automorphisms of $\mathbb{C}^n$ in the holomorphic category, with the compact-open topology. For $n=1$ this is just $\...
- 35.5k
3
votes
1
answer
3k
views
Solving Functional Equation
Continue with my previous question “Regarding Kolmogorov's Superposition Theorem”, here are some further questions:
Question-1
Is it true: for any given $C^1$ continuous real function $f(x, y): \Re^2 ...
- 103
2
votes
3
answers
5k
views
Specializing in Complex Analysis [closed]
May someone kindly provide a useful list of books on complex analysis that would be appropriate for a graduate student intending to specialize in that area.
27
votes
3
answers
5k
views
Weak and Strong Integration of vector-valued functions
This is probably an elementary question, but outside my area of expertise, and I was unable to find any suitable reference:
Suppose $f:X\to E$ is a continuous function from a compact spaces (endowed ...
- 741
4
votes
1
answer
564
views
monodromy of plane curve singularities
Are there two IRREDUCIBLE plane curve singularities having different equisingular type with the same monodromy (linear action on the first homology group of the (regular) Milnor fibre)?
- 105
5
votes
0
answers
616
views
Lebesgue measure on Frechet space?
It is well known that there are no Lebesgue measures on infinite-dimensional Banach spaces (see e.g. http://en.wikipedia.org/wiki/There_is_no_infinite-dimensional_Lebesgue_measure). However, I couldn'...
- 1,368
9
votes
2
answers
2k
views
Nice Classes of Non-Closable Operators
The only thing I know about non-closable operators can be summarised as "they exist, but they're nasty, so let's not talk about them!" This seems to be the case with everyone else I've talked to. I'd ...
- 1,411
2
votes
2
answers
882
views
What $Re(f(z))=c$ can be if $f$ is a holomorphic function ?
Suppose that $f:U\subset\mathbb{C}\to\mathbb{C}$, where $U$ is a region in the complex plane, is a holomorphic function.
Of course, if $c\in\mathbb{R}$ is a regular value for $\text{Re}(f(z))$ then $...
- 2,044
3
votes
0
answers
130
views
Positive block matrices over tensor algebras
Let $A$ be a unital C*-algebra. A positive block matrix in $M_2(A)$ must have the form
$$ \begin{pmatrix} a & a^{1/2} x b^{1/2} \\ b^{1/2} x^* a^{1/2} & b \end{pmatrix}, $$
where $a,b$ are ...
- 18.7k
5
votes
1
answer
781
views
Does a log-concave function on a convex set extend continuously to the boundary?
Let $U$ be an open convex set in a locally convex space $X$, and let $f : U \to [0,1]$ be a log-concave function on $U$ (i.e., bounded and real-valued). Under what conditions does $f$ have a ...
- 8,512
7
votes
1
answer
362
views
Nonexpansive multi-valued maps in $\ell^2$
Let $C$ be a nonempty bounded closed convex subset, say the unit ball, of $\ell^2(\mathbb{N})$. Let $T: C\to 2^C$ be a map such that $T(x)$ is nonempty closed for each $x$, and that $$D(Tx,Ty)\le \|x-...
- 744
4
votes
0
answers
257
views
A matrix minimisation problem
Feel free to edit the title!
Suppose A is a C*-algebra and $a,b\in A$ are self-adjoint. I'd be very happy with A being just $n\times n$ matrices.
Question: If there are $t\in\mathbb R$ and $\...
- 18.7k
8
votes
1
answer
458
views
Do proper polynomial mappings have a path-lifting property?
Suppose $f: \mathbb{C}^n \to \mathbb{C}^n$ is a proper polynomial mapping and $\gamma: [0,1] \to \mathbb{C}^n$ is a continuous path. Further, suppose $z_0 \in \mathbb{C}^n$ satisfies $f(z_0)=\gamma(0)...
- 325
7
votes
1
answer
481
views
Gelfand theory Problem
I have 2 problems in Gelfand theory. I shall be thankful for any
answers.
1)What is the gelfand spectrum of l^1(N)?
A few of
the elements are evaluations of functions(defined below) on closed
unit ...
- 71
2
votes
1
answer
412
views
General Sobolev Inequalities
In Partial Differential Equation by Lawerence Evan p284 there is this theorem stated:
Let $U$ be a bounded open subset of $\mathbb{R}^n$ with $C^1$ boundary. Suppose $u\in W^{k,p}$ then if $k>n/p$ ...
- 3,217
0
votes
1
answer
435
views
spectrum of the compression of a selfadjoint operator
Let T be a (unbounded) selfadjoint operator in $B(H)$, the bounded operator acting on Hilbert space $H$.
Def: A compression of T is an operator $pTp$, where $p$ is a projection in $B(H)$.
I am ...
- 31
1
vote
1
answer
2k
views
Square root of integral operator
Consider the 1-torus $\mathbb{T}$. Let $k$ be a smooth function on $\mathbb{T}^2$ and $K$ be the integral operator on $L^2(\mathbb{T})$ with kernel $k$. One can show that $K$ is of trace class, hence $...
- 1,702
1
vote
2
answers
606
views
Do separable $C^*$-algebras form a set?
The question is in subject.
Update: See Andreas Thom's answer.
- 613
1
vote
1
answer
491
views
Bounding a smooth function near the boundary
Let $f: \Omega\rightarrow \mathbb{R}$ where $\Omega\subset\mathbb{R}^d$ is bounded with lipschitz smooth boundary. Further suppose that $f\in\mathcal{H}^{\tau}(\Omega)$, $\tau>0$ and that $f$ ...
- 3,217
4
votes
1
answer
2k
views
Characterizations of a linear subspace associated with Fourier series
Let $c_0$ be the Banach space of doubly infinite sequences $$\lbrace
a_n: -\infty\lt n\lt \infty, \lim_{|n|\to \infty} a_n=0 \rbrace.$$ Let $T$ be the space of $2\pi$ periodic functions integrable ...
- 744
1
vote
0
answers
129
views
Relation between different spatial derivatives of a random field (related to complex integral and/or bessel function)
2 random fields $b$ and $c$ are derived from random field $a$ by
$b=\nabla^2a\equiv(\partial_{xx}+\partial_{yy})a $
and
$c \equiv c_1+i c_2 = (\partial_{xx}-\partial_{yy}+2i \partial_{xy}) a$.
(...
- 111
22
votes
5
answers
3k
views
Unexpected applications of Dvoretzky's theorem
Dvoretzky's theorem is a classic of convex geometry. Recently at a conference in quantum information I learned (from Patrick Hayden's talk) about a nontrivial application of the theorem to a problem ...
- 2,449
18
votes
4
answers
2k
views
Does "taking the dual space" stabilize?
Every book which treats dual spaces of normend spaces states that $(c_0)' = \ell^1$ and $(\ell^1)' = \ell^\infty$ and some also describe $(\ell^\infty)'$.
However, is anything known about higher ...
- 12.7k
3
votes
1
answer
1k
views
If $H$ is a separable Hilbert space, is its dual dense in $L^2(H)$?
Let $H$ be an infinite-dimensional, separable Hilbert space, and let $\gamma$ be a Radon probability measure on $H$ with mean zero and covariance operator the identity $I$.
Let $H^*$ denote the space ...
- 8,512
1
vote
3
answers
801
views
A simple ordinary differential equation
Consider an entire function $f : \mathbb{C} \rightarrow \mathbb{C}$! We search the function
$$ g: (a,b) \rightarrow \mathbb{C},$$ which solves the following equation locally: $g'(t)=f(g(t))$ and $g(0)=...
8
votes
3
answers
2k
views
Conformal Mappings for hyperbolic polygon
I am searching for a conformal mapping from the upper halfplane onto a hyperbolic polygon, i.e. the sides of the polygon have to be geodesics.
The classical Schwarz Christoffel theorem does the job ...
7
votes
1
answer
1k
views
If $H$ is a separable Hilbert space, is $L^2(H)$ separable?
Let $H$ be a separable Hilbert space, and let $\gamma$ be a Radon probability measure on $H$ with mean zero and covariance operator the identity $I$.
Is the Hilbert space $L^2(H,\gamma)$ separable?
- 8,512
12
votes
1
answer
2k
views
Wick rotation and the Riemann zeta function
The goal of this question is to conceptualize in some way the fact that the Riemann zeta function $\zeta(s)$, and other zeta functions like it, have analytic continuations.
Background
I have by now ...
- 118k
26
votes
2
answers
5k
views
Does Arzelà-Ascoli require choice?
Inspired by a recent Math.SE question entitled Where do we need the axiom of choice in Riemannian geometry?, I was thinking of the Arzelà--Ascoli theorem. Let's state a very simple version:
...
- 29.8k
1
vote
0
answers
439
views
Does the dual Banach space $B(\ell^\infty)$ have weak* normal structure?
Let $K$ be a bounded closed convex subset of a Banach space $E$. A point $x$ in $K$ is called a diametral point if
$$
\sup_{y\in K} \|x-y\|={\rm diam}(K).
$$
where ${\rm diam}(K)$ denotes the ...
- 1,222
23
votes
5
answers
6k
views
Hahn-Banach without Choice
The standard proof of the Hahn-Banach theorem makes use of Zorn's lemma. I hear that, however, Hahn-Banach is strictly weaker than Choice. A quick search leads to many sources stating that Hahn-Banach ...
- 1,202
8
votes
3
answers
2k
views
Harmonic level sets and boundary data
This is probably a classic problem, so a good reference book or paper to get me started on this type of question would be great:
Let $\mathbb{D} \subset \mathbb{C}$ be the unit disk with boundary $\...
7
votes
2
answers
2k
views
Heat kernel estimates and Gaussian estimates for semigroups, good reference?
Hi, it seems like a big field and I'm having trouble getting some solid/classic references to get me started.
If $U \subset \mathbb{R}^d$ is a bounded domain with, say, $C^2$-boundary $\partial U$ ...
7
votes
3
answers
1k
views
A Question concerning the Fourier Transform of $\mathbb{R}$
Consider the classical Schwartz space $\mathcal{S}(\mathbb{R})$ together with the Fourier transform $\mathcal{F} : \mathcal{S}(\mathbb{R}) \rightarrow \mathcal{S}( \mathbb{R})$.
Consider the subspace ...
- 11.2k
18
votes
1
answer
1k
views
Are there non-reflexive abelian topological groups isomorphic to their second dual?
I posted the following question in a comment at
Are there non-reflexive vector spaces isomorphic to their bi-dual? and it got one upvote, but it didn't get an answer, so I'll post it as an ...
- 50.6k
1
vote
0
answers
720
views
Does the tangent bundle of this fiber product split?
Let $\mathcal X \to S$ be the local universal family of an elliptic curve, and let $E \to S$ be a vector bundle over $S$. Then we can form the fiber product $\mathcal Y = \mathcal X \times_S E$, which ...
- 4,343
2
votes
0
answers
153
views
Holomorphic automorphism of strictly psudo-convex domain smooth on boundary
I am wondering if anything is known about this. I couldn't find anything in the literature.
In '74 C. Fefferman published a solution to the following problem.
Let $\sigma:D\rightarrow D$ be an ...
- 496
64
votes
1
answer
6k
views
Is there a "classical" proof of this $j$-value congruence?
Let $j: \mathbf{C} - \mathbf{R} \rightarrow \mathbf{C}$ denote the classical $j$-function from the theory of elliptic functions. That is, $j(\tau)$ is the $j$-invariant of the elliptic curve $\mathbf{...
14
votes
5
answers
4k
views
Is there an extension of the Arzela-Ascoli theorem to spaces of discontinuous functions?
The Arzela-Ascoli function basically says that a set of real-valued continuous functions on a compact domain is precompact under the uniform norm if and only if the family is pointwise bounded and ...
- 943
1
vote
1
answer
555
views
Conformal mapping of C \ D* onto C \ (-1, 1) [closed]
Which is the concrete formula for the conformal mapping (normalized at infinity),
acting from $\mathbb C \backslash D^*$ onto
$\mathbb C\backslash[-1, 1]$?
Here $\mathbb C$ denotes the set of all ...
- 21
3
votes
1
answer
361
views
Harmonic equivariant maps and Simpson's correspondence
Let $\Gamma\subset PSL(2,R)$ be a Fuchsian group. For which representations $\rho:\Gamma\to PSL(2,R)$ does there exist a harmonic map from the hyperbolic plane to itself satisfying
$f(\gamma z)=\rho(...
- 454
3
votes
1
answer
330
views
Is a compact subset of a Stein space admitting a fundamental system of Stein neighbourhoods necessarily holomorphically convex?
Let X be a Stein manifold and let K be a compact subset of X. Suppose that K possesses in X a fundamental system of neighbourhoods which are Stein spaces. Then, it is a result by Rossi that such a ...
- 169
2
votes
1
answer
345
views
Is the closure of an open holomorphically convex subset of a Stein space holomorphically convex?
Let X be a Stein manifold and U an open, connected, relatively compact, holomorphically convex subset of X. Is the closure of U in X holomorphically convex?
Also, if X is a Stein space with a finite ...
- 169
1
vote
1
answer
663
views
What are conditions to make f(x) defined by f(x)=f(x-1)*x + 1/e unique(for instance convex)?
[Background:]
Looking at the powerseries for the gamma-function
$ \Gamma(1+x) = 1 + a_1 x + a_2 x^2 - a_3 * x^3 + ... $
then we can arrive at a decomposition
$ \Gamma(1+x) = r(x) + g(x) $
...
- 5,342