All Questions
9,960 questions
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cardinality of discontinuity curves of BV function
If the function $f:R\to R$ is of BV class then it has at most countably many discontinuity points (since it can be represented as a sum of two monotonic function).
I am interested to know whether the ...
- 236k
3
votes
1
answer
502
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Determining continuous functions on Banach spaces
Let $X$ be a real Banach space.
For a continuous (not necessarily linear) function $g:X \to \mathbb{R}$ and a family $\mathcal{F} \subseteq X^*$, we´ll say that $\mathcal{F}$ determines $g$ if ...
CommunityBot
- 1
1
vote
0
answers
153
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The existence of the solution of the perturbed KdV Equation(semi-group operator)
Consider the perturbed KdV Equation$$u_t-6uu_x+u_{xxx}=\epsilon u,u(x,0)=f(x)$$where $f(x)=v(x,0)$，$v(x,t)$ is a soliton solution.$u$ satisfies the condition$u\to 0 $when $|x|\to\infty$
I want to use ...
- 61
1
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1
answer
4k
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how to prove the range of a closed linear operator is closed ?
The closed range theorem tells us that given two banach spaces X,Y,and a closed densely defined linear operator T：$X \to Y$. We have the following equivalence $R(T)$ is closed in $Y \iff R(T^{*})$ is ...
CommunityBot
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2
votes
1
answer
1k
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Are bounded functions L-1 compact?
Let $(X,\Sigma,\mu)$ be a finite measure space (i.e., $\mu(X) < \infty$). Let $\mathcal{F}$ be the set of $\mu$-measurable functions $f:X \to \mathbb{R}$ that are bounded in $[0,1]$, so that $0 \...
- 54.2k
1
vote
1
answer
201
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Reference request for sums of Grothendieck spaces
I much appreciate your help with my previous question. Now, I'd like to ask about a reference. I need a fact asserting that if $p\in (1,\infty]$ (with emphasis on the case $p=\infty$) then the $\ell_p$...
- 31.5k
11
votes
1
answer
504
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Do ultrapowers of classical Banach spaces have unconditional bases?
I am trying to imagine (to some extent, of course) the geometry of ultrapowers of certain 'easy-to-handle' Banach spaces. Let me start with $X = \ell_p$, $p\in (1,\infty)$ or $X=c_0$.
Since the ...
- 31.5k
50
votes
7
answers
16k
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Way to memorize relations between the Sobolev spaces?
Consider the Sobolev spaces $W^{k,p}(\Omega)$ with a bounded domain $\Omega$ in n-dimensional Euclidean space. When facing the different embedding theorems for the first time, one can certainly feel ...
- 34.7k
0
votes
0
answers
272
views
L_2-norm representation
Let
$$
f^{\alpha}_+(x)=\frac{1}{\Gamma(\alpha+1)}\sum_{k\ge 0}(-1)^k{\alpha+1 \choose k}(x-k)^{\alpha}_+,
$$
where $\alpha > -\frac 12$.
I am wondering if one can get nice representation of $L^2$-...
- 7,637
7
votes
1
answer
331
views
States/functionals on crossed product C*-algebras
Let $A$ be a C*-algebra, $\alpha$ a strongly continuous automorphic action by a locally compact group $G$ on $A$, and consider the crossed product $A\rtimes_\alpha G$. I am looking for references ...
- 11.2k
1
vote
1
answer
1k
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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 ...
CommunityBot
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7
votes
3
answers
498
views
Sums of unitaries with small norm in full group $C^*$-algebras
Suppose $G$ is a finitely generated group, with given generating set $S={g_1, \dots, g_n}$. (Assume that if $g\in S$, then $g^{-1}\notin S$. (EDIT: Also assume that $S$ is minimal in the sense that ...
- 9,486
1
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1
answer
887
views
How to explain the condition (C) in critical point theory?
Condition (C). The closure of any nonempty subset S of H on which f is bounded but on which $\|\nabla f\|$is not bounded away from zero, contains a critical point of f.
How to see the meaning of " $\|\...
- 7,583
4
votes
1
answer
882
views
What is the domain of the "average operator"?
I can try to define an averaging operator for functions, namely let
$$A: D \subset L^\infty([0,\infty]) \to \mathbb{R}$$
by
$$Af = \lim_{N\to\infty} \frac{1}{N}\int_0^N f(x)dx$$
whenever the limit ...
- 6,034
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 ...
- 23k
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.
- 34.7k
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)\...
CommunityBot
- 1
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 $m,n \in \mathbb{Z}^2$.
...
CommunityBot
- 1
22
votes
1
answer
4k
views
Image of the trace operator
It is well-known that we have the trace theorem for Sobolev spaces. Let $\Omega$ be an open domain with smooth boundary, we know that the map
$$ T: C^1(\bar\Omega) \to C^1(\partial\Omega) \subset L^...
- 39.1k
3
votes
1
answer
503
views
When an AW*-algebra is a W*-algebra
In a very old book of Kaplansky "Rings of operators", on p. 123 one can find the following sentence:
It is a standing conjecture that an AW${}^\ast$-algebra is W${}^\ast$ if its center is W${}^\ast$.
...
- 42.8k
2
votes
1
answer
624
views
The perturbed KdV Equation
I'm now studying KdV Equation$$u_t-6uu_x+u_{xxx}=0$$To solve the initial-value problem,we can use method of Lax pair,so we can alter the original problem to the problem of solving out $u$ in the ...
CommunityBot
- 1
3
votes
1
answer
254
views
gluing along a real analytic manifold
hi,
I have a general question. Assume we have a real analytic $n-$dim. manifold $X$ and $M$ a real analytic compact submanifold of $X$ (of dimension less that the dimension of $X$, say $k < n$). ...
- 2,283
1
vote
1
answer
780
views
An asymptotic series for the digamma function
As we know, there is an asymptotic series for the digamma function when $z>0$ is a real number.
$$
\psi(z)=\ln z+\sum_{n=1}^{\infty}{\frac{B_n}{nz^n}}
$$
$B_n$ is the first Bernoulli numbers.
How ...
- 1,551
4
votes
2
answers
644
views
Does there exists a necessary condition for Lp multiplier?
Let $1 \leq p \leq 2$. A measurable function $m(\xi)$ is called a $L^p(R^n)$ ($L^p$ for convenience) multiplier, if $$\|m(D)\varphi\|/\|\varphi\|_{L^p} \leq C , \varphi \in L^p
$$ for some constant $C$...
- 90
1
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0
answers
1k
views
Can you prove the monotonicity of the function (or find a counter example)?
Let $X$ be a non-negative random variable that is drawn from a cumulative distribution function $F(\cdot)$, pdf $f(\cdot)$ and mean $E[x]$. $k$, $c$, $v_l$ and $v_h$ $(v_h>v_l)$ are non-negative ...
- 11
0
votes
0
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607
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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}(...
- 39.1k
2
votes
1
answer
235
views
ODE for functions with values in locally convex TVS
Given an ODE for a function $u \in C^1(I,V)$, where $V$ is some locally convex TVS (topological vector space) and $I \subset \mathbb{R}$, i.e.
$\frac{d}{dt} u = f(t,u)$
for some function $f: I \...
- 9,650
3
votes
1
answer
177
views
If $A \subset X'$ annihilates only $0$, then $A$ is dense
Let $X$ be a Banach space with continuous dual space $X'$ with norm topology. Let us regard the following property of $X$:
Property: Any linear subset $A \subset X'$ that satisfies $\bigcap_{\alpha\...
- 18.7k
1
vote
2
answers
394
views
When LCS is isomorphic to subspace of some function space?
Updated: Following Michael's suggestion, I rephrase the question slightly.
Given a locally convex (Hausdorff) topological vector space (LCTVS), when is it isomorphic to a subspace of some function ...
- 101
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\...
0
votes
2
answers
2k
views
The exponent of self-adjoint operator
If $X$ is a Hilbert space and $A$ is an unbounded self-adjoint operator on $X$, is it necessarily that $A^k$ is self-adjoint for all positive integer $k$? (I have already known that the conclusion ...
- 4,729
5
votes
1
answer
600
views
Closed operators and duality
Usually we would define a "densely defined, closed operator" on a Banach space $E$ to be a linear map $T:D(T)\rightarrow E$, where $D(T)$ is a dense subspace of $E$, and the graph of $T$, $G(T)=\{ (x,...
- 362
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1
answer
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Quasi-nilpotent trace class operators as limits of nilpotents
In as yet unwritten work with T. Figiel and A. Szankowski we make an observation in a Banach space context that for Hilbert spaces reduces to:
If $T$ is a quasi-nilpotent (i.e., has only $0$ in its ...
CommunityBot
- 1
2
votes
1
answer
3k
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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}\...
- 16.2k
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,...
- 236k
1
vote
2
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409
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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
1
answer
348
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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 ?
- 56.9k
0
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0
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395
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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} ...
CommunityBot
- 1
5
votes
2
answers
464
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
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3
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464
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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 ...
- 15.4k
2
votes
3
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2k
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Solving $x\partial_x f = 0$ over distributions
Solving $x\partial_x f = 0$ over 'normal' functions is the same as solving $\partial_x f = 0$, i.e. one gets $f(x)=c_1$ as the complete answer. But over distributions (if my calculations are correct),...
- 16.2k
2
votes
4
answers
411
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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 $f'(x)>0$...
- 8,903
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 ...
CommunityBot
- 1
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 ...
CommunityBot
- 1
6
votes
4
answers
2k
views
Distributions more complicated than the Dirac δ and derivatives
The responses to another question clarifies that the best known examples of distributions that are not measures, are the derivatives of the delta and such. What I want to know is: Is that the only way ...
CommunityBot
- 1
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 ...
- 31.5k
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 ...
- 25.8k
0
votes
1
answer
765
views
($n$-dimensional) Inverse Fourier transform of $\frac{1}{\| \mathbf{\omega} \|^{2\alpha}}$
Note: I first posted question on math.stackexchange and I got one reply, which was a bit helpful (I'm still trying to understand it fully), but did not explore the two solution cases that I mentioned. ...
CommunityBot
- 1
14
votes
2
answers
4k
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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 ...
- 34.7k
2
votes
0
answers
807
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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