Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
5,467 questions
1
vote
0answers
33 views
Approximation of functions by tensor products
Given a function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$, can we find a sequence of functions $f_n$ of the form $f_n(x,y)=\sum_{i=1}^ng_i(x)h_i(...
2
votes
0answers
47 views
Approximation of functions in $L^p(R^d;L^\infty)$
Assume that the function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$. Can we find a class of functions $f_n\in C_b^2(R^d;L^\infty(B_R))$ such that
$$...
3
votes
1answer
48 views
Question on relation between a parabolic sobolev space and a sobolev bochner space
For parabolic sobolev spaces I follow the following definition:
According to this definition, we have that $W^{1,1,2}(I \times \Omega)=L^2(I; W^{1,2}(\Omega)) \cap W^{1,2}(I; W^{-1,2}(\Omega))$
...
3
votes
0answers
66 views
Tensor product of compact operators on Banach modules
Let $A$ and $B$ be Banach algebras. Consider a right Banach $A$-module, $E$, and a right Banach $B$-module, $F$, as well as a Banach algebra morphism $\pi\colon A\to\mathcal L_B(F)$ into the bounded $...
3
votes
1answer
262 views
Where to find the proof of this property?
I am doing some exercises in the analytic and there is a problem as following:
``Let $\{f_n\}_{n \in \mathbb{ N}}$'' to be a positive sequence such that:
$\sum\limits_{n=1}^{+\infty} f_n = 1$.
$\...
-2
votes
0answers
48 views
What is the relationship (mapping) from a reciprocal function 1/r to a exponential function exp(-r)? [on hold]
The mathematical problem:
Consider the mapping from $r$ to $u$.
for large $r$ the theory suggests a formulation like $u=a_1 e^{-a_2 r}$, which means that the function decay exponentially.
for ...
1
vote
0answers
96 views
Affine algebraic variety as a set of common zeroes of holomorphic functions on ${\mathbb C}^n$
Let $V$ be an affine algebraic variety in ${\mathbb C}^n$, i.e. a set of common zeroes of some set $S$ of polynomials on ${\mathbb C}^n$:
$$
V=\{z\in {\mathbb C}^n:\ \forall p\in S\quad p(z)=0\}.
$$
...
3
votes
1answer
61 views
Mapping inclusion theorem for the numerical range
We denote the numerical range of a complex square matrix $A \in \mathbb{C}^{n\times n}$ by $W(A)$.
Let $A \in \mathbb{C}^{n\times n}$ and let $f: \mathbb{C} \to \mathbb{C}$ be, say, an entire ...
4
votes
1answer
82 views
Complemented subspaces of Lorentz sequence spaces?
Let $d(\textbf{w},p)$, $1\leq p<\infty$, denote the Lorentz sequence space, where $\textbf{w}=(w_n)_{n=1}^\infty\in c_0\setminus\ell_1$ is a normalized decreasing weight.
Is there very much known ...
3
votes
0answers
64 views
Dual Lorentz spaces
MO seems the perfect place to ask for the following question. Denote the Lorentz spaces on an arbitrary measure space $(E,\mu)$ by $L^{p,q}=L^{p,q}(E,\mu)$, and by $p'$ the conjugate index of $p$.
...
5
votes
1answer
190 views
Quasinilpotent , non-compact operators
If $X$ is a separable Banach space and $(\epsilon_n)\downarrow 0$, can we find a quasinilpotent, non-compact operator on $X$ such that $||T^n||^{1/n}<\epsilon_n$ for all $n$? I suspect the answer ...
3
votes
3answers
140 views
Existence of solution to linear fractional equation
We consider the equation
$$ \sum_{j=1}^n \frac{\lambda_j}{x-x_j} =i$$
where $\lambda_j>0$ and $x_j$ are real distinct numbers.
I want to show that if $\lambda_k$ is small compared to the ...
0
votes
0answers
68 views
Duality mapping of a the space of continuous functions? [closed]
The duality map $J$ from a Banach space $Y$ to its dual $Y^{*}$ is the multi-valued operator defined by: $J(y)=\{\phi\in Y^{*}:\, \left< y,\phi\right >=\Vert y\Vert^{2}=\Vert \phi \Vert^{2}\},...
5
votes
0answers
147 views
+100
Approximation of a compactly supported function by Gaussians
Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function whose support is a closed interval, e.g. $\text{supp}(f)=[a,b]$. Then $f$ can be approximated (e.g. in $L^2$) by a linear combination of Gaussian ...
2
votes
0answers
120 views
+100
Open Questions about Wasserstein Space and PDE
While working on my thesis, I encountered the idea of OMT and started reading some more (like Villani's book). In particular, I came across a PhD thesis by Martial Agueh. I thought it was interesting ...
6
votes
1answer
162 views
Do analytic functionals form a cosheaf?
Let $X$ be a complex-analytic manifold.
Consider the sheaf of holomorphic functions $\mathcal{O}_X$ as a sheaf with values in the category of locally convex vector spaces. For $U\subseteq X$ open, we ...
3
votes
0answers
75 views
Lower semicontinuous and convex envelope
L.Ambrosio, in paper [1] writes:
Let $g:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$ be a function (...)
for every $s\in\mathbb{R}$, $z\in\mathbb{R}^n$; we denote, with a slight abuse ...
3
votes
1answer
104 views
Equivalent notion of approximate differentiability
Is it true that the definition of approximate differentiability presented here of a function $f: \mathbb{R}^N \to \mathbb{R}$ is equivalent to the following one?
$$\lim_{r \to 0} \rlap{-}\!\!\int_{...
1
vote
1answer
79 views
Two PDE for one unknown?
Let $x \in (0,L)$, $t \in (0,T)$, and let $f_1 = f_1(x,t) \in \mathbb{R}$, $f_2 = f_2(x,t) \in \mathbb{R}$, $u^0 = u^0(x) \in \mathbb{R}$ and $g= g(t) \in \mathbb{R}$ be continuous functions.
My ...
3
votes
0answers
59 views
Functional characterization of local correlation matrices?
Definition: A matrix $C\in\mathbb R^{m\times n}$ is local correlation matrix iff there exists real random variables $x_1,\dots,x_m,y_1,\dots,y_n$ defined on a common probability space which takes ...
6
votes
2answers
272 views
Properties of heat equation
** I simplified the question: **
On bounded domains, the maximum principle implies that the solution to the heat equation is (strictly) positive, if the initial and boundary data is positive.
I ...
0
votes
0answers
70 views
Measurability of the heat semigroup in $L^\infty$
Let $S(t)$ be the $C_0$-semigroup generated by the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega)$, where $\Omega$ is a bounded open subset of $R^n$.
It is known that $S(t)$ ...
0
votes
0answers
31 views
Prove that the Bynum space $\ell ^{p, \infty}$ is a uniformly not square space
In the book Measures of Noncompactness in Metric Fixed Point Theory, the authors prove that $\varepsilon_{0}(\ell ^{p, \infty}) = 1$, where $\ell ^{p, \infty} = (\ell ^{p}, \Vert \cdot \Vert _{p, \...
3
votes
0answers
131 views
Largest weak(-like) topology with respect to which continuous functions are dense in the space of Borel functions
Let $X$ denote the space of bounded Borel functions $f\colon [0,1] \to \mathbb{R}$. Let $M$ denote the space of finite Borel measures on $[0,1]$. What is the largest family $F \subset M$ such that for ...
6
votes
1answer
186 views
Compactness of set of indicator functions
Let $\chi_A(x)$ denote an indicator function on $A\subset [0,1]$. Consider the set
$$K=\{\chi_A(x): \text{ A is Lebesgue measurable in }[0,1]\}.$$
Is this set compact in $L^\infty(0,1)$ with respect ...
1
vote
0answers
47 views
About a class of expectations
Consider being given a $n-$dimensional random vector with a distribution ${\cal D}$, vectors $a \in \mathbb{R}^k$, $\{ b_i \in \mathbb{R}^n \}_{i=1}^k$ and non-linear Lipschitz functions, $f_1,f_2 : \...
2
votes
1answer
182 views
Can the characteristic function of a Borel set be approached by a sequence of continuous function through a certain convergence in $L^\infty$?
Want to find $f_n$ a sequence of continuous functions, so that for all Borel regular measure $\mu$, we have
$\int f_n d\mu\rightarrow\int \chi_\Delta d\mu$, as $n$ goes to infinity, where $\Delta$ ...
0
votes
0answers
34 views
Is every pair of closed linear subspaces boundedly regular?
In the context of the development of the spectral theory, J. v. Neumann showed that for every pair of closed linear subspaces $M,N$ of a Hilbert space the iterations $(P_M P_N)^nx$, where $P_M$ and $...
-1
votes
0answers
31 views
Function classes with high Rademacher complexity
My question is two fold,
Is there any general understanding of what makes a function class have high Rademacher complexity? (Sudakov minoration would say that one sufficient condition for a class of ...
4
votes
1answer
121 views
Brascamp-Lieb inequalities on the sphere
In the paper [CLL], Carlen, Lieb, and Loss demonstrate a version of the Young inequality on the sphere $S^{N-1}$ in $\mathbb{R}^N$. For positive functions $f_j$ on $[-1,1]$, the following bound holds:...
2
votes
1answer
65 views
Existence of a bounded operator which satisfies the discrete product rule
Is there a bounded self-adjoint operator $H$ acting on $\ell^2(\mathbb{Z})$ such that for all sequences $u,v\in \ell^2(\mathbb{Z})$
$$ H(uv)=(Hu)v+u(Hv)$$
where uv is the pointwise product.
This is ...
3
votes
1answer
106 views
Ultraproduct of non-commuative $L^p$-spaces
Let $1<p<\infty.$ Let $I$ be a non-empty set and $\mathcal{U}$ be an ultrafilter over $I.$ Let $M_i$ be von Neumann algebras equipped with normal faithful semifinite traces $\tau_i,$ $i\in I.$ ...
-1
votes
0answers
51 views
How to prove the binary function uniformly boundary?
Assume that $a_1<1$, $a_3<1,$ $a_1+a_2+a_5>1$, $a_3+a_4+a_5>1,$ $a_1+a_2+a_3+a_4+a_5>2.$ For $x,y>0,$ define a fucntion
$$H(u,v)=\frac{u^{\frac{1}{2}}\int_0^{\infty}\int_0^{\infty}\...
3
votes
0answers
60 views
Domain of the Generator of a Bessel process
Consider the Bessel Process of index $\nu\in (-1,0)$, or dimension $\delta=2\nu-1$
\begin{align}
\rho_{t}=x+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{\rho_{s}}\,ds+W_{t}
\end{align}
where $(W_{t})_{t\geq ...
8
votes
1answer
245 views
Laplacian spectrum asymptotics in neck stretching
Let $M$ be a compact Riemannian manifold. Let $S \subset M$ be a smooth hypersurface separating $M$ into two components. Let $g_T$ be a family of Riemannian metric obtained by stretching along $S$, i....
1
vote
0answers
34 views
The significant role of dual frames in the progress of Frame theory
For a given frame $\{\zeta_i\}_{i=1}^\infty$, any Bessel sequence $\{\eta_i\}_{i=1}^\infty$ satisfying in the following identity for every $\xi\in H$
$$\xi=\sum_{i=1}^\infty \langle \xi, \eta_i\...
2
votes
1answer
124 views
$L^{2}$ Betti number
Let $\tilde{X}$ be a non-compact oriented, Riemannian manifold adimits a smooth metric $\tilde{g}$ on which a discrete group $\Gamma$ of orientation-preserving isometrics acts freely so that the ...
2
votes
1answer
77 views
Measure on union of measure spaces and on quotient space
There are two questions about measures bothered me a lot.
Given a set X and a countable covering ${U_i}$ of $X$. Suppose that for each i, there is a measure $m_i$ on $U_i$. Is there a very general ...
1
vote
1answer
56 views
Convergence of probability density function
There are various kinds of (convergence of random variables) but I have never read about convergence of density functions.
Let $X_1, X_2, \dots, X$ be random variables $\Omega \to \mathbb{R}$ and $...
17
votes
2answers
701 views
When is $\sum_{n\in\mathbb Z} f(x+n)$ constant?
A recently asked question (linked here) deals with the remarkable identity
$$ \sum_{n\in\mathbb Z} \mathrm{sinc}(n+x)= \pi,\quad x\in\mathbb R, $$
where $\mathrm{sinc}(x)=\sin(x)/x$.
It is easy ...
3
votes
2answers
138 views
Ultraweak topology in abelian von Neumann algebras
Let $A$ be an abelian von Neumann algebra acting on the (not necessarily separable) Hilbert space $\mathcal{H}$ (with identity $I$). From the Gelfand-Neumark theorem, there is a compact Hausdorff ...
0
votes
0answers
58 views
Estimate of $L^1$-norm of semilinear elliptic inequality
Let $\Omega$ be $\mathbb R^n$ or a complete non-compact manifold, we consider
$$\Delta u+f\cdot u+u^2\leq0,$$
where $\Delta$ denotes $-\sum^n_{i=1}\partial^2_{x_i}$ and $f$ is a $C^2$ function such ...
6
votes
0answers
124 views
Can we approximate any eigenvalue of an infinite matrix via eigenvalues of some sequence of submatrices which approximates the matrix?
Let $T:\ell^2\to\ell^2$ be a compact linear operator. Let $[T]=(a_{i,j})_{i,j=1}^{\infty}$ be the representing infinite matrix of $T$ with respect to the canonical base. Let $T_n$ be the finite rank ...
2
votes
1answer
78 views
Discrete dynamical system and bound on norm
Let $z \in \mathbb R\backslash \left\{2 \right\}$ then I would like to understand the following:
Consider the dynamical system with $x_i \in \mathbb C^2:$
$$ x_{i} = \left(\begin{matrix} z &&...
2
votes
0answers
70 views
1D Schrödinger Equation with Measure-Valued Coefficients
I've been looking at one of the simplest systems I can think of: a one-dimensional infinite square well on $[0,1]$ with Hamiltonian given by the following:
$$\hat{H}=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}...
4
votes
0answers
57 views
Perturbation theory compact operator
Let $K$ be a compact self-adjoint operator on a Hilbert space $H$ such that for some normalized $x \in H$ and $\lambda \in \mathbb C:$
$\Vert Kx-\lambda x \Vert \le \varepsilon.$
It is well-known ...
1
vote
0answers
46 views
Adjoint for a non-densely defined unbounded operator on a Hilbert space
Let $\mathbf{H}$ be a Hilbert space, and $D$ an unbounded densely-defined operator on $\mathbf{H}$. As is well-known, every such operator admits an adjoint, with domain possibly different from that ...
2
votes
0answers
64 views
Off-diagonal estimates for Poisson kernels on manifolds
Let $(M,g)$ be a complete Riemannian manifold, $\Delta$ its Laplace-Beltrami operator and $T_t = (e^{t \Delta})_{t \geq 0}$ the associated heat semigroup. We can define the subordinated Poisson ...
1
vote
0answers
162 views
One question about Schrodinger Semigroups-(B. Simon)
This question comes from the paper: B. Simon, Schrodinger Semigroups, Bull. A.M.S., (1982) Vol. 7 (3).
On the Theorem C.3.4(subsolution estimate) of the paper, it says that: Let $Hu=Eu$ and $u\in L^...
0
votes
0answers
92 views
Compact operator
Let $k:[0,1]^2 \to [0,1]$ be a measurable function. Define $K:L^2([0,1])\to L^2([0,1])$ to be the operator:
$$
(Kf)(x) = \int_0^1\int_0^1 f(z) k(x,y) \mathbf{1}_{x\leq z\leq y} \ \mathrm{d}z \mathrm{d}...
