Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
3,436 questions with no upvoted or accepted answers
1
vote
0
answers
88
views
Reference request: PDE of the form $(\Delta - |u|^2)f = F(u)$
I am interested in equations of the form
$$(\Delta -|u|^2)f = F(u)$$
where $F$ depends on $u$ and preferably on its derivative, too. $u$ is supposed to be given and $f$ the unknown. More precisely I ...
- 469
1
vote
0
answers
74
views
Good source for Jordan Fréchet algebras
Is there any good source for Jordan Fréchet (or more generally, Jordan locally convex) algebras?
I'm looking for something on the level similar to the level of the book "Banach and Locally Convex ...
1
vote
0
answers
442
views
Stein's extension operator for fractional Sobolev spaces
In his book Singular Integrals and Differentiability Properties of Functions,
Stein constructs an extension operator $\mathcal{E}:W^{m,p}(\Omega)\rightarrow
W^{m,p}(\mathbb{R}^{N})$, $m\in\mathbb{N}$, ...
- 411
1
vote
0
answers
81
views
Relation between minimizer of regularized risk & risk in statistical learning theory
In supervised machine learning, we typically take a Risk Minimization (RM) point of view when formulating a problem. So, what we typically solve for is the following:
$$ R^L(h) = \underset{h\in\...
- 11
1
vote
0
answers
91
views
A bilinear estimates involving critical Sobolev norms
Given $q>1$, consider the critical Sobolev space $W^{n/q,q}(\mathbb{R}^n)$, which fails to embed in $L^{\infty}(\mathbb{R}^n)$. I'm wondering if we can recover some critical estimate by considering ...
- 943
1
vote
0
answers
203
views
Uniformly local Sobolev spaces and interpolation
Let $d\in\mathbb{N}^+$, $s\geq 0$, and consider the uniformly local Sobolev space
$$H^s_{u,loc}(\mathbb{R}^d):=\{f\in H^s_{loc}(\mathbb{R}^d)\,s.t.\,\|f\|_{H^s_{u,loc}}:=\sup_{x\in \mathbb{R}^d} \|f\|...
- 943
1
vote
0
answers
66
views
Well-posedness of hyperbolic system with constant coefficients in finite domains
I'm studying the PDE
$$
\frac{\partial u}{\partial t} + A_x\frac{\partial u}{\partial x} + A_y\frac{\partial u}{\partial y} + A_z\frac{\partial u}{\partial z} = 0
$$
with $A_x, A_y, A_z$ being ...
- 11
1
vote
0
answers
67
views
Approximate identities on the unit disk and going beyond a power series' radius of convergence
Let $\left\{ a_{n}\right\} _{n\geq0}$ be a bounded sequence of complex numbers, so that the power series $f\left(z\right)=\sum_{n=0}^{\infty}a_{n}z^{n}$ has a radius of convergence $\geq1$. ...
- 1,284
1
vote
0
answers
75
views
Basic question about convergence of top and penultimate eigenvalues of a sequence of operators
$\newcommand{\norm}[1]{\|#1\|}$ $\newcommand{\ab}[1]{\langle #1\rangle}$ $\newcommand{\C}{\mathbf C}$
Questions
Let $I$ be the unit interval.
Let $H=L^2(I)$ and $T:H\to H$ be a compact self-adjoint ...
- 113
1
vote
0
answers
67
views
Order of ultradistribution
I know that the order of any distribution of compact support is finite. Is this true in the case of ultra distribution of compact support ( dual of Denjoy-Carleman space)?
- 75
1
vote
0
answers
84
views
A Riemann Hilbert problem on the unit square
Let $p_0=0$, $p_1=1$, $p_2=1+i$ and $p_3=i$ be the four vertices of a square $Q$ on the complex plane $\mathbb C$.
Let $f \in C^{\infty}_c((0,1))$ and consider the following Riemann-Hilbert problem on ...
- 4,115
1
vote
0
answers
73
views
The $w^{*}$-convergent sequences and the Mackey topology on $X^{*}$
Let $X$ be a Banach space. Recall that the Mackey topology $\mu(X^{*},X)$ on $X^{*}$ is the topology of uniform convergence on weakly compact subsets of $X$. Let $(x^{*}_{n})_{n}$ be a sequence in $X^{...
- 3,343
1
vote
0
answers
56
views
Moduli of continuity and Wasserstein differentiability of functions between measures
Let $X=\mathbb{R}^n$; I am also interested in the general case $X$ is a metric space but for simplicity let's focus on Euclidean space. Let $\mathcal{P}(X)$ denote the space of Borel probability ...
- 11
1
vote
0
answers
72
views
Multivarate "RKHS" Examples
I've been reading about RKHSs and Hilbert spaces of functions these days a bit these days and I haven't yet come across an example of a hilbert space $H$ whose elements are all functions $f:\mathbb{R}^...
- 5,405
1
vote
0
answers
67
views
Toeplitz operators for other measures then Lebegue
In the standard setting there is a lot known about Toeplitz operators i.e that the compression of a multiplication operator restricted to the Hardy space.
Are there any results when one has a ...
- 243
1
vote
0
answers
139
views
Property $(\mathcal{L}(\phi),\phi)\geq 0$ about a operator $\mathcal{L}$
Consider the operator $\mathcal{L} : H^2(\mathbb{T}_L) \subset L^2(\mathbb{T}_L) \longrightarrow L^2(\mathbb{T}_L)$ given by
$$\mathcal{L} = -\omega \partial_x^2+3\varphi^2-1,$$
that is
$$\mathcal{L}(...
- 205
1
vote
0
answers
79
views
Conditions on triangle inequality for integral kernel
Consider $\int_RK(x,y)f(y)dy$, where $K(x,y) \in M_+(R^2)$.
Let $L(t,s)$ be an iterated rearrangement of $K$. Let also $$
A(t,v)=\int_0^{1/v}L(1/t,s)ds,
$$
which is decreasing with $v$ and ...
- 343
1
vote
0
answers
26
views
On some bounds on two constants concerning the disconnectedness of the spectra of small perturbations of operators
Let $H$ be a separable, infinite dimensional, complex Hilbert space. In the book:
Jiang, C. L.; Wang, Z. Y. (1998). Strongly Irreducible Operators on Hilbert
Space. CRC press
above the statement of ...
- 1,175
1
vote
0
answers
181
views
Eigenvalues of product of operators
Let $A,B$ be two Trace class operators with spectral decomposition $\sum_{j\geq 1} \lambda_j \phi_j(\cdot)\otimes \phi_j(\cdot)$ and $\sum_{j\geq 1} \gamma_j \psi_j(\cdot)\otimes \psi_j(\cdot)$ ...
- 519
1
vote
1
answer
294
views
Regarding subspace generated by the polynomial multiples of outer functions
Let $\mathbb{D}$ and $\mathbb{T}$ denote the open unit disk and unit circle in $\mathbb{C}$ respectively. We write $Hol(\mathbb{D})$ for the space of all holomorphic functions on $\mathbb{D}.$ The ...
- 31
1
vote
0
answers
115
views
Algorithm/iterative procedure for constructing hypercyclic vectors?
Let $B$ be a separable Banach space and let $L:B\rightarrow B$ be a hypercyclic operator; here I use the definition of hypercyclicity given implicitly by Birkhoff's Transitivity Theorem: continuous ...
- 5,405
1
vote
0
answers
353
views
Eigenvalues of convolution matrices
Let $h: \mathbb{R}\to \mathbb{R}$ be a smooth function. Fix $0\leq s_1\leq \cdots \leq s_m\leq 1$ and $0\leq t_1\leq \cdots \leq t_n\leq 1$. Construct $A\in \mathbb{R}^{m\times n}$ by letting $A_{i,j}:...
- 315
1
vote
0
answers
148
views
Using Paley-Wiener Theorem to prove the decay of $G(x-y)$
This question is related to my previous one, where I was looking for some help to prove the decay of the lattice Green function:
\begin{eqnarray}
G(x-y) = \int_{[-\pi,\pi]^{d}}\frac{d^{d}k}{(2\pi)^{d}}...
- 1,305
1
vote
0
answers
74
views
Empty Weyl/Fredholm spectrum of an operator on an infinite dimensional Banach space
Let $X$ be a complex infinite dimensional Banach space, and let $T \in B(X)$ be nonscalar. The Fredholm spectrum of $T$ is defined by:
$$ \sigma_{\Phi} (T) := \lbrace \lambda \in \mathbb{C} : T- \...
- 1,175
1
vote
0
answers
40
views
Minimax theorems in nonconvex setting
Let $X$ be a topological space, $Z$ be a compact convex subset of $\mathbb R^m$, and let $f:X \times Z \to \mathbb R$ be a continuous function (w.r.t the product topology on $X \times Z$).
Question. ...
- 6,853
1
vote
0
answers
126
views
Does my functional satisfy the Palais Smale condition?
Consider the functional
$$ I(u)=\frac{1}{2} \int_\Omega |\nabla u|^2\ dx + \frac{1}{4} \int_\Omega (1-|u|^2)^2 \ dx - \frac{c}{2} \int_\Omega \langle i\partial_1 u , u\rangle ,$$
where $u:\mathbb{R}^2 ...
- 383
1
vote
0
answers
277
views
Why the name `Lipschitz-Free Banach spaces'?
There are many names for the same objects that is known as the Arens--Eells spaces, transportation cost spaces, free Banach spaces over a (pointed) metric space, and Lipschitz-free Banach spaces.
The ...
1
vote
1
answer
379
views
Creating an inverse system which "stratifies density"
Setting:
Let $X'$ be a dense subset of an infinite-dimensional Fréchet space $X$ and suppose that $(X_n')_{n \in \mathbb{N}}$ is a nested sequence of non-empty subsets of $X'$ satisfying
$$
\bigcup_{n ...
- 5,405
1
vote
0
answers
139
views
Any reflexive space has the property of Banach-Saks?
We say that a Banach space $(X,\|.\|)$ has the Banach-Saks property if every bounded sequence $(x_m)_m$ in $X$ admits a subsequence $(x_{m_n})_n$ which converges in the sense of Cesàro, that is, there ...
- 115
1
vote
0
answers
66
views
Outer-regular product of $\tau$-additive measures
Due to the deficiencies of the simple product measure defined on measurable rectangles, there have been many different constructions of product measures in more specialized circumstances.
Originally, ...
- 3,816
1
vote
0
answers
54
views
Constrain representation of tempered distribution
This is a follow-up to this question.
Let $T$ be a tempered distribution on $\mathbb{R}^d$.
Then there is a multiindex $\alpha \in \mathbb{N}_0^d$, an $n \in \mathbb{N}_0$ and a bounded continuous ...
- 661
1
vote
0
answers
153
views
Maximal ideal space of $\ell^\infty(A)$, $A$ a commutative unital Banach algbera
Let $A$ be a commutative unital complex Banach algebra with norm $\|\cdot\|_A$, and let $\ell^\infty(A)$ denote all bounded sequences $(a_n)_{n\in \mathbb{N}}$ with $a_n\in A$, $n\in \mathbb{N}$, with ...
- 21
1
vote
0
answers
236
views
Fredholmness of elliptic operator on Hölder spaces
Let $(M,g)$ be a smooth oriented closed Riemannian manifold, $E\to M$ a smooth vector bundle, and $C^{k,\alpha}(E)$ the Banach space of sections of $E$ that are $k$-times differentiable (with respect ...
- 163
1
vote
0
answers
36
views
Does a total preorder on lotteries that preserves countable mixtures preserve arbitrary mixtures?
Let $X$ be a countable set. A lottery on $X$ is a function $\lambda: X \to [0,1]$ such that $\sum_x \lambda(x) = 1$. Let $\Delta X$ be the set of lotteries on $X$.
A total preorder $\preceq$ on $\...
- 869
1
vote
0
answers
30
views
Hypercylic operators with sets of hypercyclic vectors almost covering the space
Let $\{T_i\}_{i \in I}$ be a family of hypercylic operators on a separable Banach space $X$. From the transitivity theorem, we know that $HC(T_i)$, the set of vectors $x \in X$ with $\{T_i^n(x):n \in ...
- 5,405
1
vote
0
answers
55
views
$ \{x\in X:h(x)\leq r\} $ is sequentially compact subset of $X$?
Let $(X,\|.\|)$ be a reflexive Banach space and $(D,\|.\|)$ be a Suslin subspace of $X$ such that $D$ is weakly closed subset of $X$.
Take $h:X\to [0,+\infty]$ such that $h(x)=\|x\|$ if $x\in D$ and ...
- 117
1
vote
0
answers
106
views
strict convexity of the Legendre-Fenchel transform
Let $d$ be a positive integer.
Let $L:\mathbb{R}^d\to\mathbb{R}$ be a differentiable function with continuous derivatives.
Assume that the Legendre-Fenchel transform of $L$ exists everywhere, is ...
- 11
1
vote
0
answers
126
views
Continuity of Helmholtz-Hodge projection in $H^1(\Omega)$
Let $\Omega \subset \mathbb{R}^d$ (for simplicity $d = 2$ or $3$) be a bounded Lipschitz domain. For any vector-valued function $\mathbf{f} \in \mathbf{L}^2(\Omega):= \left ( L^2(\Omega) \right )^d$, ...
- 23
1
vote
0
answers
175
views
Compute Frobenius inner product of two tensor-trains in terms of tensor contractions
Let $p\in\mathbb N$, $n\in\mathbb N^p$ and identify the Hilbert space tensor $\bigotimes_{k=1}^p\mathbb R^{n_k}$ with $\mathbb R^{n_1\times\cdots\times n_p}$ (equipped with the Euclidean inner product)...
- 167
1
vote
0
answers
314
views
History of microlocal analysis
Was the study of pseudo-differential operators the basis for the birth of microlocal analysis? I'm trying to find out how this branch of analysis was developed...
- 97
1
vote
0
answers
231
views
Dual space of mean-free Sobolev space
I am considering the space $V:=\{v \in H^1(\Omega): \int_\Omega v = 0\}$ of mean free functions. What is the dual space of this space? Is the dual space given by $D:= \{f \in (H^1(\Omega))^*: \langle ...
- 11
1
vote
0
answers
48
views
Can we say that $f$ admits a $m(X,X^*)$-continuous extension to $X$?
Let $X$ be a Banach space equipped with the Mackey topology $m(X,X^*)$. We suppose that $\big(X,m(X,X^*)\big)$ is separable space. Let $H$ be a countable, $m(X,X^*)$-dense subset with $(H=-H)$.
Let $...
- 117
1
vote
0
answers
183
views
G-abelian systems
Let $(\mathfrak{A},\alpha,\phi)$ be a $C^*$-dynamical system made of a unital $C^*$-algebra, a $*$-automorphism and an extremal invariant (i.e. ergodic) state.
Consider the covariant GNS ...
1
vote
0
answers
55
views
Operational quantities characterizing upper semi-Fredholm operators
An operator $T:X\rightarrow Y$ is said to be upper semi-Fredholm if its range is closed and its kernel is finite-dimensional. M. Schechter (1972) introduced a quantity $$\nu(T):=\sup_{\operatorname{...
- 3,343
1
vote
0
answers
27
views
Show that a tensor-train is contained in a recursive sequence of subspaces
Let
$p\in\mathbb N$;
$n_k\in\mathbb N$ and $\left(e^{(k)}_1,\ldots,e^{(k)}_{n_k}\right)$ denote the standard basis of $\mathbb R^{n_k}$ for $k\in\{1,\ldots,p\}$;
$u\in\bigotimes_{k=1}^p\mathbb R^{n_k}...
- 167
1
vote
0
answers
97
views
Determining the behavior of a contraction mapping with undefined points
Label $X$ as the real interval $[0, a]$ where $a \in \mathbb{R}^+$, so that $\text{int}(X) = (0, a)$ labels the interior of $X$ and $\partial X$ labels the boundary of $X$. I have a function $f:\text{...
- 1,087
1
vote
0
answers
75
views
"Constructive" proof that compact sets $K\subseteq L_1(\mu\times\nu)$ are contained in products $S\widehat{\otimes} T$
A.Defant and K.Floret in chapter 7 of their Tensor Norms and Operator Ideals prove the equality
$$
L_1(\mu\times\nu)\cong L_1(\mu)\widehat{\otimes}L_1(\nu)
$$
for measures $\mu$ and $\nu$. At the same ...
- 7,426
1
vote
0
answers
148
views
Spectrum of Laplacian-like operator
Let $\kappa_1, \kappa_2>0$ be fixed.
Consider the unbounded operator $A: D(A) \rightarrow L^2(-1,1)\times\mathbb{R}$ defined by
$$
A\begin{bmatrix} y \\ h \end{bmatrix} = \begin{bmatrix} \...
- 309
1
vote
0
answers
203
views
Construction of weight function to satisfy condition on given functional
Consider the following function :
$$F(z) = \omega(z){\sin^2\left(\frac{c\Gamma(z)}{z}\right)}$$
Here, $\omega(z)$ is a weight we are going to consider
The following two conditions should meet for $\...
- 375
1
vote
0
answers
39
views
$C^2$-control using orthonormal frame on a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold. Let $E=(E_1,\dots,E_n)$ be an orthonormal frame for $M$.
So for $M$ itself we have a natural $C^k$-norm $\|f\|_{C^k_g(M)}:=\max\limits_{1\le m\le k}\sup\limits_{...
- 938