All Questions
3,508 questions with no upvoted or accepted answers
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45
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Skorohod Space with $J_1$ topology homeomorphic to Frechet Space
Is the Skorohod space $D([0,T];\mathbb{R}^d)$ equipped with the $J_1$ topology homoeomorphic to a separable Fr\'{e}chet space. In particular, is it homeomorphic to $L_{\mu}^1(\mathcal{B}([0,1])$ ...
- 5,405
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221
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Existence of the eigenvalue of the dual operator of the transfer operator
In the passage that I marked in green apparently the author uses a relationship between fixed point and eigenvalues. The result that I know of to ensure the existence of this eigenvalue requires that ...
- 677
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97
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Integral of kernel in RKHS with respect to probability measure
I'm trying to understand how to work with the expression $$ \int u (d\mu-d\pi)$$ where $u\in \mathcal{H}$, an RKHS. $\mu$ and $\pi$ here are two different probability measures. Since $\mathcal{H}$ is ...
- 383
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79
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Hausdorff distance restricted to linear subspaces
Let $V$ be a Hilbert space, $Q \subset V$ be convex and compact and $Q_n \subset V$ be convex and compact for $n\in \mathbb{N}$ such that $Q_n \rightarrow Q$ for $n\rightarrow \infty$ in Hausdorff ...
- 1,095
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149
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Compact embedding of the $\mathcal{C}^k$ norm on a compact Kahler manifold
Given a smooth complex valued function $f$ on a Kahler manifold $X$, we can define its $\mathcal{C}^k$ norm to be $\sum_{p+q \leq k, 0 \leq p \leq q} sup_{X}|\nabla^{p} \overline{\nabla^q} f|_g$, ...
- 1
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0
answers
55
views
Smooth compactly supported function with good scaling with respect to the fractional Laplacian
Is there a smooth cut off function with compact support such that $\phi: \Omega \subset \mathbb R^N \to \mathbb R$, $\mathrm{supp} \phi \subset B_R(0) \subset \Omega$ and $$(-\Delta)^s \phi \le C R^{-...
user139845
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74
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Parseval type lower bound on sum of squares of function projections
This is a followup to this earlier question
Let $f:\mathbb{Z}\rightarrow \{\pm 1\}.$ Assume that the support of $f$ is finite, say it is contained in $[1,N],$ it can even be taken to be $[1,N]$ if it ...
- 10.4k
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479
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What are the sets on which norm-closedness implies weakly closedness?
Let $X$ be a Banach space. Let $C$ be a convex, and normed-closed subset of $X$. It is well-known that $C$ becomes weakly closed subset of $X$. I want to know is there any well-know class of non ...
- 369
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44
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S-familiy induced by an operator induces a Schwartz function
Let $T:S(\mathbb{R}^d)\to S(\mathbb{R}^d)$, a continuous linear operator, where $S(\mathbb{R}^d)$ is the Schwartz space. There is a result that guarantees that the family $F=\{\delta_s\circ T\}_{s\in\...
- 3
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224
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Show convergence of a sequence of resolvent operators
Let
$E$ be a locally compact separable metric space
$(\mathcal D(A),A)$ be the generator of a strongly continuous contraction semigroup on $C_0(E)$
$E_n$ be a metric space for $n\in\mathbb N$
$(\...
- 167
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97
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Does $L^p$ contractivity imply $L^p$ dissipativity?
Does $L^p$ contractivity of an operator semigroup imply the $L^p$ dissipativity of the operator ?
Thank you in advance !
- 1
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54
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On cyclicity of a module
Let $A$ be a $\text{ von Neumann algebra }$, $\mathcal{H}$ is a cyclic $A$ module, $G$ be a finite group acting on $A$, is $\mathcal{H}$ cyclic module over fixed point subalgebra of the action? ...
- 677
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78
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$p$-summing operators space is a Banach space
Let $X,Y$ be Banach spaces and $p\geq 1$. A bounded linear operator $T$ is called $p$-absolutely summing, if there is exist $K>0$, such that for all $n\in N$ and $x_1,\dots, x_n\in X$:
$$
\left(\...
- 1
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59
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Nests on Banach spaces and their duals
Let $X$ be a Banach space and $\mathcal{E}$ a nest on $X$.
Take $f\in X^{*}$ and suppose:
$N \in\mathcal{E}$ is the largest element of the nest so that $f \in N^\bot$
$N=\bigcap_{M>N}M$
Is there ...
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63
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Coarea-like formula for BV functions (not their derivative)
Let $\Omega \subset \mathbb R^N$ and $f \in BV(\Omega)$. The coarea formula states that
$$Df = \int_{\mathbb R} D \chi_{\{f >h\}} \, dh.$$
Unfortunately, the formula
$$f = \int_{\mathbb R} \...
- 839
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153
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Equivalent Definitions of Gaussian Process?
The Gaussian process $\{X_t\}_{t \in T}$ ($T=[0,1]$ for example) is usually defined using its finite-dimensional distribution. I came across this statement many times: linear operator (not necessarily ...
- 103
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109
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Elliptic equation with Neumann boundary condition: RHS in $L^2$ implies solution in $L^\infty$?
Consider the homogeneous Neumann problem $$-\Delta u + ku = f$$ $$\partial_\nu u = 0$$
on a smooth, bounded domain $\Omega$.
If $f \in L^2(\Omega)$, do we obtain the regularity $u \in L^\infty(\...
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326
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Measurability of the heat semigroup in $L^\infty$
Let $S(t)$ be the $C_0$-semigroup generated by the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega)$, where $\Omega$ is a bounded open subset of $R^n$.
It is known that $S(t)$ ...
- 1
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156
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Function classes with high Rademacher complexity
My question is two fold,
Is there any general understanding of what makes a function class have high Rademacher complexity? (Sudakov minoration would say that one sufficient condition for a class of ...
- 2,246
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172
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Compact operator
Let $k:[0,1]^2 \to [0,1]$ be a measurable function. Define $K:L^2([0,1])\to L^2([0,1])$ to be the operator:
$$
(Kf)(x) = \int_0^1\int_0^1 f(z) k(x,y) \mathbf{1}_{x\leq z\leq y} \ \mathrm{d}z \mathrm{d}...
- 31
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56
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linear functions/hyperplanes vs. convex functions/convex sets in Hilbert space
The simplest Hahn-Banach extension theorem in Hilbert space $X$ avoids the use of the axiom of choice by virtue of the Riesz representation theorem. But what about the version of the theorem where the ...
- 1,461
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0
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105
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Spectrum of Dirac sequences
Let $\delta_n\in C^0_c(\mathbb{R})$ be a Dirac sequence approximating the Dirac delta "function" $\delta$ with support in $0\in \mathbb{R}$. Then, for each $n$ we have a compact operator $K_n:L^2(\...
- 99
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63
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Feller semigroups and fractional operators
Have Feller semigroups been used to investigate the properties of the Cauchy problem associated with the fractional Laplacian (just like they have been used to study local degenerate second order ...
user60665
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0
answers
273
views
Local "boundary comparison principle" for harmonic functions
Let $u$ be a positive solution of the elliptic equation $\mathcal Lu = 0$ on $B^+_1 \subset \mathbb{R}^n$ such that $u$ vanishes continuously on $\{x_n = 0\}$. To fix ideas, we may take $\mathcal L = ...
user123456
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117
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Harnack Inequality for uniformly elliptic PDE via constructing a singularity
I am trying to prove a Harnack inequality for a nonnegative subsolution $u \in H^1(B_2)$ to the PDE $\text{div}(A Du) \ge 0$, where $A = A(x)$ is uniformly elliptic. The proof outline I am following ...
- 1
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170
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Strong continuity (weak to strong) of $\langle Au,v\rangle=\int u^3 v dx$
I am currently trying to figure out the following. If I consider the Sobolev space $W^{1,p}_0$ is it possible to show that the operator given by
$$\langle Au,v\rangle=\int u^3 v dx$$
is strongly (weak ...
- 101
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115
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If two spheres are isometric, does there exist a bijective isometry $T:S\to S$ with $\|Tu-\alpha Tv\|_Y \leq \|u-\alpha v\|_X$ for all $\alpha>0?$
Let
$$(S,\|\cdot\|) = \{(x,y)\in \mathbb{R}^2: \|(x,y)\| =1\},$$
that is, $S$ is the collection of all norm one vectors in $\mathbb{R}^2$ with respect to the norm $\|\cdot\|.$
Question: Let $\|\...
- 623
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119
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Holder-Sobolev type inequality
Let $U$ be a bounded subset of $\mathbb{R}^n$. Let $p>n$. Let $W^{2,1}(U_T)$ be the Banach space of functions $u:U\times[0,T]\rightarrow\mathbb{R}$ with the norm $\|u\|_{W^{2,1}(U_T)}=\sum_{2s+|\...
- 307
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answers
113
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Conditions for the embedding of the space $L^\infty(I, W^{1,2}(U))$ into $L^\infty(I \times U)$
Let $I$ be a compact interval of $\mathbb{R}$ and $U$ be a bounded subset of $\mathbb{R}^2$.
If $f \in L^\infty(I, W^{1,2}(U))$, what (non-trivial) condition ($L^p$-estimate on $f$ or decay-like ...
user123457
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169
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Is there an option to handle Neumann-series when it diverges? (using infinite-sized Carleman matrices)
(I asked this in MSE but did not find resonance, there is also a relation to an older discussion here on summability see here and a followup formulating an $\text{ais}()$ already here)
...
- 5,342
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56
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is this explicit linear operator hypo-elliptic
Consider an operator of the form
$$L(\phi):=\Delta \phi + \gamma \phi_{rr}$$ here the $r$ denotes derivative with respect to the radial variable (we are in $ R^N$ say where $N \ge 3$).
I am ...
- 1,385
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0
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171
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What functions can one try employing to fit an apparently doubly-periodic real function over $[0,1]$?
I have a cosine-like data curve over $x \in [0,1]$ that I can rather well-fit by
a function of the form $a \cos{2 \pi x} +b$. Although good, the fit is still lacking,
in that the residuals from the ...
- 723
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0
answers
101
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Reference Request: Egoroff Theorem for nets
Does there exist a generalization of Egoroff theorem for nets instead of sequences of functions?
- 5,405
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75
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Dense Egoroff theorem
Suppose that $f_n:X\rightarrow V$ is a sequence of continuous functions from a compact metric space $X$ to a Banach space $V$ and let $\mu$ be a Radon measure on $X$ and $\epsilon>0$ be given.
...
- 5,405
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79
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Iterative methods for minimizing sequences
Let $\mathbb{X}$ be a Banach space equipped with some norm $||\cdot||_\mathbb{X}$ and $F:\mathbb{X}\to\mathbb{R}$ be some linear functional. Suppose we are given a set $A\subseteq\mathbb{X}$ which is ...
- 364
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87
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Uniform convergence in Hadamard derivatives
Let $T\colon X \to Y$ be a nonlinear operator between Hilbert spaces which is Lipschitz and is Hadamard differentiable. It satisfies
$$T(x+th)=T(x) + tT'(x)(h) + r(t)$$
where $r(t)=r(t,x,h)$ is the ...
- 73
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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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72
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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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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
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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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324
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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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70
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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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299
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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
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55
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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 $$\lim_{x\rightarrow \infty} f(x)=0~\mbox{and}~\lim_{x\...
- 537
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votes
0
answers
194
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Estimate of a Sobolev function after a change of variables
Let $n>0$ and $\Omega$ be a bounded domain of $\mathbb{R}^n$. Consider a smooth enough mapping $\Phi$, from $\Omega$ into $\Phi(\Omega)\subset\mathbb{R}^n$, that is orientation-preserving and ...
- 143
0
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0
answers
89
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Hausdorff methods of summation
From the book of Boss "Classical and modern methods in summability":
"The class of Hausdorff methods includes the Hölder, Cesaro and Euler methods. A large number of other matrix methods which play ...
- 109
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0
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90
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criterions for polar set of Feller processes
Suppose $X_t$ is the solution to
$$
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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0
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65
views
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
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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 ...
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