All Questions
3,601 questions with no upvoted or accepted answers
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150
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smoothness of green's function in wave equation
I have a linear acoustic wave propagation originated from a monopole source, written as
\begin{align}
\mathcal{L}p(\mathbf{x},t) = S_m(\mathbf{x},t), \quad \mbox{in } \Omega
\end{align}
where the ...
0
votes
0
answers
81
views
Approximation property of Fréchet if range is restricted to an embedded Hilbert space
Let $W$ be a separable Fréchet space, and $H\subset W$ be a separable Hilbert space that is continuously embedded (equivalently, the topology of $H$ is stronger than the subspace topology generated by ...
- 320
0
votes
0
answers
201
views
Range of a trace preserving completely positive projection
I'm working in $M_n(\mathbb{C})$, the algebra of complex $n\times n$ matrices. I managed to build a completely positive, trace preserving, star preserving, projection $P$. That is
$$\text{Tr}(P(A)) = ...
- 515
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votes
0
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256
views
Explicit formula for Bergman kernel on the unit ball
On page 173 in Krantz's book "Explorations in Harmonic analysis" in the proof of Lemma 7.1.21 there is a part that I really don't understand. What I don't understand is why is $$\sum_{\alpha}\frac{z^{\...
- 325
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114
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Recontruction of the weak topolgy from the scalar product on a subset of a Hilbert Space
Let $M$ be a set a let $K:M\times M\to\mathbb{C}$ be a positive definite kernel. By a version of Moore-Aronszajn Theorem, there is a unique (up to the unitary euivalence) Hilbert Space $X$, and a map $...
- 5,529
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votes
0
answers
104
views
Must the Lebesgue measure of a $\rho$ - neighbourhood of an $(n-2)$ - dimensional set be at least $c\rho^2$?
The Lebesgue measure of a $\rho$-neighbourhood of a point in $\mathbb{R}^2$ is of course equal to $c\rho^2$. Similar such considerations in higher dimensions lead me to the following question:
Given ...
- 1,771
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0
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127
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Approximation Property: Characterization
Problem
Given a Banach space $E$. Denote compact sets by $\mathcal{C}$, compact operators by $\mathcal{C}(X,Y)$, and finite rank operators by $\mathcal{F}(X,Y)$.
Suppose it has the approximation ...
- 598
0
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0
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156
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Compact Approximation
This thread originated from MSE: Compact Approximation
This is meant as lemma for: Approximation Property
Given a Banach space $E$.
Denote compact domains by $\mathcal{C}$.
Denote compact ...
- 598
0
votes
0
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351
views
Existence of a complementary closed subspace extending a given subspace
Let $H$ be an infinite dimensional (separable) Hilbert space (or any infinite dimensional Banach space in which every closed subspace has a complementary subspace). Suppose that $X$ and $Y$ are closed ...
- 3,115
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117
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The functional $L(\varphi)=\int_0^{2\pi}\frac{\sqrt{1-\varphi^2-(\varphi')^2}}{1-\varphi^2}d\theta$
Consider the $2\pi$-periodic inner product space $L^2[0,2\pi]$.
Let $F\triangleq\{f\in L^2[0,2\pi]|f(\theta)>0,(f(\theta),\cos\theta)=(f(\theta),\sin\theta)=0\}$.
Let $G\triangleq\{\varphi\in L^2[...
- 139
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0
answers
302
views
Banach space of discontinuous functions on a product space
Edit: According to comments of Eric Wofsy and Yemon Choi I edit the question.
For a (compact) topological space $X$, we put $A=\{f:X\to \mathbb{C}\mid f\text{ is bounded}\}$. We define a semi-...
- 356
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179
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Non-reflexive linear subspace
We know that if X infinite dimensional normed space, then weak topology smaller than normed topology.
This is my problem(from russian textbook of Bogachev-Smolyanov, Functional Analysis) :
Let X be ...
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0
answers
151
views
Notion of solution of pde
Let's consider the following Schrodinger equation
$$iu_t+\Delta u+F(u)=0$$
in $\mathbb{R}^n$. In Cazenave's book, "Semilinear Schrodinger equation", he defines $H^1$-weak solution as $u\in L^\infty(0,...
- 9
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0
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145
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A question about the duality principle
Suppose $X$ and $Y$ are finite sets and $K:X\times Y\to \mathbb R$ is some function. We get an integral transform from the space of real functions on $X$ to real functions on $Y$ given by $$\Phi_Kf(y)=...
- 133
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35
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Approximate rank of the set formed by all delayed replicas of a bandlimited signals between 0 and T
Given a complex-valued signal with a certain delay $s(t-\tau)$ for which we sample $N$ instants
$$
\mathbf{s(\tau)}=\left[s(0-\tau),\ldots,s\left(\frac{N-1}{f_s}-\tau\right)\right]^T
$$
at Nyquist ...
- 345
0
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0
answers
148
views
existence of locally translation-invariant Borel measure on Frechet manifolds
It is well known that the only locally finite, translation-invariant Borel measure on an
infinite-dimensional, separable Frechet space is the trivial measure. I am wondering about an analogous ...
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0
answers
137
views
SVD of Frechet derivative
This is mainly a reference request. Is there a particular characterization of operators A from a Hilbert space H to itself such that the Frechet derivative A'(u) exists for each $u \in H$ and for any ...
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113
views
Existence of a mapping in a nonseparable Banach space
Do there exist a nonseparable Banach space $X$, a mapping $F: X\to X$, and an open nonempty subset $D\subset X$ such that
$$
\forall\,E>0 \quad \exists\,\delta>0: \quad \forall\,x,y\in D \quad (...
- 1
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votes
0
answers
182
views
Establishing an upper bound for a dyadic average of a function in $ {L^{p}}([0,1)) $
Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1)) $, where $ p > 1 $. Let
$$
(D_{n})_{n \in \mathbb{N}_{0}} =
\left( \left\{
I^{n}_{j} \stackrel{\text{df}}{=} \left[ \frac{j}{2^{...
- 1
0
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0
answers
77
views
How to generalize balanced and absorbing sets to R-modules?
I'm looking for generalizing the notions of balanced set and absorbing set. The goal is using them for analyzing topological R-modules with R being a unit ring.
It's easy to generalize balanced and ...
- 29
0
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153
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extension of function in an abstract metric space
my question is the following.(Maybe my title is not quite proper for this question):
Let $(E,d)$ be a Polish space (or a separable metric space), let $\xi: E\to R_+$ be a Lipschitz function. Now set $...
- 1,835
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votes
0
answers
103
views
The trivility of Besov space for large parameter
For all $s>0$, $1\le p<+\infty$ and $u\in L^p(\mathrm{d}x)$, we define
$$D_{s,p}(u)=\sup_{r>0}\frac{1}{r^{sp+n}}\int_{\mathbb{R}^n}\int_{B(x,r)}|u(y)-u(x)|^p\mathrm{d}y\mathrm{d}x$$
and
$$W^{...
- 369
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0
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79
views
Stable analytic manifold under simple action
For an integer $m > 1$, let us define the action
$$
f: X_i \to (1+X_i)^{m} - 1
$$
on $C[[X_1,...,X_N]]$, where $C$ is the complex number field. Consider the analytic manifold $V(I)$ defined by the ...
- 1
0
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0
answers
145
views
Does there exist this special kind of homeomorphism?
Let $A,B\subset\mathbb{R}^n, n\geq 2,$ are two different shaped spindles. One is thick and one is thin. (Sorry for my unprofessional statements. Unsure about how to say it rigorously.) So there are ...
- 183
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135
views
uniform continuity of a function in ultrametric spaces
Consider $[0,1]$ with the metric $d_1(x,y)=\left\{\begin{array}{cc}
0&x=y,\\
\max\{x,y\}&x\ne y.
\end{array}\right.$. Moreover let $(M,d_2)$ be an ultrametric space. Let
$f:(M,d_2)\to([0,1],...
- 1
0
votes
0
answers
184
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Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$
My question is motivated by an optimal transportation approach to PDE's and gradient flows in metric spaces (see e.g Otto's geometry of dissipative evolution equations: the porous media equation and ...
- 5,380
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votes
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213
views
Equivalent Gaussian measures
Let $\mu$ be a gaussian measure with eigenpair $\{e_k,2^{-k}\}$ and $\nu$ with eigenpair $\{ Te_k,2^{-k}\}$. Here, T is the unitary operator given by $Tx = x - 2\left\langle x,v \right\rangle v$. Let $...
- 33
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200
views
Want to show rigorously $\frac{d}{dt}\int_{\Omega}|u(t)|^r = r\langle u_t(t), |u(t)|^{r-2}u(t)\rangle_{H^{-1}(\Omega), H^1(\Omega)}$
We have a bounded domain $\Omega$ of $\mathbb{R}^n$. Let $$u \in L^2((0,T);H^1(\Omega)) \cap H^1((0,T);H^{-1}(\Omega))\cap L^\infty((0,T);L^\infty(\Omega)).$$
I want to show for $r \geq 2$ that
$$\...
- 1
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0
answers
606
views
$b_n \rightharpoonup b$ in $L^q(Q) \forall q < \infty$, $b_n \to b$ in $C^0([0,T];H^{-1})$ implies $b_n(t) \rightharpoonup b(t)$ in $L^q(\Omega)$
This question stems from the proof of Theorem A.1 on page 425 of this paper.
Let $Q=(0,T)\times \Omega$. Suppose $b_n \rightharpoonup b$ in $L^q(Q)$ for any $q < \infty$ and $b_n \to b$ in $C^0(...
- 266
0
votes
0
answers
88
views
References for LWP of a NLS Equation
I am studying the LWP of $$i \partial_t \psi + \Delta \psi = \left| \psi \right|^{p-1} \psi + \frac{1}{\left| x \right|^2} \psi$$ in $\mathbb{R}^{1+2}$ given appropriate Cauchy data. It will probably ...
- 516
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votes
0
answers
206
views
About approximate eigenvalue
I am in trouble when read the book "D.Henry, Geometric Theory of Semiliner Parabolic Equations". The question is relate to Page 104,proof Lemma 5.1.4.
Suppose $X$ is a real Banach Space, $M$ is a ...
- 103
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votes
0
answers
184
views
Can I define Fredholm Index using $\dim \ker ST - \dim \ker TS$?
$X$, $Y$ are Banach spaces.
Let $S \in L(X, Y)$, $T \in L(Y, X)$, where $L(X, Y)$ denotes the Banach algebra of bounded linear operators from $X$ to $Y$. If we have that $Id_Y - ST \in \mathbb{K}(Y)$ ...
- 365
0
votes
0
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216
views
Bound on integral of elliptic theta function
I need to prove that the following bound is true. I thought this might follow from the inversion property of the theta function, as the infinite sum in the integrand is precisely $\theta_3(0,\mathrm{e}...
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0
answers
74
views
Weak convergence of 4-th degrees
Good day!
We have an equation $y'+Ay=Bu$ where $y=\{\theta,\varphi\}$, $A, B$ are nonlinear operators.
$u \in L^\infty(\Gamma)$, $\theta, \varphi \in W = \{y \in L^2(0,T;V) : y'\in L^2(0,T;V')\}$, $...
- 243
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0
answers
65
views
Interpolation with time continuity
If $u(x,t)$ is a function depends on $x\in\Omega$ and $t\in[0,T]$. The following result could be found in L.C. Evans's book "PDE".
Suppose $u\in L^2(0,T;H_0^1(\Omega))$, with $u_t\in L^2(0,T;H^{-1}(\...
- 103
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votes
0
answers
83
views
Comparison between operators
I have found the following two concepts:
$\bullet$ Let $L$ be a linear operator in a Hilbert space $H$. The
operator $B$ is said to be $L$-compact if $D(L)\subset D(B)$
and for any $\varepsilon>0$,...
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votes
0
answers
210
views
Weak derivatives and dual of Hölder functions
Let $0<\alpha<1$ and $f \in C^{\alpha}$ be a Hölder function (either with compact support on $\mathbb R^n$ or on a closed Riemaniann manifold).
From what I understand, the derivative of $f$ in ...
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0
answers
428
views
Given an even function how to obtain the most close odd function and vise versa?
Given an even function $f(x)$, how to obtain the most close to it continuous odd function $g(x)$?
By most close I mean that $\int_0^\infty |f(x)-g(x)| dx$ be the minimum possible and the difference $|...
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0
answers
178
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$2$-normed Spaces
Someone suggested today that $2$-normed spaces are actually equivalent to normed spaces. Can anyone who's familiar with the topic provide a counterexample? (I can't access Gähler's original paper ...
- 1
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0
answers
474
views
Does the Laplace-Beltrami/surface gradient commute with orthogonal projection? (related to Galerkin method)
Let $\Gamma$ be a $C^k$ $(n-1)$-dimensional hypersurface embedded in $\mathbb{R}^n$. Let $H=L^2(\Gamma)$ and $V=H^1(\Gamma)$.
Suppose that $\{v_j\}$ is a basis for $H$ and $V$ (not necessarily ...
- 85
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votes
0
answers
45
views
compactness related to some distance defined on the space of increasing functions2
Let $I=[0,1]$ and denote by $C^{+}(I)$ the space of continuous increasing functions. Can we find a distance $d$ for $C^+(I)$ such that the set of the form
$$d(f,g)\rightarrow 0\Longrightarrow f(1)\...
- 1,835
0
votes
0
answers
537
views
matrix Khintchine inequality
The usual Khintchine inequality says that if $\{\epsilon_n\}_{n = 1}^N$ are i.i.d. random variables with $\mathbb{P}(\epsilon_n = \pm 1) = \frac{1}{2}$ for each $n$ then
\begin{equation*}
\left( \...
0
votes
0
answers
405
views
Dual of the space of vector valued Borel measures
What is the dual of the space of all vector valued Borel measures?
- 21
0
votes
0
answers
533
views
Constructing an $\epsilon$-net for a Lipschitz subspace of $L^2$
Let $X$ be a subset of $L^2([0,1])$ which contains only Lipschitz function.
Also, the member of $X$ are uniformly bounded
$$
|x(t)| < M, \text{ for all $x \in X$ and $t \in [0,1]$}.
$$
Let $F: X \...
user24451
0
votes
0
answers
156
views
Can a function be constructed from the direction of its gradient?
Let $\Omega$ be a bounded region in $R^n$ and $J\in (L^2(\Omega))^n$ with $|J| \leq 1$ a.e. in $\Omega$. Under what conditions the equation
$Du=J|Du|$, $u|_{\partial \Omega}=f$
has a solution in a ...
- 23
0
votes
0
answers
362
views
Banach-Mazur distance estimate finite-dimensional $\ell_p$ spaces
Hey my fellow Banach space guys. Sorry for another elementary question, and yes I have looked for the past hour and a half to see if I can find it on Google or in my books at home.
Fix $n\in\mathbb{...
- 1,591
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votes
0
answers
68
views
can we say fixed point existance of a set valued map over a compact set is homotopy invariant?
Consider two set valued maps over different compact sets as $F(\mathbf{x}):D\rightarrow\rightarrow D$, $G(\mathbf{x}):E\rightarrow\rightarrow E$ where $D,R\subset Y$. Assume there is a homotopy pair $(...
- 383
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votes
0
answers
515
views
If $\phi_n$ is a sequence of mollifier converging to the identity, does $\inf f\ast \phi_n \to \inf f$?
Let $\phi_n$ be a sequence of mollifier converging to the identity
$$
\phi_n(x) \to \delta_{0}(x), \text{pointwise},
$$
with $\delta_{0}(\cdot)$ the delta function at zero, and $\phi_n \in C^\infty_{\...
user24451
0
votes
0
answers
100
views
Two distribution spaces ${\mathcal S}'/{\mathcal P}$ and ${\mathcal S}_\infty'$
Let ${\mathcal S}'$ be the set of all distributions.
Denote by ${\mathcal P}$ the set of all polynomials,
which is embedded into ${\mathcal S}'$ as a closed subspace.
Equip ${\mathcal S'}/{\mathcal P}$...
0
votes
0
answers
149
views
Does this sequence of H\"older functions have a limit?
Let $\left\{\alpha_{n}\right\}_{n\in \mathbb{N}}$ a sequence of positive real numbers with
$$\alpha_{n}\in (0,1)\quad \textrm{and}\quad \alpha_{n}>\alpha_{n+1}$$
Moreover suppose
$$\lim_{n\...
- 91