All Questions
3,508 questions with no upvoted or accepted answers
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votes
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298
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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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votes
0
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467
views
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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votes
0
answers
56
views
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, $0<Re(\alpha)<1$):
$$...
- 1,199
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votes
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152
views
Continuity under various topologies for positive linear functionals
It is known that if $\mathcal A$ is a unital $\mathbb C$-$*$-algebra and $A$ is a unital subalgebra closed under $*$, and if $f : A \to \mathbb C$ is linear, then $f$ is positive if and only if $f$ is ...
- 5,407
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votes
0
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227
views
Negative Sobolev norm of non-zero mean non-periodic function on bounded space
The usual formulation of $H^{-1}$ norm for a zero-mean periodic function on some domain $\Omega\in\mathbb{R}$ is as follows:
$\|f\|^2_{H^{-1}}=\sum\limits_{k\in Z, k\neq 0}\dfrac{\hat{f}^2_k}{k^2}$, ...
- 318
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votes
0
answers
90
views
Discrete approximations to $\nabla^2$
I found this formula in an engineering textbook (image processing). It is an approximation of the Laplacian on flat space $\mathbb{R}^2$.
\begin{eqnarray*} \nabla^2 f &\approx& -20 f(\vec{x})...
- 22.8k
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266
views
Embedding for the Bourgain spaces $X^{s,b}$
Where can I find embedding results for the Bourgain spaces $X^{s,b}$ (for a definition see the bottom of page 2 here).
In particular, I'd like to know if, for $s$ sufficiently large, it is contained ...
user60665
0
votes
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answers
72
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Sobolev embedding for a specific family of weighted Sobolev spaces
Consider the weighted Sobolev-type space
$$
W_\alpha:=\{f\in L^2(0,\infty):\hbox{id}^\alpha\cdot f'\in L^2(0,\infty)\}.
$$
Are there any known embeddings? Ideally, I am looking for an embedding of the ...
- 3,388
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96
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Does adding a compact operator change the symbol of a pseudodifferential operator?
Suppose $X$ is a non-compact manifold. Let $P$ be an order-$0$ pseudodifferential operator on $X$ and $f:L^2(X)\rightarrow L^2(X)$ a compact operator. I'm wondering:
1) Is $P + f$ always a ...
- 1,903
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votes
0
answers
81
views
Differential operator and equivalence
Here is the problem:
I have a certain PDE and there is the nonlinear terme $h$, I have as data:
$f \in H_0^2(0,L)$,,,$g \in {H^1}(0,L)$ with ${g_x}(0) = {g_x}(L) = 0$
Now on consider the fnction $$h(...
- 617
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votes
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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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308
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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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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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59
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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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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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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
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156
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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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188
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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?
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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 $$H:=\overline{\mathcal V}^{\left\|\;\cdot\;\right\...
- 167
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107
views
Is $(u\cdot\nabla)v\in H^1$, if $u,v\in H^2$?
Let
$d\in\left\{2,3\right\}$ with
$\Lambda\subseteq\mathbb R^d$ be bounded and open with $\partial\Lambda\in C^1$
In Lemma 6.1 of Navier-Stokes Equations and Nonlinear Functional Analysis by Roger ...
- 167
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votes
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answers
96
views
Non-B-completeness of finest locally convex topology
For an index set $A$ consider the locally convex direct sum $X_A := \bigoplus_{\alpha \in A} \mathbb{R}_\alpha$ of $|A|$-many lines $\mathbb{R}_\alpha = \mathbb{R}$. Then $X_A$ is complete. It is ...
- 1,773
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votes
0
answers
76
views
Measure on infinite dimesional $L^p$ space relating size in norm to size in measure
Let $A$ be a bounded set in an infinite dimensional $L^p$ space. Fix an $\epsilon>0$. Is there a Borel measure $M$ such that
$$ M(B(x,\epsilon)) \geq C, \quad \forall x \in A$$
for some $C>0$ ...
- 111
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answers
46
views
The Minkowski $(N-1)$- dimensional upper constant of a closed curve?
Let $\Omega\subset \mathbb R^N$ be open bounded smooth boundary. Let $S\subset \Omega$ be a $N-1$ rectifiable set with $\mathcal H^{N-1}(S)<+\infty$. It is well know that if $S$ is not closed, then ...
- 679
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73
views
Continuously varying operators defined by a strange formula
Take $2n$-tuples of bounded positive operators $x_1,\dots x_n$ and $a_1,\dots a_n$ on a Hilbert space $H$ which have zero kernel and dense image and which satisfy the condition that (1)
$$
x_1^* x_1+\...
- 1,143
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votes
0
answers
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
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103
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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
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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
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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 $$\...
- 157
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votes
0
answers
59
views
Restriction to Basis of Cadlag function
If $f \in L^2([0,T])$ then it can be written as
$$
f(t) \triangleq \sum_{i \in \mathbb{N}} c_i e_i(t),
$$
for some sequence $\{c_i\}$ of real numbers and a Schauder basis $\{e_i(t)\}$ of $L^2([0,T])$ ...
- 5,405
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votes
0
answers
68
views
duals of subspaces of DF-spaces
Let $X$ be a complete barrelled DF-space and $Y$ its closed subspace. As can be seen the dual $(Y',\beta(Y',Y))$ is metrizable. Does it follow it is also complete?
- 375
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votes
0
answers
119
views
Gauge Fixing Problem on Cylindrical
For Cylindrical $Y\times\mathbb R$, where $Y$ is a closed oriented 3-manifold.
If it is necessary, we could consider the $b_1(Y)=0$ case.
Fix a Line bundle $L\to Y\times \mathbb R$ and a Hermitian ...
- 1,915
0
votes
0
answers
120
views
A topology on the product space of Euclidean space and smooth functions space
I'd like to know if there is a well-known topology on the space $S := \mathbb R \times C^\infty(\mathbb R)$, such that $(x_n, f_n) \to (x, f)$ in $S$ with respect the topology is equivalent to
$$(x_n,...
- 1,399
0
votes
0
answers
378
views
compact injection
Put:
$D=\{u\in L^{2}(\mathbb{R}^{n})| x^{\alpha}D^{\beta}_{x}u\in L^{2}(\mathbb{R}^{n}), \forall \alpha,\beta \in \mathbb{N}^{m}:|\alpha|+|\beta|\leq 2 \}$
Why $D \hookrightarrow L^{2}(\mathbb{R}^{n}...
- 135
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votes
0
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252
views
Hadamard product (Schur product) in $L^2[0,1]$
Let's consider the separable Hilbert space $\mathcal{H} = L^2[0,1]$ of square-integrable functions on the interval $[0,1]$ with orthonormal basis $(e_j)$. For $x,y \in \mathcal{H}$, the Hadamard ...
- 195
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132
views
approximating smooth functions by non-smooth ones, in the distribution topology
The classical Stone-Weierstrass theorem gives a necessary and sufficient condition for a class of continuous functions on a compact to approximate a larger class of continuous functions in $C^0$ ...
- 4,070
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votes
0
answers
194
views
Johnson's Theorem - Proof (Runde) Clarification
I am reading Runde's Lectures on Amenability. In the proof of Johnson's theorem where he proves "$L^{1}(G)$ is amenable Banach algebra implies $G$ is amenable" : he defines a $L^1(G)$ bimodule action ...
- 185
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votes
0
answers
271
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Convolution Integral involving an unknown function
I've got the following problem I'm working on which is related to some of my research.
I am trying to solve the following equation for the function $f$.
$$t^{-\alpha} \exp{ \left(- \beta x^2 t^{-2 \...
- 281
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votes
0
answers
85
views
Some problems about symmetric convolution semigroup on the unit circle
These are problems from Example 1.4.2 of Fukushima's book "Dirichlet forms and symmetric Markov processes".
Let $\Lambda$ be the set of all real sequences $\left\{\lambda_n\right\}_{n\in\mathbf{Z}}$ ...
- 369
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votes
0
answers
64
views
Approx the jump point of a $BV$ function from both hand side
Let $I=(-1,1)$ be an interval in one dimension. Let $u\in BV(I)$ be defined as
$$
u(x)=
\begin{cases}
0,&\text{ if }x\in(-1,0)\\
1,&\text{ if }x\in(0,1)
\end{cases}
$$
Clearly, we have $u\in ...
- 679
0
votes
0
answers
322
views
Comparison of Parameter estimation using maximum likelihood and Maximum entropy
I am not sure if the question is appropriate but I want to try my luck. One can estimate a parameter using maximum likelihood and we know it is optimal. On the other hand there are methods which uses ...
- 495
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0
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470
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Derivatives of Mollified functions
I'm reading Controlled Diffusion Process by N.V. Krylov. On page 87-88, in the proof of theorem II.6.1, it says the following:
Let $\sigma(t,x)$ be a matrix of dimension $d\times d$, and let $b(t,x)$ ...
- 131
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votes
0
answers
123
views
On the operators from $l_{p}$ into Tsirelson's space $T$
Let $1<p<2$. My question is: Is any operator from $l_{p}$ into Tsirelson's space $T$ compact?
- 3,343
0
votes
0
answers
54
views
Left introversion operators associated to function spaces on semigroups
I am stuck on the following question for quite sometime now. Please help, any comment is welcome.
Let $S$ be a topological semigroup and $\mathcal{F}$ be a translation invariant, conjugate closed ...
- 1
0
votes
0
answers
371
views
Harmonic function with Dirichlet boundary condition
Consider the domain $D = \{(x_1, x_2,.., x_n) \in \mathbb{R}^n : 0 \leq x_i \leq 1\}$. Let $D$ be divided into two parts $D_1$ and $D_2$ by the hyperplane $H = \{x_1 = \frac{1}{2}\}$. My question is: ...
- 1
0
votes
0
answers
510
views
Composition of upper semi-continuous real valued function with upper semi-continuous matrix valued function
Say that a matrix valued function $A: \mathbb{R} \rightarrow \mathbb{R}^{n \times n}$ is upper semi-continuous at $x_0$ if
$$ \limsup_{x \rightarrow x_0} A(x) \preceq A(x_0), $$
where $\preceq$ ...
- 1,590
0
votes
0
answers
115
views
When do block sequences yield disjoint subspaces?
Let $X$ be a Banach space having a (unconditional, normalized) Schauder basis $(e_n)_n$. Suppose that $Y$ and $Z$ are (closed) block subspaces of $X$ having normalized block bases (with respect to $(...
- 3,115
0
votes
0
answers
63
views
The union of weighted compact supported continuous function
Let $\Omega\subset \mathbb R^N$ be open. Given a weight function $v\geq 1$ such that $v\in L^1_{\text{loc}}(\Omega)$ and $l.s.c$. Also supposethere exists a Lipschitz continuous sequence $v_n$ such ...
- 679
0
votes
0
answers
107
views
Under which conditions is the union of conic hulls of sets in a cartesian product equal to $\mathbb{R}^N$?
Question: Under which conditions on $A, B\in\mathbb{R}^{N\times N}$ is the function $f: \mathbb{R}^N\mapsto \mathbb{R}^N$,
$$f(v) = A[v]_+ + B[-v]_+$$
surjective? Here $[.]_+$ is an elementwise ...
- 123
0
votes
0
answers
188
views
One parameter family differentiable dependence for linear parabolic pde's
Consider for example, the Black Schole's equation
$$
\partial_tu+0.5\sigma^2s^2\partial_{ss}u+rs\partial_su-ru=0
$$
on $[0,T]\times[0,\infty)$ subject to boundary conditions $u(s,T)=f(s)$.
The ...
- 383
0
votes
0
answers
173
views
Is this has anything to do with Riesz representation?
The Riesz representation is very useful in study BV space. There is a lot of version of it and one of the good one can be found in this book, page 49.
Here I come up with a question which has similar ...
- 679