All Questions
3,842 questions with no upvoted or accepted answers
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237
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Spectrum of a Hamiltonian on the real line
Consider the following linear (Hamiltonian) operator on functions on the real line $\mathbb{R}$
$$H\psi(x)=-\frac{d^2}{dx^2}\psi(x)+V(x)\psi(x).$$
Assume that $V$ is a smooth function and $V(x)\to +\...
- 21.8k
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votes
0
answers
72
views
weakly amenable weighted sequence algebras
Let $v=(v_n)_{n\in\mathbb{N}}$ be a positive weight with $\inf_nv_n>0$ (for convenience we may take $v_n\geqslant1$). Then $\ell_{\infty}(v)$ is a Banach algebra with coordinate-wise multiplication....
- 375
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0
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537
views
Positive Convolution Root
I try to compute the convolution root of a symmetric, positive definite, nonnegative, one dimensional function $f: \mathbb R\to \mathbb R^+_0$. Furthermore I assume $f$ is bounded and $\int_{\mathbb R}...
- 163
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161
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Topologies corresponding to norm, SOT and WOT under duality
This is a question from MSE which has not received any attention so far.
Let $X$ be a Banach space with norm dual $X'$. (I am mostly interested in the case $X = \ell^1$.)
For a linear mapping $T : X \...
- 1,773
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60
views
Solution of a functional equation with cosine transform
What are the functions verifying:
$$\int_0^{\infty} f(t) \cos(2\pi xt)=\lambda \frac{1}{x} f(\frac{1}{x})$$
With $\lambda$ a constant ?
(Functions $x^{-\alpha}$ with $0<\alpha<1$ are solutions ...
- 1,199
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votes
0
answers
324
views
Adjoint of differential equation
Motivation: Consider the ODE
$$y'(t)=Ay(t)$$ then it is true that the flow satisfies $\Phi(t)y_0=e^{tA}y_0$ and the adjoint of the flow is a solution to the adjoint equation
$$y'(t)=A^*y(t).$$
I ...
- 83
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0
answers
70
views
A question about an irreducible ultra-power
Let $A$ be a Banach algebra and $E$ be an irreducible Banach $A$-module. Is there a countably incomplete ultra filter $\mathcal U$ on $\mathbb N$, the set of natural numbers, such that the ultra power ...
- 2,118
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votes
0
answers
467
views
Is the product of two Sobolev functions in L^p?
Assume that $f\in W^{\alpha-1,p}(R^n)$ with $0<\alpha<1$ and $p>2n/\alpha$.
Given another function $ g\in W^{\beta,p}(R^n)$ with $\beta>0$.
Under what conditions on $\beta$ can we get ...
- 411
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0
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299
views
When convolution with exponential kernel is bounded
Let $g(t)=e^{-\omega t}$, $\omega>0$. What is, in terms of well-known function spaces, the space $X$, $L_{loc}^2(0,\infty)\subset X$, of all functions $f:\mathbb{R}^+\to \mathbb{R}^+$, satisfying
$...
- 395
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0
answers
55
views
Continuity of a composite function
Let $n=2$ or 3 and let $\Omega$ be a bounded domain of $\mathbb{R}^n$. Let $T>0$ and $f \in L^2([0,T],H^1(\mathbb{R}^n))$.
Is the mapping
\begin{equation}
\begin{array}{rcl}
C^0([0,T],C^1(\bar{\...
- 143
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votes
0
answers
58
views
Dense integer translates of a real-valued function with unequal limits at infinity
This is a follow up on a Previous question. Let $W$ be the space of continuous functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $$\lim_{x\rightarrow \infty} f(x)=0~\mbox{and}~\lim_{x\...
- 537
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votes
0
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194
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Estimate of a Sobolev function after a change of variables
Let $n>0$ and $\Omega$ be a bounded domain of $\mathbb{R}^n$. Consider a smooth enough mapping $\Phi$, from $\Omega$ into $\Phi(\Omega)\subset\mathbb{R}^n$, that is orientation-preserving and ...
- 143
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89
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Hausdorff methods of summation
From the book of Boss "Classical and modern methods in summability":
"The class of Hausdorff methods includes the Hölder, Cesaro and Euler methods. A large number of other matrix methods which play ...
- 109
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90
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criterions for polar set of Feller processes
Suppose $X_t$ is the solution to
$$
d X_t=b(X_t)dt+dL_t,\quad X_0=x.
$$
where $L$ is a rotational symmetric $\alpha-$stable process with $\alpha\in (0,1]$, $b$ is Lipchitz.
Assume $\Gamma\subseteq ...
- 515
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answers
65
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Does $\{ x^* \circ \psi_t:x^*\in ext(E^*), t\in K \}\subset ext(X^*)$ hold?
Notations: Let $K$ be a locally compact Hausdorff space and $E$ be a real normed linear space.
Recall that $C_0(K,E)$ is the set of $E$-valued continuous functions $f$ on $K$ such that $f$ vanishes at ...
- 623
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0
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57
views
A question on order unbounded sequences in Banach lattices
Let $E$ be a Banach lattice. It is well-known that every norm convergent sequence in $E$ admits an order convergent subsequence and hence admits an order bounded subsequence. But it seems that a norm ...
- 3,343
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0
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298
views
Generalization of the Chinese remainder theorem
Let $A$ be a Banach algebra and $\{I_{\alpha}\}_{\alpha}$ be a collection of closed two-sided pairwise coprime ideals of $A$. Is the Chinese remainder theorem true for $A$ and $\{I_{\alpha}\}_{\alpha}$...
- 565
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467
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Intersection of two subspaces of a Hilbert space
Background:
Let $D$ be a Klein Four group and consider free product $Z/2Z\star D=<a,b,c,d|a^{2}=b^{2}=c^{2}=d^{2}=bcd=1>$. Now we consider group algebra generated by $Z/2Z\star D$ with inner ...
- 407
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answers
56
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Existence of a couple of functions solution of a differential equation (with additional constraint)
I would like to know if we can find a real function $v(x)$ and a complex function $f(x)$, such that they solve the following differential equation (with $\alpha$ a complex, $0<Re(\alpha)<1$):
$$...
- 1,199
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0
answers
152
views
Continuity under various topologies for positive linear functionals
It is known that if $\mathcal A$ is a unital $\mathbb C$-$*$-algebra and $A$ is a unital subalgebra closed under $*$, and if $f : A \to \mathbb C$ is linear, then $f$ is positive if and only if $f$ is ...
- 5,407
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0
answers
227
views
Negative Sobolev norm of non-zero mean non-periodic function on bounded space
The usual formulation of $H^{-1}$ norm for a zero-mean periodic function on some domain $\Omega\in\mathbb{R}$ is as follows:
$\|f\|^2_{H^{-1}}=\sum\limits_{k\in Z, k\neq 0}\dfrac{\hat{f}^2_k}{k^2}$, ...
- 318
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votes
0
answers
90
views
Discrete approximations to $\nabla^2$
I found this formula in an engineering textbook (image processing). It is an approximation of the Laplacian on flat space $\mathbb{R}^2$.
\begin{eqnarray*} \nabla^2 f &\approx& -20 f(\vec{x})...
- 22.8k
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answers
266
views
Embedding for the Bourgain spaces $X^{s,b}$
Where can I find embedding results for the Bourgain spaces $X^{s,b}$ (for a definition see the bottom of page 2 here).
In particular, I'd like to know if, for $s$ sufficiently large, it is contained ...
user60665
0
votes
0
answers
72
views
Sobolev embedding for a specific family of weighted Sobolev spaces
Consider the weighted Sobolev-type space
$$
W_\alpha:=\{f\in L^2(0,\infty):\hbox{id}^\alpha\cdot f'\in L^2(0,\infty)\}.
$$
Are there any known embeddings? Ideally, I am looking for an embedding of the ...
- 3,388
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0
answers
96
views
Does adding a compact operator change the symbol of a pseudodifferential operator?
Suppose $X$ is a non-compact manifold. Let $P$ be an order-$0$ pseudodifferential operator on $X$ and $f:L^2(X)\rightarrow L^2(X)$ a compact operator. I'm wondering:
1) Is $P + f$ always a ...
- 1,903
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votes
0
answers
81
views
Differential operator and equivalence
Here is the problem:
I have a certain PDE and there is the nonlinear terme $h$, I have as data:
$f \in H_0^2(0,L)$,,,$g \in {H^1}(0,L)$ with ${g_x}(0) = {g_x}(L) = 0$
Now on consider the fnction $$h(...
- 617
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0
answers
97
views
Is there any concise sufficient condition for the dual space to have Kadec property?
A normed space $E$ has a
Kadec property if the norm- and weak topologies coincide on the unit sphere of $E$.
Kadec-Klee property if any sequence on the unit sphere, that is weakly convergent is also ...
- 5,529
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votes
0
answers
308
views
Invertible operator
We consider the operator $$T=I + {{{\partial ^2}} \over {\partial {x^2}}}:{H^2}(0,L) \cap H_0^1(0,L) \to {L^2}(0,L)$$
We hope to prove that $T$ is invertible if and only if $L = n\pi $.
and for this ...
- 617
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0
answers
84
views
Under what conditions on $\mu^{\beta}$ we have $L_1(\beta X,\mu^{\beta})$ isometrically isomorphic to $L_1(X,\mu)$?
Let $X$ be a locally compact Hausdorff space, $\beta X$ its Stone-Cech compactification and $\Delta: X\to\beta X$ the inclusion map. Given a Borel probability measure $\mu^{\beta}$ over $\beta X$, is ...
- 2,044
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0
answers
59
views
Differential operator
One define the operator $T$ as :$$T: = (I - {{{\partial ^2}} \over {\partial {x^2}}}):H_0^1(0,L) \cap {H^2}(0,L) \to {L^2}(0,L)
$$ let $f \in H_0^2(0,L) \cap {H^4}(0,L)$. What can we say about ${T^{ - ...
- 617
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votes
0
answers
58
views
in search of convergent daughter sequences
Let $\{f_n\}\subset L^1(\Omega,\mu)$, where $\mu$ is the Lebesgue measure, and $\Vert f_n\Vert_1\leq M$ and $\Vert Df_n\Vert_{1/2}\leq C$ uniformly in $n$.
Question. Is there a subsequence $\{f_{...
- 43.2k
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votes
0
answers
263
views
Does AX+XA=0 have any non-trivial solutions?
Let $X$ be a continuous linear self-adjoint operator on some Hilbert space $H$ and for arbitrary compact operators $A$ we have: $XA+AX=0.$ Does this imply that $X=0$ or can there be non-trivial ...
- 305
0
votes
0
answers
156
views
amenable locally compact group
Let $\tau_1,\tau_2 $ be topologies on group $G$ such that $(G,\tau_1),(G,\tau_2)$ be a locally compact group. Let $\tau_1\subseteq\tau_2$ and $(G,\tau_2)$ be an amenable group, when $(G,\tau_1)$ ...
- 565
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0
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188
views
semi simple Banach algebra
Let $G$ be a non-abelian locally compact group, $M(G)$ be the measure algebra and $B(G)$ be the Fourier Stieltjes algebra of $G$..
Question. When are $M(G)$ and $B(G)$ semi-simple?
- 565
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0
answers
89
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If $H$ is the closure of the set of solenoidal smooth vecor fields in $L^2$ and $P_H$ denote the orthogonal projection onto $H$, then $P_HH_0^1⊆H_0^1$
Let
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be open
$\mathcal V:=\left\{\phi\in C_c^\infty(\Lambda,\mathbb R^d):\nabla\cdot\phi=0\right\}$ and $$H:=\overline{\mathcal V}^{\left\|\;\cdot\;\right\...
- 167
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votes
0
answers
107
views
Is $(u\cdot\nabla)v\in H^1$, if $u,v\in H^2$?
Let
$d\in\left\{2,3\right\}$ with
$\Lambda\subseteq\mathbb R^d$ be bounded and open with $\partial\Lambda\in C^1$
In Lemma 6.1 of Navier-Stokes Equations and Nonlinear Functional Analysis by Roger ...
- 167
0
votes
0
answers
96
views
Non-B-completeness of finest locally convex topology
For an index set $A$ consider the locally convex direct sum $X_A := \bigoplus_{\alpha \in A} \mathbb{R}_\alpha$ of $|A|$-many lines $\mathbb{R}_\alpha = \mathbb{R}$. Then $X_A$ is complete. It is ...
- 1,773
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votes
0
answers
76
views
Measure on infinite dimesional $L^p$ space relating size in norm to size in measure
Let $A$ be a bounded set in an infinite dimensional $L^p$ space. Fix an $\epsilon>0$. Is there a Borel measure $M$ such that
$$ M(B(x,\epsilon)) \geq C, \quad \forall x \in A$$
for some $C>0$ ...
- 111
0
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0
answers
46
views
The Minkowski $(N-1)$- dimensional upper constant of a closed curve?
Let $\Omega\subset \mathbb R^N$ be open bounded smooth boundary. Let $S\subset \Omega$ be a $N-1$ rectifiable set with $\mathcal H^{N-1}(S)<+\infty$. It is well know that if $S$ is not closed, then ...
- 679
0
votes
0
answers
73
views
Continuously varying operators defined by a strange formula
Take $2n$-tuples of bounded positive operators $x_1,\dots x_n$ and $a_1,\dots a_n$ on a Hilbert space $H$ which have zero kernel and dense image and which satisfy the condition that (1)
$$
x_1^* x_1+\...
- 1,143
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votes
0
answers
68
views
What can be said about moments of probability distribution if it satisifies hemholtz equation?
From physical considerations I have observed, that probability density in region of interest satisfies
$$
\Delta u(x) + \phi(x)u(x) = f(x),
$$
where $\phi(x)$ and $f(x)$ are both given functions and $...
- 355
0
votes
0
answers
216
views
Intersection of weighted Sobolev spaces
Consider the Sobolev spaces with $p=2$, defined for $s \in \mathbb{R}$ as
\begin{equation}
W^{s} = \left\{ u \in \mathcal{S}', \ (1 + \lvert \cdot \rvert^2)^{{s}/{2}} \widehat{u} \in L_2 \right\}.
\...
- 2,306
0
votes
0
answers
103
views
Multiplier of Banach algebras
Let $A$ be a Banach algebra and $M(A)$ be its multiplier Banach algebra. Is there any correspondence between closed two sided idaels of $A$ and closed two sided idaels of $M(A)$?
Can we see that ...
- 565
0
votes
0
answers
272
views
Fixed-point iteration depending on a parameter
Let $f\colon X\times \mathbb{R}\to X, (x,\varepsilon)\mapsto y$, with $X$ open, be a continuous function in both arguments. Consider the following fixed-point iteration
\begin{align}
x_{k+1} = f(x_k,\...
- 2,712
0
votes
0
answers
343
views
A question on weak formulation of the p-laplacian operator
Can it be said that $$\int_{\Omega}\Delta_p u |\phi|^{p-2}\phi dx=\int_{\Omega}\Delta_p \phi |u|^{p-2}u dx\qquad\forall \phi\in C_0^2(\overline{\Omega})$$ is the generalized weak formulation of $$\...
- 157
0
votes
0
answers
59
views
Restriction to Basis of Cadlag function
If $f \in L^2([0,T])$ then it can be written as
$$
f(t) \triangleq \sum_{i \in \mathbb{N}} c_i e_i(t),
$$
for some sequence $\{c_i\}$ of real numbers and a Schauder basis $\{e_i(t)\}$ of $L^2([0,T])$ ...
- 5,405
0
votes
0
answers
68
views
duals of subspaces of DF-spaces
Let $X$ be a complete barrelled DF-space and $Y$ its closed subspace. As can be seen the dual $(Y',\beta(Y',Y))$ is metrizable. Does it follow it is also complete?
- 375
0
votes
0
answers
119
views
Gauge Fixing Problem on Cylindrical
For Cylindrical $Y\times\mathbb R$, where $Y$ is a closed oriented 3-manifold.
If it is necessary, we could consider the $b_1(Y)=0$ case.
Fix a Line bundle $L\to Y\times \mathbb R$ and a Hermitian ...
- 1,915
0
votes
0
answers
144
views
Optimality of Exponent in $\ell^{2}$ Decoupling Theorem
For $\delta>0$, let $\mathcal{N}_{\delta}$ denote the $\delta$-neighborhood of the truncated paraboloid $P^{n-1}$ and let $\theta$ denote a $\delta^{1/2}\times\cdots\times\delta^{1/2}\times\delta$ ...
- 873
0
votes
0
answers
120
views
A topology on the product space of Euclidean space and smooth functions space
I'd like to know if there is a well-known topology on the space $S := \mathbb R \times C^\infty(\mathbb R)$, such that $(x_n, f_n) \to (x, f)$ in $S$ with respect the topology is equivalent to
$$(x_n,...
- 1,399