All Questions
10,199 questions
1
vote
0
answers
693
views
A question about an equivalent definition of the Schwartz distribution
Hello,
Does anyone know a reference or proof of the "if" part of the following statement?
$$
\mu\in \mathcal{S}'(R)\quad\text{if and only if}\quad \mu*\alpha\in\mathcal{S}(R),\forall \alpha\in C_c^\...
- 1,649
4
votes
0
answers
189
views
Boundedness criterion for operators on mixed Lebesgue spaces
Define the mixed Lebesgue space $l_{p,q}$ as the space of all doubly indexed sequences
${\bf a}= (a(i,j))_{i,j\in\mathbb{Z}}$ such that
...
- 979
14
votes
0
answers
3k
views
Tanh version of a Fourier Transform?
I am trying to perform some computations in an environment where it is much easier to compute the hyperbolic tangent function (tanh) than cosines or sines. This prevents me from performing Fourier ...
- 3,979
20
votes
3
answers
8k
views
Why do inner products require conjugation?
For Hermitian matrices and operators, the most "natural" inner product is $f^H \cdot g$ or $\int f^* g\; dx$. A similar situation holds interpreting Fourier transforms as the inner product of ...
- 757
3
votes
1
answer
785
views
Maximal ideals of some algebras
This question comes up after I read the chapter 7 : Banach algebras and spectrum theory from Conway's book. As we have known that if $X$ is compact space, then all the maximal ideals of $C(X)=\{ f : X\...
- 281
7
votes
0
answers
161
views
Seeking reference - criterion for the existence of a positive linear functional on an ordered vector space below a given function
The following surely appears somewhere, I would greatly appreciate a reference. (The aim is to get a measure via Riesz representation, but that has nothing to do with the statement.)
Let $X$ be an ...
- 577
3
votes
2
answers
418
views
Subspaces of duals
It is an easy undergraduate exercise to show that (finite) direct sums are preserved under dualisation. Thus, it is natural to ask if we the following holds:
is it true that if $X$ is a subspace of $...
- 11.3k
7
votes
2
answers
7k
views
Dual operators between Hilbert spaces: with or without Riesz representation
Let $X$ and $Y$ be Hilbert spaces over the real numbers (so complex conjugation plays no role, and everything will be linear in the strict sense). Let $f : X \rightarrow Y$ be a linear continuous ...
- 5,327
31
votes
3
answers
5k
views
When is an integral transform trace class?
Given a measure space $(X, \mu)$ and a measurable integral kernel $k : X \times X \rightarrow \mathbb{C}$, the operator
$$ K f(\xi) =\int_{X} f(x) k(x,\xi) d \mu(x),$$
the operator $K$ is Hilbert ...
- 11.2k
4
votes
1
answer
690
views
What does $L^\infty_\varepsilon$ mean?
In Volume 4 of Reed and Simon on page 83 the authors refer to the space $(L^\infty(\mathbb{R}^3))_\varepsilon$,
and later on page 119 they use $L^\\infty_\varepsilon$.
Are these two spaces the same? ...
4
votes
1
answer
471
views
Embeddings for spaces of maximal regularity
Let $T\in(0,\infty)$ and $\Omega\subset\mathbb R^n$ be a smooth domain. In terms of maximal regularity it can be very beneficial to know for which $s_i,p,n$ the following holds true
$W^{s_1,p}(0,T;L^...
- 225
-3
votes
2
answers
768
views
Question on Linear Operators
Let $V$ be a normed infinite dimensional vector space. Let $L: V \longrightarrow V$ be a bounded linear operator. Moreover assume that $L$ is 'locally nilpotent' that is:
$$ \forall v \in V \quad \...
- 741
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
$$ T: C^1(\bar\Omega) \to C^1(\partial\Omega) \subset L^...
- 39.1k
15
votes
2
answers
3k
views
What do we actually know about logarithmic energy ?
In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by
$$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course it is not well ...
- 2,135
5
votes
3
answers
931
views
References for "different" proofs of the spectral theorem for compact operators
It is with some sort of reverential fear that I've come here to write. I've been reading you for a long time, but writing is another story... In any case, I suppose it is too late now to back out!
...
- 10.5k
3
votes
2
answers
949
views
Reference for proof that $C_b^* = rba$
The following theorem seems to have folk status:
The topological dual of the space $C_b(X)$ of bounded continuous functions on a topological space $X$ is isomorphic to the space $rba(X)$ of finite, ...
- 459
13
votes
2
answers
2k
views
What is the relationship amongst all the different kinds of spectra?
The word "spectrum" gets tossed around a lot in mathematics, and there seem to be a number of different concepts to which it applies. There is of course a physical connotation to the word which is ...
3
votes
2
answers
1k
views
Is there a corresponding Hahn decomposition theorem for the real-valued Radon measures?
Hello,
As we know that a signed measure $\mu$ on $R$ can be decomposed to the positive part $\mu_+$ and negative one $\mu_-$ by the Hahn decomposition theorem.
My question is whether each real-...
- 1,649
-1
votes
1
answer
696
views
Can singular measures be viewed as vanishing distributions? (Answer No!)
Hello,
Here is my original question: let $\mu$ be a singular measure with respect to the Lebesgue's measure on $R$. Is it true that $\int \psi \mu(d x)=0$ for any test function $\psi\in C_c^\infty(R)$...
- 1,649
7
votes
2
answers
3k
views
Distributions and measures
Hello,
After reading the previous post, I still have some doubts. Let's consider everything on $R$ to avoid complications.
Can we say that any distribution $\mu\in\mathcal{D}'(R)$ of zero order is ...
- 1,649
4
votes
1
answer
1k
views
Doubts on Reproducing Kernel Hilbert Spaces and orthogonal decomposition
I'm a CS student and I'm trying to learn RKHS theory to understand the passages made in this paper .
Among the bibliography I'm using there are "On the mathematical fundamentals of learning" and "...
- 151
3
votes
1
answer
1k
views
Long time behavior of the heat equation on R
Let $\mu\in\mathcal{S}'(R)$ be a Schwartz distribution. The solution of a heat equation with $\mu$ as the initial data is
$$
u(t,x)= \int_R \frac{e^{-\frac{(x-y)^2}{2t}}}{\sqrt{2\pi t}} \mu(d y)
$$
...
- 1,649
0
votes
1
answer
503
views
When are operators extended by linearity bounded?
Greetings.
Suppose that $H$ is a separable infinite-dimensional Hilbert space and that $M$ is an infinite
dimensional closed subspace of $H$. Suppose that {$v_{n}: n\ge 1$} is an infinite linearly ...
- 129
8
votes
0
answers
349
views
Finding a dimension-free bound for a certain multiplier on Euclidean space
The following question is indirectly motivated by strong type maximal function estimates. Let $f\in L_{p}(\mathbb{R}^{n})$. For $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$ define $m(\xi)$ so ...
- 141
3
votes
1
answer
338
views
Ultraproduct of n-dimensional Banach spaces and algebras
Hi, I am interested in the following question:
Fix $n$.
Let $M_n$ be matrix algebra over the field of complex numbers with normalized trace $tr_n$. Let $M_n^{\omega}$ be an ultrapover of $M_n$, ...
7
votes
2
answers
1k
views
Carleson's Theorem (on the Adeles and other exotic groups)
I have redone this question:
On $\mathbb R^n$ the Carleson Operator if defined by
$$Cf(x) = \sup_{R>0} \left \vert \int_{B_R(0)} e^{2\pi i x\cdot \xi} \widehat{f}(\xi) d \xi \right \vert. $$ (...
8
votes
2
answers
2k
views
Uniformly Convex spaces
My first question here would fall into the 'ask Johnson' category if there was one (no pressure Bill). I'm interested in constructing a uniformly convex Banach space with conditional structure ...
- 1,073
8
votes
1
answer
1k
views
Spectra of a Symmetric Toeplitz Operator
For a physics application, I would like to be able to compute the eigenvalues of the linear operator (acting on the Hilbert space $\ell^2$) given by an infinite matrix of the form
$\begin{bmatrix}
...
- 81
7
votes
3
answers
814
views
Preduals of B(E)
For a Hilbert space $H$ it is well known that the algebra $B(H)$ has a unique predual; the Banach space of trace class operators.
If $E$ is a Banach space then is it known whether
$B(E)$ is always a ...
- 1,411
1
vote
1
answer
470
views
Extensions of Banach spaces
I am looking for an answer to the following questions:
Are there infinite-dimensional Banach spaces $X$ and $Y$ for which there are non-split extensions $0 \to X \to E_1 \to Y \to 0$ and $0 \to X \...
- 125
5
votes
3
answers
1k
views
Integration by parts for a general negative-definite self-adjoint operator.
I suspect I am asking a very stupid question.
Suppose you have self-adjoint negative-definite operator $L$ densely defined on a space $L^2(\pi)$, with $Lf = \nabla \cdot ( A(x)\nabla f)$, for some ...
- 617
9
votes
1
answer
1k
views
Distributions on product spaces
I hope this is suitable to MO.
Question. Let $X$ and $Y$ be two open sets in $\mathbb R^n$ and $\mathbb R^m$, respectively. In what sense can we consider $\mathcal{D}^{\prime} \left(X\times Y\right)$ ...
- 33.8k
3
votes
0
answers
498
views
PDES - from Vector fields whose inner product with their vector Laplacian equals norm of the vector field
Let $g(x_{1},........,x_{n}) = \sum_{i=1}^{n}g_{i}(x_{1},\cdots,x_{n})e_{i}$ be a function in $\mathbb{C}^n$ ($e_{i}$ are the standard bases).
Let $\nabla^{2}$ be the vector Laplacian. Let $<\cdot,...
- 800
1
vote
2
answers
177
views
Restriction on the coefficients for an operator in the free group factor $ L(\mathbb{F}_2) $
Let $\mathbb{F}_2$ denotes the free group generated by a,b, denote this group by $G$. Then consider the von Neumann algebra $L(G)$ generated by the family
$\{L_{x_g} : g \in G\}$, here, with $g \in G$...
- 1,528
2
votes
1
answer
695
views
Is the F.T of $\operatorname{Span}(\mathscr S(\mathbb R^2)\otimes\mathscr D(D))$, $D\subset\mathbb R^2$ dense in $L^2(\mathbb R^4)$?
Let $K$ be the real vector space generated by elements $f$ in $\mathscr S(\mathbb R^2,\mathbb R)\otimes\mathscr D(D, \mathbb R)$, where $D$ is any bounded subset of $\mathbb R^2$. Let $\hat K$ be the ...
- 417
5
votes
3
answers
4k
views
Under what conditions a bounded linear map can be extended ?
I have two questions after reading the Hahn-Banach theorem from Conway's book ( I have googled to know the answer but I have not found any result yet. Also I am not sure that whether my questions have ...
- 281
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
1
vote
2
answers
2k
views
Why is the output of an LTI system the convolution of the input funtion and the impulse response?
I am looking at the description of LTI systems in the time domain.
Intuitively, I'd have guessed it would be the composition of the input function and some "system function".
$$ y(t) = f(x(t)) = (f\...
- 113
23
votes
4
answers
2k
views
Are almost commuting hermitian matrices close to commuting matrices (in the 2-norm)?
I consider on $M_n(\mathbb C)$ the normalized $2$-norm, i.e. the norm given by $\|A\|_2 = \sqrt{\mathrm{Tr}(A^* A)/n}$.
My question is whether a $k$-uple of hermitian matrices that are almost ...
- 9,486
4
votes
1
answer
783
views
Dependence of norm of extension map on Sobolev spaces and $(\epsilon,\delta)$ domains
Let $D\subset \mathbb{R}^n$ be a bounded domain.
An extension map is $E_D: W^{p,k}(D)\to W^{p,k}(\mathbb{R}^n)$ satisfying:
...
- 175
0
votes
0
answers
238
views
Square variation norm and non-negative, non-increasing sequences
I am trying to understand the properties of square variation, namely, the possibility of preserving it under certain operations. I am following Albiac & Kalton's book: Let $J$ stand for the usual ...
0
votes
1
answer
2k
views
What is the orthonormal basis for the Bergman space on the disk?
[EDIT by YC: the original question's title asked about a basis for the Hardy space on the disk. It is clear from the actual question that what was meant was the Bergman space.]
In arXiv:0310.5297, ...
- 22.8k
1
vote
1
answer
636
views
Does anyone know an example of non-separable $L^1$ of a probability space?
It is easy to give examples of non-separable $L^p$ spaces by considering a measure on a big space. If one adds the condition that the space has to have total measure 1, the problem is not as easy.
...
- 96
9
votes
3
answers
3k
views
The classical Krein-Rutman theorem
The classical Krein-Rutman theorem states that any positive compact linear endomorphism $T:X \to X$ on a Banach space $X$ with positive spectral radius $r(T)$ has an eigenvalue $r(T)$ with a positive ...
- 1,202
2
votes
2
answers
2k
views
Separable quotients of non-separable Banach spaces?
I am reading the Functional Analysis book of Conway, one question from the book is find a closed subspace M of $l^{\infty}=l^{\infty}(\mathbb{N})$ with the property that $l^{\infty}/M$ is separable. I ...
- 281
-2
votes
1
answer
3k
views
Separability of continuous functions with compact support [closed]
Hi,
is the space $C_0(\mathbb{R}^m)$, $m \in \mathbb{N}$ of continuous functions with compact support separable? If yes: where can I find a proof for that?
Please note: this is not a duplicate of ...
- 9
1
vote
1
answer
342
views
Compact sets in TVS
Let $K$ be a compact subset of a Hausdorff topological vector space. Is it true that
$\bigcap_{n\in \mathbb{N}}\frac{1}{n}K$ is either empty of or is a set consisting of the origin only?
- 11
2
votes
2
answers
1k
views
Characterization of Weakly measurable functions
I wonder if we can characterize weak measurability of a function taking values in a Banach space using sequence of step functions (functions that have finite range) just like how we define strong ...
- 23
2
votes
2
answers
667
views
Computing Bochner integrals with values in L^p-spaces by Lebesgue integrals?
Let $f: \mathbb{R}^n \to L^2(\mathbb{R}^d) $ be a Bochner-integrable function (all measures are the Lebesgue measure). Does then $ \int_{\mathbb{R}^n} f(x) d\lambda^n (y) = \int_{\mathbb{R}^n} f(x)(y) ...
- 403
2
votes
1
answer
208
views
Expanding Measurable Sets
Let $S,T \subset \mathbb{R}^n$ be measurable sets, and suppose that there exists a measurable bijection $f\colon S\to T$ so that
$$
\|f(x)-f(y)\| \;\geq\; \|x-y\|
$$
for all $x,y \in S$. Does it ...
- 8,493