All Questions
10,448 questions
0
votes
1
answer
130
views
Extending affine maps defined on weakly closed sets to the whole topological space
Given $C$ a weakly closed convex subset of a (real) Banach space $B$, with $0\in C$ and $\varphi:C\longrightarrow \mathbb{R}$ weakly continuous, with $\varphi(0)=0$, can we extend $\varphi$ to a $\...
- 647
0
votes
1
answer
241
views
Nonlocal (parabolic) PDEs in the Sobolev space setting
Can someone recommend me some literature on nonlocal parabolic problems (eg. of the form
$$u_t + (-\Delta)^s u = f$$
where the nonlocal operator is the fractional Laplacian)
in the setting of Sobolev ...
- 155
0
votes
2
answers
797
views
If $u_m \rightharpoonup u$, how to show using monotonicity that $f(u_m) \rightharpoonup f(u)$?
Let
$$u_m \rightharpoonup u \quad \text{(weakly) in $L^\infty(0,T;L^2(\Omega)) \cap L^2(0,T;H^1(\Omega))$}.$$ We are given $f:\mathbb R \to \mathbb R$, a Lipschitz continuous invertible map which is ...
- 193
0
votes
1
answer
212
views
Some convergence similar to weak-$\ast$ convergence on the space of finite measures
I have a question:
Let $D$ be the space of cadlag functions defined on $[0,1]$ and $V$ be its subspace consisting of $x$ with finite variation and $x(0)=0$.
Define $TV(x)$ as the total variation ...
- 1,835
0
votes
1
answer
715
views
The dual space of the Dirac measures on an Abelian group
Let $G$ be a Hausdorff locally-compact Abelian group and $L^2(G)$ the Hilbert space of two-integrable complex functions on the group.
Question. What would be natural vector space $\mathcal{R}$ of ...
0
votes
2
answers
664
views
Defining surface integral on boundary of $C^1$-domain
Let $\Omega$ be a bounded $C^1$ domain with bounded boundary $\partial\Omega$. Can someone point me to a reference where the surface integral of a measurable function $f\colon \partial\Omega \to \...
0
votes
1
answer
162
views
Extracting moments from a special Z-transform
Suppose I have a sequence of positive continuous random variables $\{X_k\}_{k=1}^\infty$ with (unknown) MGF's $M_{X_k}(s)$. Furthermore, it is known that
\begin{equation}\frac{X_n-n\mu}{\sqrt{n}\sigma}...
- 41
0
votes
1
answer
343
views
Norm bound on eigen-vector change caused by rank-one update
Suppose $A$ is a positive semi-definite, Hermitian matrix with a unit-norm eigen vector $\textbf{v}$ corresponding to its largest eigen value $\lambda$. Let $B = A + \alpha \textbf{z}\textbf{z}^H$, ...
- 1
0
votes
2
answers
319
views
Fixed point theorem that does not require the hemi-continuity of the set valued map?
All of the fixed point theorem I have seen (like Kakutani and Brower, Browder) required the set valued map to be hemi-continuous (lower). Is any fixed point theorem that can assure the existence of ...
- 383
0
votes
1
answer
496
views
Trace, eigenvalues and functional calculus
Let $T$ be a (possibly unbounded) self-adjoint operator on a Hilbert space. Assume that we for some reason know that the point spectrum of $T$ consists of a finite number of eigenvalues $\lambda _1, \...
- 450
0
votes
1
answer
168
views
local moments of measures whose Fourier transform vanish in an interval
Assume h is a measure whose Fourier transform vanishes in an interval $[-\Omega,\Omega]$. I'm interested in obtaining inequalities of the form
\begin{equation*}
\int_{-\delta}^{+\delta}|h|(dt)\le C(\...
- 859
0
votes
1
answer
216
views
Find a special element in group algebra
Let $$G=\langle x, y, z\mid xyx^{-1}=zy, xzx^{-1}=z, yz=zy\rangle,$$ denote $l^1(G)^{\times}$ to be the set of units in $l^1(G)$, which we have considered as a ring with multiplication defined by the ...
- 1,528
0
votes
1
answer
321
views
How to handle a scalar product in an integral?
I am having a problem with a certain inequality I try to understand. I think it's just a basic idia (/trick) I'm missing, but I can't seem to find it.
Here's a simplification of the problem:
$ \...
0
votes
1
answer
319
views
continuty of volume of a convex set in Rn [closed]
Let O(X) be the metric space of all compact subsets of a compact set X in Rn and let L be an element of O(X). Let vol(L) be the volume of L. How do we prove that vol(L) is a continuous function on O(X)...
- 1
0
votes
1
answer
321
views
Increasing regularity for $L^2$ function
Suppose that we have a function $u$ on $\mathbb{R}^2$ such that $r\frac{\partial}{\partial\theta}u \in L^2(\mathbb{R}^2)$. Suppose it is also given that $u$ lies in some fractional Sobolev space $H^s(\...
- 1
0
votes
1
answer
305
views
Embedding a semigroup into a divisible semigroup
The following is motivated by the fact that I'd like to have a way, much better if canonical, to isometrically embed a normed group into a normed divisible group. But semigroups are a much more ...
- 10.5k
0
votes
1
answer
2k
views
How to compute difference between 2 similarity matrices?
Hello,
I have two n*n correlation matrices with values ranging between -1,1. (2 correlation matrix because I have the same n terms under 2 different conditions)
I then transformed the correlation into ...
- 1
0
votes
1
answer
499
views
How identify bounded Borel measurable functions
Let $S$ be a topological semigroup, and $M(S)$ be bounded, regular complex Borel measures on $S$. How can we identify bounded Borel measurable functions with elements in $M^*(S)$?
- 109
0
votes
1
answer
813
views
a problem in functional analysis that erdos solved in 2 lines
https://math.stackexchange.com/questions/261685/paul-erdoss-two-line-functional-analysis-proof .
does anyone know about what the problem was and what was his solution.
[Edit by quid:] please follow ...
- 2,106
0
votes
1
answer
483
views
Absolute norms and 1-unconditional sums
Absolute norm
Let $X$ and $Y$ be Banach spaces. Let $Z=X\times Y$ a norm $\|\cdot\|_N$ on $Z$ is called absolute if there is a function $N\colon R^2\rightarrow R$ such that
$$
\|(x,y)\|_N=N((\|x\|, \|...
- 3
0
votes
1
answer
873
views
Bounding a determinant ratio
Let $A=[A_{0}\ E;E^{T} \ B]$ be a real positive definite matrix and let $B$ be a principal submatrix. I am interested in tightly bounding $\frac{|B|}{|A|}$ from below in some "explicit" way that will ...
- 6,962
0
votes
1
answer
549
views
One-dimensional Hausdorff measure of preimages
Let $\Omega$ be an open subset of $\mathbf{R}^n$. For a mapping $f: \Omega\to \bf{R}^n$, what kind of condition ensures that the one-dimensional Hausdorff measure of $f^{-1}(E)$ is zero whenever $E$ ...
- 1,881
0
votes
1
answer
913
views
Bounding derivative of a function
Consider $a(t)\in\mathbf{L}^{2}(\mathbb{R})$ and $a(t)>0$, is a low pass smooth function with $\hat{a}(f)=0, |f|>f_{max}$. Can we have a upper bound on the following,
$\Big|\frac{a'(t)}{a(t)}\...
- 151
0
votes
2
answers
765
views
About a generalization of the Radon Nikodym Theorem
Im trying to prove a generalization of the Radon Nykodym theorem, but im having troubles even for finite measures, could someone help?
Let $\mu$ and $\nu$ two $\sigma$-finite measures in $\(X,\...
- 11
0
votes
1
answer
1k
views
Showing a coercivity condition for this bilinear form
Suppose $\Omega \subset \mathbb{R}^n$ is a compact domain. Let $f$ and $J$ (and also $\frac 1J$) be $C^1$ functions on $\Omega$. Consider the bilinear form $a:H^1(\Omega) \times H^1(\Omega) \to \...
- 107
0
votes
1
answer
229
views
Complemented subspaces of $\ell_p(I)$ for uncountable $I$
I was looking for an article mimicing result of Pelczynski for $\ell_p$. I have found this one
Rodriguez-Salinas, B. (1994). On the Complemented Subspaces of $c_0(I)$ and $\ell_p(I)$ for $1 < p &...
- 1,697
0
votes
1
answer
372
views
Does this sequence converge to zero?
Description
Let $\{e_n\}$, $e_n\in \mathbb{R}^p$ be a sequence of vectors, $\{U_n\}$, $U_n\in\mathbb{C}^{p\times p}$ be a sequence of unitary matrices (that is $U_i^*=U_i^{-1}$, $^*$denonts conjugate ...
- 418
0
votes
1
answer
488
views
Discrete Sobolev space of $R^n$ valued maps
Can some one tell me the reference or any idea how to take the Discrete Sobolev space work defined for a scalar valued map to the space of maps which are vector valued.Let's say
$f:\Omega \...
- 17
0
votes
1
answer
156
views
Does homeomorphism preserves the family of cones?
Let me state my problem. Suppose we have a ball $B$ in standard $\mathbb{R}^3$, that is a $\varepsilon$-neighbourhood of $0$ point. Suppose we have a family of cones $X_C = \lbrace C > 0 \vert x^2 +...
- 165
0
votes
1
answer
238
views
A property of a quasiperiodic function
Let F be a continuous periodic function on R^N. Let a,b be vectors in R^N. Also assume a is not parallel to b.
Does the limit of
$\varepsilon \int_0^{1/\varepsilon} F(as+b/\varepsilon) ds$
Exist ...
- 213
0
votes
2
answers
818
views
Application of inverse function theorem to get short time existence
I am reading a book on curve shortening flow. Optionally, please see this image for the page that is confusing me (I am not allowed to include it in this post since I'm new): https://i.sstatic.net/...
- 45
0
votes
2
answers
146
views
representation of compact supported distribution
Is this true?
Any compact supported distribution can be represented as finite sum of partial derivatives of functions.
- 9
0
votes
1
answer
365
views
How to estimate the norm of a matrix
There is a matrix as following,
\begin{eqnarray}
A = \left (
\begin{array}{l}
0 \quad \quad \quad \quad \quad \quad \quad ~~ 1\\
b \quad ~~~0 \quad \quad \quad \quad \quad a\\
ab \quad ~~ b \...
0
votes
1
answer
338
views
The part of an operator as an analytic generator
Let the operator $A$ be the generator of an analytic semigroup on a Banach space $X$.
Let $Y$ be another Banach space embedded in $X$. We consider$A_Y$, the part of $A$ in $Y$, defined as the ...
- 271
0
votes
1
answer
864
views
Sequence of smooth functions converging to sgn(x)
I'm looking for a sequence of smooth functions $f_i(x)$ converging to Sign$(x)$, each of which additionally have the following property:
\begin{equation}
f_i(x_1+x_2) = g_i(x_1, f_i(x_2))
\end{...
- 561
0
votes
1
answer
666
views
A Cauchy–Schwarz Type Inequality Involving Scaled Distributions
I have stumbled upon a rather intriguing inequality involving the product of the scaled distribution and the scaled density of a random variable. The inequality has a very attractive form, and it ...
- 197
0
votes
1
answer
437
views
Möbius Transform of a Continuous Possibility Function
In order to be able to use a basic possibility function as a Body of Evidence in the Dempster-Shafer Theory of Evidence, it is needed to transform the function to its Möbius representation.
There is ...
- 103
0
votes
1
answer
498
views
Quotient of \ell_1 by space of finite sequences
The following question came up during a reading of Rudin's functional analysis. I have not been able to find any information through searching online, but I apologise if the answer is obvious, or the ...
- 11
0
votes
2
answers
415
views
Commutative *-subrings of the noncommutative C*-algebra $B(l^2)$
A $\star$-ring is a ring with an involutive anti-automorphism. The simplest example of a noncommutative $\star$-ring is perhaps $B(l^2)$, the ring of bounded linear functions on the sequence space $l^...
user2529
0
votes
1
answer
340
views
Reference for spectral theory of group of linear operators
It is not hard to find the spectral theory of a single unitary operator $U$. This is the spectral theory for a $\mathbb{Z}$-action because we consider $U^n$ for $n\in\mathbb{Z}$. This is clear with ...
- 163
0
votes
1
answer
611
views
Linear functionals and continuous functions on open intervals
Let $Q$ be an open interval of ${\mathtt R}$ and $E$ be the space of continuous and bounded functions in $Q\to \mathtt{R}$.
I call $E^*$ the set of linear functionals over $E$ and $E_+^*$ the subset ...
- 13
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
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
0
votes
1
answer
384
views
spectral measure
how to calculate spectral measure for a given normal operator for example right shift operator?
- 1
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
0
votes
1
answer
263
views
Separability of inner product to a product of Minkowski function and norm
I’ve encountered the following assumption:
Let D be a set such that there exists a Minkowski function $f(u)$ on $\mathbb{R}^l$ and norm $g(v)$ on $\mathbb{R}^m$ such that
$\forall u\in \mathbb{R}^l, \...
- 1
0
votes
1
answer
297
views
Continuity of cylindrical functions.
Let $C_c^\infty(\mathbb R^n)$ be the functions from $\mathbb R^n$ to $\mathbb R$ with compact support, further let $X$ be a separable Hilbert space with a fixed orthonormal basis $(e_n)_n$. Define the ...
- 455
0
votes
0
answers
10
views
Fractal dimension using wavelets
I'm trying to estimate the fractal dimension of a function.
I created the log(energies) Vs log(scales) plot and I'm computing the Fractal Dimension (D) from the slope using the relation
$$
\alpha = -...
0
votes
0
answers
67
views
$L_1$ norm of $f\in L^1(\mathbb{R}^n)$ compactly supported and its change of variable
Let $M\in\mathbb{R}^{n\times n}$ be an invertible matrix, denote its induced linear map on $\mathbb{R}^n$ also by $M$, and let $f\in L^1(\mathbb{R}^n)$ be compactly supported.
I am wondering if we can ...
- 91
0
votes
0
answers
46
views
Amenability of locally convex algebras
Let $A$ be an amenable Banach algebra, and let $A_w$ denote $A$ with the weak topology. Clearly, $A_w$ is a Hausdorff locally convex algebra (l.c.a.).
Q0: Is $A_w$ amenable as a l.c.a. in the sense ...
- 2,605