All Questions
12,780 questions
2
votes
1
answer
3k
views
The fourier transform of homogeneous distribution and related topics
When we have a distribuion $u\in \mathcal{D}'(R^n)$,and the restriction to $R^{n}\backslash{0}$ is homogeneous of degree a,we have $u \in \varphi'$ and $\widehat u$ is of degree(-n-a) in $R^{n}\...
- 1,644
4
votes
1
answer
360
views
Density character of $\ell_\infty(\kappa, S)$
Let $S$ be an uncountable set. Consider the subspace $\ell_\infty(\kappa, S)$ of $\ell_\infty(S)$ formed by all functions with support of cardinality at most $\kappa$ (here $\kappa<|S|$). Certainly,...
- 537
0
votes
0
answers
395
views
The ratio of two strictly increasing functions
Given:
\begin{equation}
f_1(a)=\sum_{i=1}^{k^*-1} \left(\begin{array}{c}
K \\\
i \\
\end{array} \right) \left(-1-\frac{1}{ar}\right)^i
\end{equation}
\begin{equation}
f_2(a)=\sum_{i=1}^{k^*-1} ...
2
votes
1
answer
349
views
Atomic enveloping von Neumann algebra
Let $A$ be a $C^*$-algebra. If the second dual of $A$, which is the enveloping von Neumann algebra of $A$, is atomic, can we deduce that $A$ is an ideal in its second dual ?
- 195
0
votes
0
answers
607
views
partial differential equations with mixed boundary conditions
hi,
does anyone know some good references (books, papers) on partial differential equations with mixed boundary conditions ?
actually I am intrested in the following: Let $f(x)=(f_{1}(x),...,f_{n}(...
- 89
7
votes
6
answers
2k
views
Fractional Leibniz formula
Let $T=(-\Delta)^{1/2}$.
Can we have estimates, similar to the one below
$$
\| T^{\alpha}(fg)-(T^{\alpha}f)g-f(T^{\alpha}g) \|_p \leq \|T^{\alpha-1}f\|_p \|T^{\alpha-1}g\|_p,
$$
hold in $L^p$, where $...
- 1,644
5
votes
2
answers
465
views
A name for a weak topology
Let $V$ be a real vector space and let $V'$ be the algebraic dual of $V$, i.e. the space of all the linear functionals $V\to\mathbb{R}$. Then there exists the weakest topology $\tau$ which makes all ...
- 3,749
1
vote
2
answers
409
views
Does these commutator estimates bound in $L^{2}$
According to the basic rules of symbolic caculus,$[a(x,D),x_{j}]=-ia^{j}[x,D]$.So we have $[(1-\triangle)^{\frac{1}{2}},x_i]=\partial_i(1-\triangle)^{-\frac{1}{2}}$ which is $L^2$ bounded.
It's also ...
- 1,644
2
votes
3
answers
941
views
The topology of $C_0^\infty(M) $
I have read definitions in my PDE book as follows: If $M$ is a smooth paracompact manifold, the space of all linear functional on $C^\infty(M)$ is denoted by $E'$ and the space of all linear ...
- 1,441
4
votes
3
answers
464
views
What classes of functions are closed under all rescalings?
Let us denote by the symbol $\mathcal{G}$, a group of functions $f: \mathbb{R} \rightarrow \mathbb{R}$ (with the composition operation) that is additionally closed under all affine change of variables ...
- 81
3
votes
0
answers
289
views
How well do continuously differentiable functions behave from R^2 to R^2 ?
The behaviour of complex smooth vs 1-dimensional real smooth functions is discussed in a previous question.
In "Complex Analysis as Catalyst" by Steven G. Krantz, the Cauchy integral formula is ...
- 529
2
votes
1
answer
96
views
Are the real components of s-roots subharmonic?
Suppose $f(z)$ is an analytic function on a domain $D$ which maps negative axis to negative axis. For $s>1$ consider the function $$u(z)=\Re \sqrt[s]{f(z)}$$ with the branch cut along the negative ...
- 1,306
2
votes
2
answers
665
views
Finite imensional subspaces of $L^\infty.$
This is the question I had meant to ask when I asked this question.: Is there a concise characterization of finite dimensional subspaces of $L^\infty?$ (that's what the discussion with @fedja was ...
- 96.4k
5
votes
3
answers
512
views
Finite dimensional subspaces of $L^1.$
This question is motivated by my discussion (via comments) with @fedja regarding this earlier question. In any case the question is whether there is any concise characterization of finite dimensional ...
- 96.4k
6
votes
2
answers
487
views
Induction theorems for finite-dimensional complex representations of infinite groups
Let $G$ be a group, usually infinite. I am interested in finite-dimensional complex unitary representations of $G$, i.e. group homomorphisms $G \rightarrow U_n(\mathbb{C})$. The category of these ...
- 976
42
votes
4
answers
3k
views
What is the Krull dimension of the ring of holomorphic functions on a complex manifold?
Consider a connected holomorphic manifold $X$ and its ring of holomorphic functions $\mathcal O(X).$
My general question is simply: in which cases is the Krull dimension $\dim \mathcal O(X)$ known?
...
- 47.5k
15
votes
2
answers
1k
views
Is zero a hydrogen eigenvalue?
This question has been bugging me for some time.
Take the hamiltonian for the hydrogen atom: $$\hat{H}=-\frac{1}{2}\nabla^2-\frac{1}{r},$$ acting on (a domain contained in) $L^2(\mathbb{R}^3)$. It is ...
- 2,358
2
votes
4
answers
411
views
A Fractional Linear Transformation Class Property
Let $\mathcal{S}$ be the class of Fractional Linear Transformations (or FLT's) consisting of $f: [-1,0] \rightarrow [-1,0]$ such that $f(x) = \frac{ax+b}{cx+d}$ where
$a,b,c,d \in R$, and $f'(x)>0$...
- 81
5
votes
1
answer
508
views
Projections which are not completely bounded
There are 'canonical' examples of maps on operator spaces which are not completely bounded. Nevertheless, I couldn't produce any examples of bounded projections on relatively easy to understand ...
- 181
5
votes
2
answers
404
views
Do (Banach) ultrapowers carry some sort of 'elementary equivalence'?
The (model-theoretic) ultrapowers had been used for studying elementary equivalnce of first-order structures. Then, they have been adapted to Banach spaces, which are, let me say, second-order ...
- 181
5
votes
2
answers
4k
views
sparsity of QR decomposition
Hi, everyone!
I have a sparse $n \times n$ matrix $A$ with $nnz(A)$ denoting the number of non-zero entries in $A$. Now I use QR factorization to decompose $A$ into an orthogonal matrix $Q$ and ...
- 51
7
votes
1
answer
588
views
A characterization of Lagrange multiplier. Where to find a proof?
Let $F,G\in C^1(\mathbb{R}^n,\mathbb{R})$. Assume for
$s\in(s_0-\varepsilon,s_0+\varepsilon)$,
\begin{align}
E(s) = \min F\quad\mbox{subject to}\quad G=s
\end{align}
is achieved at some $x(s)\in\...
- 305
3
votes
1
answer
3k
views
Approximate point spectrum
I have a question concerning the relation between the approximate point spectrum and the spectrum of an operator.
Let $T$ be a bounded linear operator of a complex Hilbert space $H$. The approximate ...
- 758
2
votes
0
answers
807
views
Why groups that admit Folner Sequences are amenable
I've been looking at Folner's Condition recently, and I'm struggling to find a proof for why the existence of a Folner sequence on a locally compact group implies that it is amenable (and the converse ...
- 21
6
votes
0
answers
291
views
What is the status of the subadditivity problem for analytic capacity?
Hi,
Here is another question that concerns analytic capacity. For a compact set $K$ in the plane, define the analytic capacity of $K$ by
$$\gamma(K):=\sup|f'(\infty)|,$$
where the supremum is taken ...
- 2,154
8
votes
4
answers
744
views
j-invariant fixed point?
If we view the j-invariant of a lattice as a map from the upper-half plane to the complexes by $\tau\mapsto j([1,\tau])$, then it is surjective, holomorphic, and has quite a number of other wonderful ...
- 1,261
3
votes
1
answer
598
views
is a non-invertible operator a boundary point of the group of invertible operators?
Good evening,
I have a question concerning non-invertible operators.
Let $H$ be a Hilbert space and $T$ a non-invertible bounded operator on $H.$ Is it true that $T$ is the limit of some sequence ...
- 758
14
votes
2
answers
4k
views
What is a good reference that compact resolvent implies Fredholm operator?
Suppose $A \in \mathcal{L}(E_1, E_0)$ is a bounded linear operator between Banach spaces $E_1$ and $E_0$, and we also have that $E_1$ is densely, continuously embedded in $E_0$ (i.e. $A$ can be ...
- 191
1
vote
1
answer
775
views
Weak* continuity of linear maps
I consider a linear map $T\colon X^*\to Y^*$, where $X^*$ and $Y^*$ are duals of Banach spaces. I would like to know if I can deduce that $T$ is weak* continuous (I consider the weak* topologies on ...
- 113
5
votes
2
answers
2k
views
Fundamental Theorem of Algebra, Theorems of Brouwer and Borsuk
Several months ago I was browsing through a question posted here ("Applications of Brouwer’s fixed point theorem"); amongst the comments attached it was mentioned that one could derive both Borsuk's ...
25
votes
2
answers
4k
views
Dual of the space of Hölder continuous functions?
Let $X=C^{\alpha}(\Omega,\mathbb{R})$ be the space of Hölder continuous functions. What is its dual?
- 1,256
1
vote
1
answer
565
views
Cesaro means for $\alpha<1$ and Banach limits
I am interested in conditions in terms of standard scales of summation methods that guarantee the existence of an averaged limit for all almost convergent sequences. For the Cesaro summation method $(...
- 217
5
votes
4
answers
6k
views
General procedure for inverse of an integral transform
Is there a general inversion formula or procedure for an integral of the form (where f is the function being transformed and g depends on the type of transform) $\int^{a}_{b} f(x) g(x,\xi) dx $ ?
...
1
vote
1
answer
1k
views
Prokhorov theorem
Hi there. It is known that on a polish space, if a family of bounded positive measures (no need to be probabilities) is tight, then it is relatively compact in the space of positive measures with ...
- 11
10
votes
2
answers
3k
views
Cesaro means and Banach limits
Consider the class of bounded sequences to which every Banach limit (non-negative shift-invariant continuous functional on $l^\infty$ taking convergent sequences in the usual sense to their limits) ...
- 217
5
votes
2
answers
889
views
Fatou Coordinate for function with rationally indifferent fixed point, and repelling fixed point
Lets say I have $f(z)=z^2+c$, with $c=0.35676274578 + 0.32858194507i$. Then $f(z)$ has a fixed point $\kappa_0=0.15450849719 + 0.47552825815i$, which is rationally indifferent with a period $m=5$. ...
- 159
8
votes
1
answer
2k
views
Fastest decay of Fourier Transform for Generalized Functions of compact support
What is the fastest decay possible for the Fourier transform of a generalized function with compact support and finite value at the origin? I know that regular functions cannot attain exponential ...
- 83
10
votes
1
answer
1k
views
Separating vectors for C$^*$-algebras
(I asked this on math.stackexchange, without response).
Let $A$ be a C$^*$-algebra, concretely acting on a Hilbert space $H$. Suppose that $\xi_0\in H$ is cyclic and separating for $A$ (that is, the ...
- 18.7k
5
votes
1
answer
699
views
Integer polynomials mapping the unit disk into itself
Is there a complete characterization of those integer polynomials, that is $P\in{\mathbb Z}[X]$, such that $P(D)\subset D$, where $D$ is the unit disk ? At least, $P(X)=\pm X^k$ works, when $k\in\...
- 52.3k
3
votes
2
answers
221
views
Do infinite products commute with functor of smooth sections?
Similarly to my previous question about direct limits, I have now basically the same question about inverse limits. It seems in fact, that I only need the result for products.
Question: Is there a ...
- 8,597
0
votes
1
answer
2k
views
When are conformal maps holomorphic?
It is a standard fact from elementary complex analysis that a holomorphic function $f:\mathbb{C}\to \mathbb{C}$ is a conformal mapping. Now, suppose I have a map $f':\mathbb{R}^2\to \mathbb{R}^2$ ...
- 640
6
votes
1
answer
1k
views
A question about the Beurling-Selberg majorant
Beurling's majorant is defined as the unique entire function $B(z)$ such that the Fourier transform of $B(x)$ is compactly supported in the interval $[-1;1]$, $B(x) \geq \text{sgn}(x)$ and $B(x)$ ...
- 63
5
votes
1
answer
941
views
When quotient of a $k$-algebra by any maximal ideal is $k$?
Let $k$ be a valued field.
Is there a special term for a commutative (Banach) $k$-algebra $A$ such that for any maximal ideal $m$ we have $A/m=k$?
Is there an easy to check criterion that would imply ...
- 378
4
votes
1
answer
276
views
Abelian sub-W*-algebras
Let $M$ be a von Neumann algebra which acts faithfully on a Hilbert space of density character $\kappa$ but does not on a Hilbert space of density character $\lambda<\kappa$ (that is, the density ...
- 41
8
votes
1
answer
1k
views
derivative in the Wasserstein space
Villani gives the following formula to find the gradient of a function $F$ of a probability density function $\rho$ in the Wasserstein space :
$$\nabla_W F(\rho) = -\nabla.(\rho \nabla \frac{\delta F}{...
- 481
2
votes
1
answer
372
views
Complementable subspaces of $(c_{00}(S),\Vert\cdot\Vert_1)$
Let $\ell_{1,0}(S)=(c_{0,0}(S),\Vert\cdot\Vert_1)$ be a space of functions on a set $S$ with finite support, endowed with $\ell_1$ norm. Could you answer the at least one of the following questions
...
- 1,697
2
votes
2
answers
171
views
approximation of holomorphic functions on a halfplane.
Let $\mathbb {C} _ + $ denote the right halfplane and $A$ the algebra
$$
A = \{ f \in H^\infty({\mathbb C} _ +) \cap C(\overline{{\mathbb C} _ +}): \;
|f(z)| \le M (1+|z|)^{-\epsilon} \text{ ...
- 21
1
vote
0
answers
136
views
Boundedness of Integral
Suppose we operate on the unit simplex $\Delta \subset \mathbb{R}^d$ with $0$ as a corner point.
Define the integral
$$
Iu(x):=\int_0^1 t^{|\beta|-1}x^\beta u(tx)\mathrm{d}t,\quad x\in \Delta
$$
and ...
- 233
11
votes
6
answers
6k
views
Why $\partial$ and $\bar{\partial}$ defined in that way (the Wirtinger derivatives)?
For $\mathbb{C}$-valued functions, why are $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial \bar{z}}$ defined as
$$
\frac{\partial}{\partial z}=
\frac{1}{2}\left(
\frac{\partial}{\...
- 1,111
0
votes
2
answers
424
views
Unbounded sequences in Banach spaces
Let $X$ be a Banach space and let $T$ be a bounded operator acting on $X$. Suppose for each linearly independent unbounded sequence $(x_n)$ in $E$, the sequence $(Tx_n)$ is unbounded. Must $T$ be ...
- 181