All Questions
10,050 questions
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
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
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
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 :
$$\nabla_W F(\rho) = -\nabla.(\rho \nabla \frac{\delta F}{...
- 481
2
votes
1
answer
372
views
Complementable subspaces of $(c_{00}(S),\Vert\cdot\Vert_1)$
Let $\ell_{1,0}(S)=(c_{0,0}(S),\Vert\cdot\Vert_1)$ be a space of functions on a set $S$ with finite support, endowed with $\ell_1$ norm. Could you answer the at least one of the following questions
...
- 1,697
1
vote
0
answers
136
views
Boundedness of Integral
Suppose we operate on the unit simplex $\Delta \subset \mathbb{R}^d$ with $0$ as a corner point.
Define the integral
$$
Iu(x):=\int_0^1 t^{|\beta|-1}x^\beta u(tx)\mathrm{d}t,\quad x\in \Delta
$$
and ...
- 233
0
votes
2
answers
424
views
Unbounded sequences in Banach spaces
Let $X$ be a Banach space and let $T$ be a bounded operator acting on $X$. Suppose for each linearly independent unbounded sequence $(x_n)$ in $E$, the sequence $(Tx_n)$ is unbounded. Must $T$ be ...
- 181
4
votes
1
answer
521
views
Basic sequences in $\ell_p$
Let $p\in [1,\infty)\setminus\{2\}$. Suppose $(e_n)$ is a basic sequence in $\ell_p$ (or $L_p$) equivalent to the basis of $\ell_p$ ($L_p$). Is there a subsequence $(e_{n_k})$ such that $[e_{n_k}]$ is ...
- 181
6
votes
0
answers
98
views
Do the translates of integrable function approximate its radial part?
For an integrable function $f$ on $\mathbb R^n$ we consider its ``radial'' part
$$R(f)(x)=\int_{\mathrm{SO}(n)} f(kx)dk.$$ What is the minimal condition on $f$ so that the span of translates of $f$ (...
- 415
12
votes
1
answer
838
views
A measure theory question
Here's an interesting problem one can formulate for a student. This problem arises when considering special ergodic theorems:
On a finite dimensional manifold $M$ with a Lebesgue measure $\mu$, does ...
- 1,143
3
votes
2
answers
340
views
Perturbing upper-semi Fredholm operators
Let $T\colon X\to X$ be an upper-semi Fredholm operator acting on a $B$-space $X$ (the range of $T$ is closed and kernel is finite-dimensional) with complemented range. Suppose $S\colon X\to X$ is ...
- 181
2
votes
1
answer
2k
views
If any perfect set is uncountable in a metric space which is not complete?
We know that every ferfect set $E$ in a complete metric space $X$ is uncountable. My question is if there exists a metric space which is not complete, but every ferfect set in it is uncountable. The ...
- 21
1
vote
2
answers
139
views
Given f(t) = \sum_k C_k exp(2 pi i w_k t ) + noise. Need to estimate C_k and w_k .
Simpliefied setup.
Assume I am given some function f(t).
I know that it is constructed as $f(t) = \sum_{k=1...M} C_k exp(2 \pi~ i~ w_k t ) + noise(t)$.
where $noise(t)$ is some random set of numbers ...
- 24.9k
1
vote
1
answer
287
views
General compactness criterion in functional spaces
What follows is a total boundness criterion in the space $L^1(X)$, where $X$ is arbitrary space with probabilistic continuous measure (Lebesgue space). Of course, all such spaces $X$ and hence $L^1(X)$...
- 109k
2
votes
1
answer
214
views
union of Stone-Cech remainders
Can anyone point me to a reference or further information on the following construction? Let $X$ be a compact metric space such as $[0,1]$. Let $A$ be the commutative pre-C*-algebra consisting of [...
- 607
3
votes
1
answer
393
views
A Sobolev-type inequality with weights
In the study of a particular PDE I found myself wanting to prove the following inequality:
$( \int_0^{\infty} r^{-3} |f|^6 \; dr )^{1/6} \leq C ( \int_0^{\infty} [ r^{-1} |f|^2 + r |f'|^2 + r |f''|^2]...
- 143
20
votes
1
answer
994
views
Which spaces are characterized by functions with compact support ?
It's well known that two locally compact Hausdorff spaces $X, Y$ are homeomorphic iff the rings $C_0(X), C_0(Y)$ (continuous functions vanishing at infinity) are isomorphic.
Is there a class $\...
- 16.2k
2
votes
2
answers
408
views
Elliptic function with constant real part on the unit square diagonals?
Consider the following even meromorphic doubly periodic function with poles at the gaussian integer lattice.
$H(z) = \prod_{n \in \mathbb{Z}} {1 \over{ 1 - {1 \over{\cosh\left(2\pi\left(z-n\right)\...
- 181
3
votes
0
answers
409
views
Continuous function sort
If you have a real-valued function f(x), positive, continuous and bounded on some interval, then what kind of transform would convert this to a monotonic function g(x) on that interval analogously to ...
- 529
3
votes
2
answers
771
views
Special values of a doubly periodic meromorphic function
Consider the following function: $G(z) = \prod_{n \in \mathbb{Z}} {1 \over{\tanh^2\left(\pi\left(z-n\right)\right)}}$.
By constuction, it has poles at $z=m+in$ with $m,n \in \mathbb{Z}^2$.
...
- 181
13
votes
2
answers
776
views
Properties of orthogonality-preserving c.p. maps between $C^*$-algebras
Suppose that $A,C$ are $C^*$-algebras and $\phi:A \to C$ is a completely positive, orthogonality-preserving linear map.
(Orthogonality preserving means: if $a,b \in A$ satisfy $ab=0$ then $\phi(a)\phi(...
- 1,786
2
votes
1
answer
373
views
Is it true that $c_0(X)^* = \ell_1(X^*)$ ?
I'm trying to prove this that but I can't . Any help/reference ?
- 151
1
vote
0
answers
180
views
iterated traces for sobolev functions
It is well known that if $M$ is a smooth $(n-1)$-dimensional surface in $\mathbb R^n$ (e.g. a subspace) then there is a continuous trace operator $W^{s,p}(\mathbb R^n)\to W^{s-1/p,p}(M)$. Now suppose ...
- 2,041
5
votes
1
answer
664
views
Are piecewise linear curves dense among Hölder curves?
Consider for some $0 < \alpha \leq 1$ the space functions $x:[0,1] \to \mathbb{R}^n$ such that $x(0) = 0$ and
$\sup_{s,t} \frac{\|f(t)-f(s)\|}{|t-s|^{\alpha}}$
is finite.
There are at least two ...
- 4,304
1
vote
1
answer
686
views
analytic continuation of a Laplace transform from a countably infinite set of points?
Let $f(\lambda)=\int_0^\infty e^{-\lambda s} F(ds)$, where $F$ is the distribution of a positive random variable. Suppose I know the value of $f(n)$ for $n=0,1,2,\cdots$. Is this enough to uniquely ...
- 351
7
votes
2
answers
920
views
Exotic spectrum of Laplace operator
Given a closed Riemannian manifold and a generalized Laplace $\Delta$ operator,
it is well known that $\Delta$ has discrete spectrum $(\lambda_n)_n$ (arranged in a increasing way, not counting ...
- 101
2
votes
0
answers
564
views
Young inequality in weighted spaces
Let $U$ be a bounded open set in $\mathbb{R}^2$, $g\in L^1_{\mathrm{loc}}(\mathbb{R}^2)$.
Let moreover $w$ be a weight (i.e. a non vanishing locally integrable function) on $U$ and $p\geq2$.
Does ...
- 1,205
24
votes
3
answers
4k
views
Self-dual normed spaces which are not Hilbert spaces
Are there any examples of non-Hilbert normed spaces which are isomorphic (in the norm sense) to their dual spaces? Or, is there any result in Functional Analysis which says that if a space is self-...
- 2,239
2
votes
2
answers
2k
views
A point in the weak closure but not in the weak sequential closure
I'm trying to find a proof of this counterexample by von Neumann:
Let $x_{mn}\in \ell^2$ be defined by
$$x_{mn}(m)=n \quad,\quad x_{mn}(n)=m \quad,\quad x_{mn}(k)=0 \hbox{ otherwise,} $$
and let $S=\...
- 131
-2
votes
1
answer
3k
views
Multiplying two Fourier series gives one Fourier series, but what are the new coefficients? [closed]
If I have $A(x)=B(x) C(x)$ (sine periodic from 0 to 1) rewritten as
$\sum_n A_n \sin(n \pi x)=\sum_m B_m \sin(m \pi x)\sum_p C_p \sin(p \pi x)$
is there any easier way to compute $A_n$ from $B_m,...
- 149
8
votes
2
answers
1k
views
Approximation by polynomials
Let $f:[a,b] \rightarrow \mathbb{R}$ be of class $C^n$. Let $ x_0, ..., x_m$ be different numbers from $[a,b]$.
Does for each $\varepsilon >0$ there exist a polynom $P$ such that $P^{(k)}(x_i)=f^{...
- 277
4
votes
1
answer
266
views
Exotic uniform algebras
The first non-trivial example of a uniform algebra which comes to mind is the disc algebra $A(\mathbb{D})$. In a similar manner one can define its relatives $P(U)$ and $R(U)$, where $U$ is any region ...
- 43
5
votes
3
answers
821
views
are the smooth vectors of a Frechet space dense?
Given an action $\alpha$ of $V$ a Lie group on $B$ a Fréchet space with seminorms $ \{ \| \cdot \|_j \} $, let $B^\infty$ be the space of smooth vectors. Is this dense in $B$? Can I guarantee it is ...
- 342
7
votes
2
answers
484
views
Extension of weakly compact operators from $\ell_1$ into $c_0$
Is every weakly compact operator from $\ell_1$ into $c_0$ extendible
to any larger space? Equivalently, is every weakly compact operator from $\ell_1$ into $c_0$ extendible to $\ell_\infty$?
7
votes
0
answers
266
views
Problem with Shelah and Stern's paper on the Hanf number of the theory of Banach spaces
I have been trying to understand "The Hanf number of the first order theory of Banach spaces" by Shelah and Stern (Trans. AMS 244 (1978) 147-241). They construct a normed space $M$ from a Hilbert ...
- 879
15
votes
3
answers
1k
views
Extreme points of unit ball in tensor product of spaces
Let $B_1, B_2$ be unit balls in finite-dimensional normed spaces $X_1, X_2$ respectively.
Let $e(B_1), e(B_2)$ be corresponding extreme points sets.
Consider the unit ball $B$ in tensor product $...
- 580
23
votes
8
answers
8k
views
Grothendieck on topological vector spaces
In a short biography article on Alexander Grothendieck, it is mentioned that after Grothendieck submitted his first thesis on topological vector spaces (TVS), apparently, he told Bernard Malgrange ...
7
votes
3
answers
4k
views
Is a semicontinuous real function Borel measurable?
Let $f(x,u): [0,1]^2 \mapsto \mathbb{R}$ be a continuous
function.
[Q] Is $g(x) = \inf_{u\in [0,1]} f(x,u)$ always Borel measurable?
If not, can one find a counter-example?
Note that, for any $c$,
...
- 1,399
5
votes
1
answer
3k
views
Inner product of linear bounded operators between Hilbert spaces
Let $X$ and $Y$ be Hilbert spaces, and let $L(X,Y)$ be the set of bounded linear operators between Hilbert spaces.
Can we equip $L(X,Y)$ with a natural inner product? I think it should look like
$\...
- 5,327
0
votes
1
answer
221
views
Sort-of extension of Young inequality to arbitrary measures
Hello folks,
Suppose we're working in ${\mathbb T}^q$, the $q$-times Cartesian product of $[-\pi,\pi]$ with the endpoints identified. So, $ L^p := L^p({\mathbb T}^q) $.
The Young inequality may be ...
- 1
9
votes
2
answers
1k
views
polynomials with minimal $L_\infty$ norm on multiple disjoint intervals
It is well-known that Chebyshev polynomials are the polynomials of minimal $L_\infty$ norm on [-1,1] with leading coefficient 1. But what if you want the minimal $L_\infty$ polynomial on two disjoint ...
- 223
1
vote
1
answer
138
views
Estimating norms of derivatives
Let $(M,g)$ be a closed, compact Riemannian manifold. Let $u$ be a smooth function. Let $H^{-k}(M)$,, $k$ is a positive integer, be the dual Hilbert space of $H^{k}(M)$. Does it follow that $|| |\...
- 1,701
2
votes
3
answers
3k
views
Extension of pointwise convergence of a sequence of uniformly continuous functions that converges on a dense set
It is known that a sequence of continuous functions on a metric space that converges pointwise on a dense subset need not converge pointwise on the full space. But what about if one assumes uniform ...
- 777
1
vote
0
answers
178
views
Inequalities between self-adjoint operators
Let $T_s$ ($s\ge0$) be a smooth family of non-negative self-adjoint operators in a separable Hilbert space $H$. Suppose that, for some $C'>C>0$, we have $T_0+Cs^2\le T_s\le T_0+C's^2$ for all $s$...
- 329
10
votes
0
answers
509
views
Lacunary hyperbolic groups and weak amenability
In the paper called Lacunary Hyperbolic group, Y. Ol'shanskii, D. Osin and M. Sapir define and characterize the lacunary hyperbolic groups, which contains the hyperbolic groups but also Tarski's ...
- 195