All Questions
3,628 questions with no upvoted or accepted answers
1
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50
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Comparison of (square) of a function and its Fourier transform in an integral
I am completely stuck on a comparison between $f(t)^2$ and $\hat{f}(t)^2$ in an integral.
Considering $f(t)$ of rapid decrease at infinity such that near zero: $f(t) \sim_0 t^{-\frac{1}{2}- \alpha}+o(...
- 1,199
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0
answers
60
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Is there a vector-valued trace such that $\text{tr}((L\otimes_π\text{id}_H)T)=LT$ for all $L∈\mathfrak L(H,\mathfrak L(H))$ and $T∈H\hat\otimes_πH$?
Let
$H$ be a separable $\mathbb R$-Hilbert space
$L\in\mathfrak L(H,\mathfrak L(H,\mathbb R))$
$T\in\mathfrak L(H)$ be nonnegative, self-adjoint and nuclear (trace-class)
Note that$^1$ $$\...
- 167
1
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0
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134
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Operator-valued stochastic integral and quadratic variation for operator-valued processes
Let $U$ be a separable $\mathbb R$-Hilbert space and $W$ be a $Q$-Wiener process on a complete and right-continuous filtered probability space. Let $H$ be a separable $\mathbb R$-Hilbert space and $X$ ...
- 167
1
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0
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74
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If $f$ takes values in $L(H,L(H,\Bbb R))$ and $μ$ is a $H\hat ⊗_πH$-valued measure, how are $\int f\:dμ$ and $\int f⊗_π\text{id}_Hdμ$ related?
Let
$H$ be a separable $\mathbb R$-Hilbert space
$H\:\hat\otimes_\pi\:H$ denote the projective tensor product of $H$ and $H$
$(\Omega,\mathcal A)$ be a measurable space
$\mu$ be a $H\:\hat\otimes_\pi\...
- 167
1
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0
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181
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Hölder's inequality for Hilbert-Schmidt operators which are also trace class
Do Hilbert-Schmidt operators which are also trace class, satisfy Hölder's inequality? That is, we have two Hilbert-Schmidt operators $A$ and $B$. Is the following true?
$$\langle A, B \rangle \leq \...
1
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0
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84
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Coercivity of $\int (\Delta u + u)^2$ on a subspace of $H^2$?
Let $\Omega = [0,L] \times [0,2\pi]$ and split its boundary into $\Gamma_d = \{0,L\} \times [0,2\pi]$, $\Gamma^1_p = [0,L] \times \{0\}$, $\Gamma^2_p = [0,L] \times\{2\pi\}$. Consider the following ...
- 111
1
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0
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101
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Non standard Lipschitz extension
Consider a ball B and let $f(x) \in L^1(B)$ such that $\int_B f(x) dx = 0$. Furtheremore, there exists a closed set $E \subset B$ such that $f|_E$ is Lipschitz. The standard Lipschitz extension ...
- 455
1
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0
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877
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Changing the order of integration of double integral: references and theorems
The Fubini's theorem states that if we have $ \int_0^{\infty} \int_0^{\infty} |f(t,x)| dt dx$ well defined (i.e. function is absolutely integrable) then we can interchange order of integration:
$$ \...
- 1,199
1
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0
answers
63
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Martingale covariation operator in infinite-dimensions
Let
$(\Omega,\mathcal A,(\mathcal F_t)_{t\in[0,\:T]},\operatorname P)$ be a filtered probability space
$U,H$ be separable $\mathbb R$-Hilbert spaces
$(e_n)_{n\in\mathbb N}$ and $(f_n)_{n\in\mathbb N}$...
- 167
1
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0
answers
77
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Zero energy resonances for scaling critical Schrodinger operators
Given a real valued potential $V\in L^1(\mathbb{R}^3)$, we say that the Schrodinger operator $-\Delta + V$ has a zero-energy resonance if there exists $\psi\in L^2_{loc}(\mathbb{R}^3)\setminus L^2(\...
- 943
1
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0
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43
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Hidden regularity for the coupled wave equation with dynamaic boundary condition
We have the equation
\begin{equation}
\left\{
\begin{array}{rrrr}
u_{tt}-\Delta u=0,&\text{in} &
\Omega \times ]0,T[ & \left( 1.1\right) \\
u=0, & \text{on
} & \Gamma _{0}\...
- 617
1
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0
answers
137
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Is there an analysis theorem analogous to Kuznetsov/Petersson trace formula?
I am thinking about general differential operator acts on a compact manifold. Is there something similar to Kuznetsov trace formula?
For example, let $f_i $ be the eigenfunctions of an operator $D$, ...
- 3,804
1
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0
answers
110
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Trace embedding and unbounded domain
Let $D\subset\mathbb{R}^d$ be an open domain and let consider the open cylinder $D\times (0,T)\subset\mathbb{R}^{d+1}$ where $T\in (0,+\infty)$ arbitrary. Let $H^{1}(D\times (0,T))$ be the Sobolev ...
- 33
1
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0
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141
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Characterisation of functions for which the Fourier transform commutes with a particular operator
Defining the operator $\phi$ by: $\phi(f(x))=\frac{1}{|x|} f(\frac{1}{x})$, and noting $\mathcal{F}$ the Fourier transform on the real line, can we characterize all the functions (with real variable ...
- 1,199
1
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0
answers
51
views
Extension of $\sigma$-additive vector measures on a ring and the relationship between the corresponding total variation functions
Let
$\Omega$ be a set
$\mathcal S\subseteq2^\Omega$ be a set with $\emptyset\in\mathcal S$
$\mathcal S_{\text{loc}}:=\left\{A\subseteq\Omega:A\cap S\in\mathcal S\text{ for all }S\in\mathcal S\right\}$...
- 167
1
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0
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115
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Why the weighted fractional Sobolev semi-norm si not defined for $p=1$?
I already ask the question on MSE (here), but I didn't had any answer, so I try here.
We defined the Weighted Sobolev semi-norm by $$[u]_{W^{s,p,\alpha }(\Omega )}^p=\iint_{\Omega \times \Omega }\...
- 103
1
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0
answers
304
views
Harmonic coordinates on asymptotically flat manifold
I am studying the existence of harmonic coordinates at infinity on an asymptotically flat manifold. My Reference papers are, The Mass of Asymptotically Flat Manifold, by Bartnik [B] and The Yamabe ...
- 914
1
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0
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107
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Laplacian on squashed spheres
Is anything known about the Laplacian on squashed spheres $S^{2n-1}_\omega$, where the ambient $C^n$ coordinates satisfy
$$ 1= \sum_{i=1}^n \omega_i |z_i|^2 $$
for fixed real numbers $\omega_i$? for ...
- 533
1
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0
answers
213
views
Restriction of a Sobolev function to a straight line
I have been asked the following question, and I have to admit that I have no idea about the answer.
Assume that $f \colon (a,b) \to \mathbb{R}$ is a function. Assume also that there exists a ...
- 459
1
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0
answers
127
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A point in Ion Suciu's paper on semigroups of isometric operators
My question is concerned a point in this 1968 paper by Ion Suciu which is given in Theorem 2. In the last paragraph of page 104, it is claimed that $N$ (given in the formula 2.5) is a wandering ...
- 4,058
1
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0
answers
180
views
Positive square roots of inverse operators on different Sobolev spaces
Let $D$ be a self-adjoint (in the $H^0$-inner product) first-order differential operator on a manifold $M$, where $H^i$ stands for the $i$-th Sobolev space on $M$. Then $D$ extends to a bounded ...
- 1,903
1
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0
answers
235
views
Associative law of the stochastic integral in Hilbert spaces
Let
$(\Omega,\mathcal A,\operatorname P)$ be a complete probability space
$T>0$
$I:=(0,T]$
$(\mathcal F_t)_{t\in\overline I}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A)$
...
- 167
1
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0
answers
60
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Existence of solutions to $\lambda u-\frac{1}{(1+(u')^2)^2} \, \Delta u = f$
I'm looking for existence results for the equation
$$\lambda u-\frac{1}{(1+(u')^2)^2} \, \Delta u = f \quad \text{on the domain $[a,b]$}$$
for $u:[a,b] \to \mathbb{R}$, with either zero Dirichlet or ...
- 11
1
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0
answers
233
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Bochner integrals with values in a Hilbert $A$-module
I'm wondering whether there exists a generalisation of Bochner integration with values in a Hilbert $A$-module $M$, where $A$ is a general $C^*$-algebra rather than $\mathbb{C}$ (and whether there are ...
- 1,903
1
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0
answers
68
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Working in coordinates with topologies on the algebra of continuous functions
Let $X$ be a Hausdorff completely regular topological space, and let $C_b (X)$ be its algebra of continuous bounded functions. Endow $C_b (X)$ with a topology given by some seminorms, that contains ...
- 5,407
1
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0
answers
102
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domain dependence of best constant in inequality
Take $N \ge 3$ and consider the inequality
$$ \| \nabla u\|_{L^N} \le C(\Omega) \| \Delta u \|_{L^\frac{N}{2}} $$ for all $ u \in W^{2,\frac{N}{2}}(\Omega) \cap W^{1,N}_0(\Omega)$ where $ \Omega$ is ...
- 1,385
1
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0
answers
128
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Decomposition of Banach bimodules of Banach algebras
Let $A$ and $B$ be Banach algebras, $\theta:A\rightarrow \mathbb{C}$ be a character (i.e., a multiplicative linear functional) and $A\oplus _{\theta} B$ be the $l^1$-direct sum of $A$ and $B$ equipped ...
- 167
1
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0
answers
50
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Characterizing (minimal) tensor product inside Hilbert C*-module
Let $A$, $B$ be C$^*$-algebras, $\mu$ be a state on $B$ and $\mathcal{I}$ be a family of ideals in $A$. Let $I_0:=\cap_{I\in\mathcal{I}} I$ and put $A_0:=A/I_0$. Consider the minimal tensor product on ...
- 111
1
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0
answers
82
views
Hankel operator with symbol a Blaschke product
If $B={\prod}_j \varphi_j$ is a Blaschke product (finite or infinite) of Blaschke factors $\varphi_j(w)=\frac{w-\alpha_j}{1-\overline{\alpha_j}w}$ with $|\alpha_j|>1$, is it true that the norm of ...
- 11
1
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0
answers
233
views
Fubini: can we interchange integration order on this double integral (with Fourier series product)
Can we interchange the order of integration of following double integral ?
$$I = \int_{0}^{1} \int_{0}^{\infty} F(x,y) \overline{R(x,y)} - R(x,y) \overline{F(x,y)} \; dx \; dy$$
Where $F(x,y)= \...
- 1,199
1
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0
answers
198
views
Morrey space is Banach space
I'm working with Morrey spaces, which are the spaces
$$L^{p,\lambda}(\Omega):= \left\{ u \in L^1_{loc}(\Omega): \sup_{x \in \Omega, r > 0} r^{-\lambda}\int_{B(x,r)\cap \Omega}|u(y)|^pdy< \infty\...
- 121
1
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0
answers
48
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Analogues of properties (DN) and (Ω) for more general locally convex spaces
In the structure theory of Fréchet spaces, especially results around splitting short exact sequences, the properties (DN) and (Ω) play a major rôle. There are many variants, but they are phrased in ...
- 35.5k
1
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0
answers
47
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Elliptic boundary regularity for vectorial neumann problem on semi-infinite domain with radiation condition?
I'm studying the regularity of time-harmonic linear elasticity with a traction free boundary condition and an appropriate radiation condition. I want to prove that the solutions that I have found have ...
- 256
1
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0
answers
100
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Convergence and boundedness in $L^\infty([0,T]\times \Omega)$ of Karhunen-Loeve expansion
Let $X:[0,T]\times\Omega\rightarrow\mathbb{R}$ be a stochastic process in $L^2([0,T]\times\Omega)$. Consider the Karhunen-Loeve expansion of $X$:
$$ X(t,\omega)=\mu_X(t)+\sum_{n=1}^\infty \sqrt{\nu_n}\...
- 141
1
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0
answers
130
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Spectrum of an operator
Let $L$ an operator self-adjoint acting on $L^2(\Bbb{R}^{2})$ such that :
$L(\phi_{\alpha,\beta})=(|\alpha|-|\beta|)(\phi_{\alpha,\beta})$ where $(\phi_{\alpha,\beta})$ is an orthonormal basis for ...
- 11
1
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0
answers
36
views
Roberts orthogonality and $\alpha$-Isosceles orthogonality
The definitions of Roberts orthogonality (B D Roberts) and $\alpha$-Isosceles orthogonality (Alonso & Benitez) seems to be identical to me. Can anyone point me out the difference between the two ...
- 149
1
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0
answers
66
views
Fractional Leibniz rule with Lorentz spaces
The "fractional Leibniz rule" asserts that
$$\Vert D^s(fg)\Vert_{L^r}\lesssim\Vert D^sf\Vert_{L^{p_1}}\Vert g\Vert_{L^{q_1}}+\Vert f\Vert_{L^{p_2}}\Vert D^sg\Vert_{L^{q_2}}$$
where
$$\frac{1}{p_i}+\...
- 943
1
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0
answers
63
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Supnorm problem involving kernel of Cauchy problem
Let $M$ be the $2$-dimensional hyperbolic manifold. Let $K(t,x,y)$ be the kernel appearing in the fundamental solution of the Cauchy problem
$$(\partial^2_t-\Delta_M)u=0,\text{ on }\mathbb{R}^+\times ...
- 1,692
1
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0
answers
90
views
Compactness of a lifted multiplication operator
If $M$ is a non-compact smooth manifold, then an analogue of Rellich's lemma states that the operator of multiplication by a compactly supported function $f:M\rightarrow\mathbb{C}$ is a compact ...
- 1,903
1
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0
answers
259
views
An estimate for the solution of an elliptic PDE depending on a parameter
Let $\Omega\subset\mathbb R^n$ be a bounded domain with a sufficiently smooth boundary $\partial\Omega$.
We assume $\lambda_1\in\mathbb R$ is the principle eigenvalue of the operator
$$
-\Delta:\ H^...
- 375
1
vote
0
answers
79
views
Time dependent Hamiltonians
I'm studying time dependent perturbation theory on Reed-Simon book "Method of modern mathematical physics, II". If one considers an Hamiltonian of the form
$$H(t)=H_0+V(t)$$
the corresponding formal ...
- 11
1
vote
0
answers
124
views
Inequality about the Fourier transform: $\Vert u \Vert_{L^k} \le \Vert \mathcal{F}(u) \Vert_{L^m}$ (where $1 \le m \le 2$ and $m,k$ Holder conjugates)
How can I prove the following inequality about the Fourier transform?
$$\Vert u \Vert_{L^k(\mathbb{R}^N)} \le \Vert \mathcal{F}(u) \Vert_{L^m(\mathbb{R}^N)}$$ for $1 \le m \le 2$ and $m,k$ Holder ...
user89890
1
vote
0
answers
85
views
A kernel on the d-dimensional flat torus with smoothing properties in the $L^{\infty}$-norm
Let $\rho: \mathbb{R}^d\rightarrow \mathbb{R}_+$ be smooth, symmetric, of compact support, and satisfy $\int_{\mathbb{R}^d}\rho(x)dx=1$. For each $\epsilon>0$, set $\rho_{\epsilon}(x)=\epsilon^{-d}\...
- 11
1
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0
answers
69
views
minimize with orthogonal constraint
Let $\mathcal{L}=(L^2(\mathbb{R}^3))^N$ be the product space with the associated norm
$$
\Vert U\Vert_0=\left(\sum^N_{i=1}\Vert u_i\Vert_0^2\right)^{1/2}
$$
where $U=(u_1,u_2,...,u_N)\in\mathcal{L}$. ...
- 163
1
vote
0
answers
66
views
The infimum over Sobolev norms of compactly supported functions which are 1 on an interval
Let $n\in \mathbb{N}_{0}$. I am interested in the quantity
$\inf\{\|\psi\|_{W^{1,n}(\mathbb{R})}\mid \psi\in W^{1,n}(\mathbb{R}), 0\leq \psi \leq 1, \psi\equiv 1 \text{ on }[-1/2,1/2], \text{ supp}(\...
- 107
1
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0
answers
80
views
Estimate a projection from a product space of $H^1(\mathbb{R}^3)$ to a finite dimensional space
Let $\mathcal{H}=(H^1(\mathbb{R}^3))^N$ be the product space with the associated norm
$$
\Vert U\Vert_1=\left(\sum^N_{i=1}\Vert u_i\Vert_1^2\right)^{1/2}
$$
where $U=(u_1,u_2,...,u_N)\in\mathcal{H}$. ...
- 163
1
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0
answers
72
views
What kind of null set is this?
Let $H$ be a Hilbert space.
Suppose the following holds: for every orthonormal basis $\{e_n\}$ of $H$ and sequence $\{\epsilon_n\} \in \ell^2(\mathbb{N})$, with $C:= \prod_n [-\epsilon_n e_n, \...
- 11
1
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0
answers
76
views
Which sets support which spectra?
I know (and this is of course rather elementary) that an isolated point in the spectrum of a self-adjoint operator $T$ always belongs to the point-spectrum.
I would like to ask: Are there similar ...
- 173
1
vote
0
answers
53
views
Asymptotics of $K$-functional between $\ell_1$ and $\ell_2$ for a specific sequence
I originally had posted this question on Math.SE, two weeks ago. Since it is research-based (even though I am not 100% confident it fits the bill for MathOverflow) and didn't receive any answer on ...
- 1,372
1
vote
0
answers
105
views
The inverse image of a Noetherian topological space
A topological space $X$ is called Noetherian if
closed subsets satisfy the descending chain condition, equivalently,
the open subsets satisfy the ascending chain
condition.
Let $A$ and $B$ be ...
- 11