All Questions
3,859 questions with no upvoted or accepted answers
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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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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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273
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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 $\|\...
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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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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)
...
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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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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).$$
Assume that $V$ is a smooth function and $V(x)\to +\...
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72
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weakly amenable weighted sequence algebras
Let $v=(v_n)_{n\in\mathbb{N}}$ be a positive weight with $\inf_nv_n>0$ (for convenience we may take $v_n\geqslant1$). Then $\ell_{\infty}(v)$ is a Banach algebra with coordinate-wise multiplication....
- 375
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161
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Topologies corresponding to norm, SOT and WOT under duality
This is a question from MSE which has not received any attention so far.
Let $X$ be a Banach space with norm dual $X'$. (I am mostly interested in the case $X = \ell^1$.)
For a linear mapping $T : X \...
- 1,773
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60
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Solution of a functional equation with cosine transform
What are the functions verifying:
$$\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).$$
I ...
- 83
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70
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A question about an irreducible ultra-power
Let $A$ be a Banach algebra and $E$ be an irreducible Banach $A$-module. Is there a countably incomplete ultra filter $\mathcal U$ on $\mathbb N$, the set of natural numbers, such that the ultra power ...
- 2,118
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467
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Is the product of two Sobolev functions in L^p?
Assume that $f\in W^{\alpha-1,p}(R^n)$ with $0<\alpha<1$ and $p>2n/\alpha$.
Given another function $ g\in W^{\beta,p}(R^n)$ with $\beta>0$.
Under what conditions on $\beta$ can we get ...
- 411
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299
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When convolution with exponential kernel is bounded
Let $g(t)=e^{-\omega t}$, $\omega>0$. What is, in terms of well-known function spaces, the space $X$, $L_{loc}^2(0,\infty)\subset X$, of all functions $f:\mathbb{R}^+\to \mathbb{R}^+$, satisfying
$...
- 395
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55
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Continuity of a composite function
Let $n=2$ or 3 and let $\Omega$ be a bounded domain of $\mathbb{R}^n$. Let $T>0$ and $f \in L^2([0,T],H^1(\mathbb{R}^n))$.
Is the mapping
\begin{equation}
\begin{array}{rcl}
C^0([0,T],C^1(\bar{\...
- 143
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58
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Dense integer translates of a real-valued function with unequal limits at infinity
This is a follow up on a Previous question. Let $W$ be the space of continuous functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $$\lim_{x\rightarrow \infty} f(x)=0~\mbox{and}~\lim_{x\...
- 537
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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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76
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An ideal in Fourier–Stieltjes algebras $B(G)$
Let $G$ be a locally compact group and $R$ be any family of representations of $G$. Let $A_R(G)$ be the closed linear span in Fourier–Stieltjes algebras $B(G)$ of the coefficient functions of all
...
- 565
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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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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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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 ...
- 3,343
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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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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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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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152
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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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227
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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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90
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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
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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
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84
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Consequence of John and Nirenberg's lemma?
The lemma I'm referring to in the title is the following:
John and Nirenberg's lemma: Let $C_0 \subset \mathbf{R}^n$ a finite cube. Let $u \in L^1(C_0)$ and assume there exists a constant $k$ such ...
- 109
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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)\}.
$$
Are there any known embeddings? Ideally, I am looking for an embedding of the ...
- 3,388
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124
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Reference for the Hardy maximal function on the torus
I am searching for a reference for the (sharp) Hardy maximal function on the torus $\mathbb{T}^2:=\mathbb{R^2}/\mathbb{Z}^2$, for instance I would need result result of the following type : if $g\in H^...
- 3,425
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96
views
Does adding a compact operator change the symbol of a pseudodifferential operator?
Suppose $X$ is a non-compact manifold. Let $P$ be an order-$0$ pseudodifferential operator on $X$ and $f:L^2(X)\rightarrow L^2(X)$ a compact operator. I'm wondering:
1) Is $P + f$ always a ...
- 1,903
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0
answers
81
views
Differential operator and equivalence
Here is the problem:
I have a certain PDE and there is the nonlinear terme $h$, I have as data:
$f \in H_0^2(0,L)$,,,$g \in {H^1}(0,L)$ with ${g_x}(0) = {g_x}(L) = 0$
Now on consider the fnction $$h(...
- 617
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97
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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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0
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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
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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
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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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0
answers
58
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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
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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 ...
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