Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
3,435 questions with no upvoted or accepted answers
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Is every bijective analytic map bi-analytic?
Suppose that
$E$ and $F$ are complex Banach spaces and $U\subset E$ and $V\subset F$ are open subses.
$f\colon U\to V$ is analytic
$f\colon U\to V$ is bijective
Is $f$ bi-analytic? (i.e. is its ...
- 131
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110
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If $f_j\to f$ in $L^1(\Bbb R^n)$ then $Tf_j\to Tf$ in $L^{1,\infty}(\Bbb R^n)$
Let's define $A:=\{f\in L^1(\Bbb R^n)\cap L^2(\Bbb R^n)\;:\;f\;\mbox{has compact support}\}$. So $A$ is dense in $L^1(\Bbb R^n)$.
Given then $f\in L^1(\Bbb R^n)$; by density there exists $\{f_j\}_j\...
user76025
2
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0
answers
84
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Implicit function theorem and selfadjoint operators
Is there a version of the implicit function theorem in some space of functions from $[0,1]$ to a Hilbert space $H$ that contains as a special case the unique solvability of the initial-value problem $\...
- 3,873
2
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63
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How do spectrums interact with bi-Lipschitz maps?
If it makes things simple, we can just stick to bi-Lipschitz maps from $S^k \rightarrow \mathbb{R}^d$ (w.r.t geodesic distance on the sphere with the standard round metric and the $2-$norm on the ...
- 617
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86
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I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection
I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...
user74319
2
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350
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Dual space of vector fields with null divergence
Let us consider $\mathscr D(\mathbb R^3;\mathbb R^3)$ the space of smooth compactly supported vector fields in $\mathbb R^3$ and let us define
$\widetilde{\mathscr D}=\mathscr D(\mathbb R^3;\mathbb R^...
- 16.2k
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201
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Reflexive subspaces of dual spaces
If $X$ is a Banach space, is it true that $X^{*}$ must contain an infinite dimensional reflexive subspace? I know of Gowers' example of a Banach space not containing $c_0$, $l_1$, or reflexive, but I ...
- 1,361
2
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106
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Hilbert Transform and multiplier in $\mathbb{C}(X)$
I found myself trying to solve an equation of that kind :
$$
H f= R f,
$$
where $f$ has to be found in $L^2(\mathbb{R})$, $H$ is the Hilbert transform and $R$ is a rational function having no poles ...
- 3,425
2
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320
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Banach-Mazur distance from finite-dimensional subspaces of $\ell_p$ to the Hilbert space
I am reading a paper http://www.math.tamu.edu/~johnson/TF3.4.pdf by Bill Johnson and Andrzej Szankowski and having trouble grasping why
$d_n(Z_m) \leq d_n(\ell_{p_{m+1}} ) = n^{|p_{m+1}-2|}$ in the ...
2
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346
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When is the sum of complemented subspaces complemented?
Let $X$ be a Banach space.
Question. Suppose $X_1,...,X_n$ are complemented subspaces of $X$. When is the sum $X_1+...+X_n$ complemented? Further, suppose we know some projections $P_1,...,P_n$ onto $...
- 935
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149
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Question about continuity in the "complete Skorohod Topology"?
I am reading the book in progress of Timo Seppäläinen about the "Translation Invariant Exclusion Process"
https://www.math.wisc.edu/~seppalai/excl-book/ajo.pdf
In one of the exercises, exercise 8.9 ...
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210
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The functional equation $T(x\otimes y)=T(x)\otimes T(y)$ on certain $C^{*}$ algebras
Is there a name for the following property of a $C^{*}$ algebra $A$?
$$A \simeq A \otimes A\;\;\; \text{The spatial tensor product}$$
Example of this situation is $A=C(X)$ where $X$ is the ...
- 366
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463
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Conditions for continuity of non-simple eigenvectors
Here, https://math.stackexchange.com/a/1146455, it is noted that eigenprojections are continuous, but eigenvectors are not. Are there any conditions where the eigenvalues are not simple, but the ...
- 37
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1D inhomogeneous linear Schrodinger equation
I have the following problem:
$iu_t - u_{xx} = f$ on the interval $[0,L]$ with $u(0,t)=u(L,t)=0$ and $u(x,0)=0$. I can show that $\|u\|_{L^2(x,t)}$ (space-time) is controlled by the norm $\|f\|_{L^2(...
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166
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Getting an a priori energy estimate from PDE weak formulation
On a bounded domain $\Omega$, I have two functions $u$ and $v$ in $L^2(0,T;H^1(\Omega))\cap H^1(0,T;(H^1(\Omega))^*)$ satisfying
$$\frac{d}{dt}\int u^2 + c_1\int |\nabla u|^2 + n\int u^2 \leq n\int uv$...
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154
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The boundedness of an entire function along the imaginary axis
I am looking for an answer in the affirmative or the negative concerning the asymptotics of an entire function. The question, relatively simple to state, has proven very taxing to solve and requires ...
user78249
2
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68
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Topologies with the same convex closed sets
Let $\tau_1$ and $\tau_2$ be locally convex Hausdorff topologies on vector space $X$ such that $(X,\tau_1)^\ast = (X,\tau_2)^\ast$. It is well known that $(X,\tau_1)$ and $(X,\tau_2)$ have the same ...
- 105
2
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134
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Restricted weak type bound at the endpoint
We know that if we have an operator that is (restricted) weak type $(p,p)$ and (strong) type $(\infty,\infty)$ with norm 1, then it's also of strong type $(q,q)$ for all $p<q<\infty$ by the real ...
- 319
2
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43
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Specific type operators and basic sequences
Let $s$ be the space of rapidly decreasing sequences, i.e.
$s=\{\xi=(\xi_j)_j\colon\,\,\sup_j|\xi_j|j^n<\infty\,\,\text{for all}\,\,n\in\mathbb{N}\}$ and $s'$ its topological dual, i.e.
$s'=\{\eta=(...
- 375
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82
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Properties of a Sobolev bound
I am interested in computing
$$
A:=\inf_{f\in L^{2}(\mathbb{R}^3)}\frac{||K^{\frac{1}{4}}f||_2^2}{||f||_{\frac{5}{2}}^2}
$$
where $K:=-\Delta+1$. We call $f_c$ the function that saturates the bound.
...
- 193
2
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answers
135
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Extending continuous functions from $\partial X$ to $X\cup \partial X$
Consider a proper geodesic hyperbolic space $X$ (in the sense of Gromov). Let $\partial X$ be its Gromov boundary. Consider a complex-valued continuous function on the boundary $f\colon\partial X\to\...
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156
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The minimum value of a energy integral
Let $D \subset {\mathbb{R}^3}$ a simple connected open domain with volume $\int_{\bar D} {dV = 1} $. $\varphi :{\mathbb{R}^3} \to \mathbb{R}$ is ${C^1}$, $\varphi (\infty ) = 0
$ and
$${\nabla ^2}\...
- 273
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answers
166
views
Must rows of a transition matrix be distinct?
Is it true that for all continuous time Markov processes on a countable state space $S$, we have
all rows of the transition matrix $\mathbf{P}_t$ are distinct for all time $t\in[0,\infty)$ ?
This ...
- 165
2
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184
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Modify the jump set of $BV$ function
Let $u\in BV(\Omega)$ be a function of bounded variation where $\Omega\subset \mathbb R^N$ is open bounded with smooth boundary. We use $Du$ to denote the weak derivative of $u$. (So $Du$ is a Radon ...
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125
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When is an selfadjoint operatorvalued matrix with positive semidefinite diagonal elements positive semidefinite as well?
We have $p \in \mathbb{N}$ and $\mathcal{H}$ is a Hilbert space.
let's consider a matrix $\boldsymbol{\Gamma}_p := (C_{i-j})_{i,j=1, ..., p} \in \mathcal{S_H}^{p\times p}\!\!\,,$ that is a $p\times p$...
- 195
2
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168
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Interchanging integrals and continuous linear forms in RKHS
I am reading Reproducing kernel Hilbert spaces in probability and statistics by A Berlinet, C Thomas-Agnan.
In Chapter 5 INTEGRATION OF $\mathcal{H}$-VALUED RANDOM VARIABLES they write One of the ...
- 619
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110
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Biharmonic Equation in a Rectangle with Some Uncommon Boundary Conditions
I asked this question on Mathematics network but it didn't receive any answers. So I assume it is just beyond the classic things in PDEs and I decided to ask it here too.
Consider the following ...
- 271
2
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108
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Finite dimensional subrepresentations in a Fréchet space
The Lie group $G := SO(n, 1)$ acts on the hyperbolic space $\mathbb{H}^n$ by isometries. In particular, we have a representation of $G$ on $F := C^\infty(\mathbb{H}^n)$. I assume that $F$ is endowed ...
- 1,263
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266
views
Can we do better than zero padding of FFT?
My background is in signal processing, and never took any course related to functional analysis or even advanced algebra. But I have a strong conviction (may be wrong) that we may be do better then ...
- 495
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552
views
Normed space between $H^{0+}$ and $L^2$
Cosider a function $f\in L^2(\mathbb{R}^3)$ with consider the following condition.
$$\int_{\mathbb{R}^3}\frac{|\widehat{f}(x)|}{1+|x|^{3/2}}dx<+\infty \, .\qquad\mbox{(*)} \, $$
Of course if $f\in ...
- 943
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answers
93
views
Multiplier operators on anisotropic weighted $L^2$ spaces
Suppose $\mathcal{M}$ is a multiplier operator on $L^2(\mathbb{R})$, in the sense that, for any $u(x)\in L^2(\mathbb{R})$,
$\widehat{\mathcal{M}u}(k)=m(k)\hat{u}(k),$
where the scalar complex function ...
- 21
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answers
232
views
Kirillov orbit Method for Complex nilpotent groups
Let $G$ be a nilpotent simply connected real Lie group. From the classical work of Kirillov, it is well-known that the irreducible unitary representations of $G$ are in a canonical bijective ...
- 2,508
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votes
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828
views
Dirichlet-to-Neumann Map is selfadjoint
Let $\Omega$ be a compact, riemannian manifold with non-empty smooth boundary $\partial \Omega = \Gamma$.
For a smooth function $u \in C^\infty(\Gamma)$ we define the harmonic extension $\hat{u}$ as ...
- 501
2
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150
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Completion of $C_{0,rad}^{\infty}(\Omega)$ with respect to the norm $\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $
I have a question that it seems simple but I can not solve it.
Let $\Omega$ be the unit ball centered at zero in $\mathbb{R}^N$, $N>4$. Assume that $C_{0,rad}^{\infty}(\Omega)$ is the space of all ...
- 371
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234
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Concentration compactness on a compact setting
Consider a compact Riemannian manifold $M$ of dimension $n$ and a sequence of positive functions $\{\varphi_k\}_k \in C^\infty(M)$ such that $\{\varphi_k\}_k$ satisfy the basic concentration ...
- 1,407
2
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491
views
$L^\infty-L^1$ smoothing effect for the heat equation
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$.
Let $u \in L^2(0,T;V)$ be the weak solution of the heat equation
$$u_t - \Delta u = 0$$
$$u(0) = u_0$$
where $u_0$ is bounded initial data. Here ...
- 21
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69
views
Doubts regarding pre-compactness of bounded sequence of measure valued functions
Given:
$\phi(x,\lambda) : \Omega$X$\mathbb R \to \mathbb R^{n}$ be a Caratheodory vector such that for each $M \gt 0 $, $\alpha_{M}(x) = max_{|u| \leq M} |\phi(x,u)| \in L^{2}_{loc} (\Omega)$ .
...
- 121
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125
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Constant periodic Sobolev embedding
Dear mathoverflowers,
I would like to have a reference regarding the optimal constant in the Sobolev embedding
$$
\|u\|_{L^q}\leq C_{s,q}\|u\|_{\dot{H}^s},
$$
($H^s$ denotes the standard L^2 ...
- 843
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votes
0
answers
112
views
Is the Wave Function a "Smooth" Function of the Potential?
Consider the Schroedinger equation in a spherically symmetric system. In the unit system under which energy is measured in Hartree and length in Bohr radius $a_0$, the schroedinger equation can be ...
- 183
2
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answers
619
views
Laplace transform of a integral function of CIR/CEV process
The Cox–Ingersoll–Ross model (or CIR model) describes the evolution of interest rates. Constant elasticity of variance model (CEV) is a stochastic volatility model, which attempts to capture ...
- 323
2
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0
answers
114
views
What is $(L^2(M), H^1_0(M))_{\frac 12}$ on a smooth manifold with boundary?
Let $M$ be a smooth compact manifold. If $M$ is closed, we have that the interpolation space
$$(L^2(M), H^1(M))_{\frac 12}=H^{\frac 12}(M)$$
(see Taylor's book on PDE for example). Suppose $M$ has a ...
- 251
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69
views
How sensitive are the n-th step transition probabilities of the simple random walk to a small perturbation of an infinite graph?
Suppose that $G$ is an infinite, locally finite, connected graph.
Fix a vertex $o$ in the graph and for each $n$ and $x$ let $p(n,o,x)$ be the probability that a simple random walk (at each step a ...
- 4,304
2
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0
answers
244
views
A modification of Minty's trick?
I have the following result:
$$0 \leq \int_0^T (a(t)- |w(t)|)(b(t) - g^{-1}(|w(t)|))\quad\forall w \in L^2(0,T)$$
where $a$ and $b$ are both non-negative.
Does it follow that $b(t) = g^{-1}(a(t))$? ...
- 21
2
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0
answers
116
views
Factorization of the Fokker-Planck semigroup
I have posted this question on stackexchange over four month ago, but didn't get an answer: "In the classical theory of Markov processes, the Fokker-Planck semigroup $\{T_t:t\ge 0\}$ can be factored ...
2
votes
0
answers
136
views
Is $C^{\gamma}(\Sigma)$ dense in $C(\Sigma)$?
Consider $\{0,1\}$ with the discrete topology and $\Sigma=\{0,1\}^{\mathbb{N}}$ with the product topology. We know that this
product topology is generated by the metric
$$
d(x,y)=\sum_{n=1}^{\infty}\...
user11178
2
votes
0
answers
153
views
Factorization of linear bounded operators in Banach spaces
I am looking for a reference, if any, to the following statement: "Let $X$ and $Y$ be
Banach spaces, $A\in\mathcal{B}(X,X)$ be a linear bounded operator, and $B\in\mathcal{B}(X,Y)$ be surjective. Then,...
2
votes
0
answers
86
views
Regularity of $u$ in $u_t - \Delta \beta(t,u) = f$, can we get $u_t$ is a function?
I'm looking for reference discussing the regularity of the weak solution $u$ to the equation
$$u_t - \Delta \beta(t, u) = f$$
$$u(0) = u_0$$
where $\beta(t,\cdot)$ is a nonlinear function depending ...
- 51
2
votes
0
answers
139
views
Power-spectrum of quasi-periodic functions
From Scholarpedia:
Quasiperiodic oscillation is an oscillation that can be described by a quasiperiodic function, i.e., a function $F$ of real variable $t$ such that $$F(t)=f(\omega_{1}t,\dots,\...
- 297
2
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0
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148
views
Green's functions on linear subspaces and relations to boundary conditions
Consider the Laplacian $-\Delta$ on (in a suitable sense) twice differentiable functions subject to homogeneous Dirichlet boundary conditions $\mathscr{H}=\{f : f(0)=f(1)=0\}$. We can identify the ...
- 151
2
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214
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About optimizing a convex function on a hypercube
Given a real valued convex function $g$ on $[-1,1]^n$, let $f$ be the restriction of it on the hypercube $\{-1,1\}^n$. I want to find a vertex on the hypercube $\{-1,1\}^n$ on which either (1) $f$ ...
