All Questions
1,533 questions with no upvoted or accepted answers
2
votes
0
answers
158
views
Estimate involving Besov norm
When reading some old notes of my advisor on interpolation spaces, I bumped into a problem I can't quite wrap my head around. Here are the details.
For $p\in(0,\infty)$ a $p$-variation semi-norm of a ...
- 421
2
votes
0
answers
100
views
What is the weak limit of $f_n \ \mathrm{sign}(f_n - 1)$ if $f_n \to f$ weakly in $L^p([0,1])$?
Let $f_n: [0,1] \to \mathbb R$ be a uniformly bounded sequence in $L^p$. Then there exists a subsequence such that $f_{n_k} \to f$ weakly in $L^p([0,1])$. What is the weak limit of the sequence of ...
- 217
2
votes
0
answers
42
views
Analysis of coefficients for quickly vanishing analytic vector field
Let $u = (u_1, u_2, u_3): \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a divergence-free analytic vector field for $n =3$ or $n =4$, i.e., $u_i : \mathbb{R}^n \rightarrow \mathbb{R}$ are analytic ...
- 749
2
votes
0
answers
85
views
Are a map with constant singular values and its inverse always conjugate through isometries?
Let $U \subseteq \mathbb R^2$ be open, connected and bounded, and let $0<\sigma_1<\sigma_2$ satisfy $\sigma_1 \sigma_2=1$.
Suppose that $f:U \to U$ is a diffeomorphism whose singular values (of $...
- 6,741
2
votes
0
answers
160
views
Approximation in fractional Sobolev space
Assume $\Omega\subset \Bbb R^d$ is Lipschitz open set. Let $p\geq 1$ and $0<s\leq 1/p$.
How to prove that $C_c^\infty(\Omega)$ is dense in $W^{s,p}(\Omega)$?
Recall that,
$$|u|^p_{W^{s,p}(\Omega)}= ...
- 1,101
2
votes
0
answers
196
views
Have you seen this PDE before?
Consider the second-order nonlinear PDE
$$(\partial_x\partial_y\varphi)\cdot\varphi = \partial_x\varphi\,\partial_y\varphi.$$
This PDE is solved by all ('separable') functions $\varphi\in C^2(\Omega)$ ...
- 463
2
votes
0
answers
186
views
Bounding the condition number of a matrix associated with an even symmetric positive definite function
Define a set $A = \{x_i/x_i\in\mathbb{R}^m, i = 1,2,3..n\}$. Let $f:\mathbb{R}^m\to(0,\infty)$ be an even symmetric positive definite function.
Let $D = [d_{i,j}]$ be an $n\times n$ matrix such that $...
- 698
2
votes
0
answers
190
views
What is the smallest dimension that allows finding $n$ points at distances $|x_i-x_j|^{\delta/2}$, where $0<\delta<1$, and $x \in \mathbb{R}^n$?
Let $x_1,\cdots,x_n \in \mathbb{R}$, are there $\xi_1,\cdots,\xi_n \in \mathbb{R}^s$, such that
$|x_i-x_j|^{\delta}=||\xi_i-\xi_j||^2$, $0<\delta<1$, what is the smallest $s$ to guarantee the ...
- 178
2
votes
0
answers
162
views
Conditions for absolute continuity in the Bochner-Schwartz theorem
Suppose that $f$ is a positive-definite Schwartz distribution, that is,
$$\langle\phi,f*\phi\rangle\geq0\qquad\text{for every }\phi\in C_0^\infty(\mathbb R^n).$$
By the Bochner-Schwartz theorem, there ...
- 891
2
votes
0
answers
176
views
Approximate identities: a converse question
Let $K(x)$ be a positive-valued function such that
$$\int_{-\infty}^{\infty} K(x) \ dx = 1$$
and let
$$K_{\lambda}(x) = \lambda K(\lambda x), \ \ \ (\lambda > 0);$$
that is to say, the family of ...
- 71
2
votes
0
answers
107
views
Proof that Littlewood-Paley vertical square function is NOT bounded on L^infinity
The classical heat semigroup on $\mathbb{R}$ is given by
$$
W_t f(x)=\frac{1}{t}\int_{\mathbb{R}}e^{-\pi (\frac{x-y}{t})^2}f(y)dy, \qquad t>0.
$$
Then the Littlewood-Paley vertical square ...
- 421
2
votes
0
answers
56
views
A question about Holder exponents of a function at different points in its domain
Suppose that $f(x)$ is continuous on $[0,1]$. We make an agreement that if there exists an interval $[a,b]\subseteq[0,1]$ including point $y$ such that $f(x)$ satisfies $\alpha$-Holder condition on $[...
- 419
2
votes
0
answers
65
views
Measure of the convex hull of a ball and a point
I need to prove the following statement:
Let $B_s(z)$ be a ball centered at $z$ of radius $s$ s.t. $0\not\in B_s(z)$. Moreover let $K_s(z)$ the convex hull of $\{0\}\cup B_s(z)$.
Then
$$ \...
- 73
2
votes
0
answers
77
views
Second derivative estimates for a subsolution of linear elliptic equation
Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$.
Note. Saying that $u$ is semiconvex is ...
2
votes
0
answers
92
views
First Dirichlet eigenvalue below second Neumann eigenvalue?
Let $\Omega$ be a bounded domain in $\mathbb R^n $ with smooth boundary.
I was wondering if there exist any known conditions on $\Omega$ such that the 1st Dirichlet eigenvalue of the (positive) ...
- 173
2
votes
0
answers
130
views
Partition of unity in $\mathbb{R}$ with additional conditions on the derivatives
Let $K\subseteq \mathbb{R}$ be locally compact without isolated points and $X$ an infinite dimentional Banach space. Then
$$C_{0}^{(1)}(K,X)=\{ f\in C_{0}(K,X): \text{$f$ is continuously ...
- 121
2
votes
0
answers
66
views
Properties of solution to Burger's equation using Cole-Hopf transformation
I am currently looking at a $1$D-Burger's equation defined by
\begin{equation} \label{ex burgers}
\left\{
\begin{array}{ll}
{} & \frac{\partial V_m}{\partial t} (t,x) = \frac{\sigma^2}{2} \...
- 357
2
votes
0
answers
189
views
Point wise convergence of Laplace transform and convergence of functions
Assume that functions $f_n(t), f(t)\in C_b(R_+)$. For every $\lambda >0$, we have
$$
\bigg|\int_0^\infty e^{-\lambda t}f_n(t)d t-\int_0^\infty e^{-\lambda t}f(t)d t\bigg|\leq C_\lambda n^{-1},
$$
...
- 411
2
votes
0
answers
370
views
Extension of subharmonic function: can someone explain the details?
In this paper we have the following situation on page 60. $E$ is a compact subset of $\mathbb{R}^\tau\cup\{\infty\}$ (one point compactification) for $\tau\geq2$, $M_0$ is a point in the boundary of $...
- 411
2
votes
0
answers
200
views
The collection of mean value abscissas in the Mean value theorem
The integral mean value theorem for continuous f on [0,b] and finite positive continuous measure $\mu$ we have
$$\frac{1}{\mu[a,b]}\int_{a}^{b}f(x)d\mu(x)=f(c)(*)$$
for at least one $c\in [a,b]$. We ...
- 5,474
2
votes
0
answers
93
views
Are locally integrable functions almost completely determined by their approximate modulus of continuity?
This is a follow up to this question, which was answered in the affirmative: Are continuous functions almost completely determined by their modulus of continuity?
Note: We do not identify functions ...
- 2,069
2
votes
0
answers
97
views
Prove that this integral operator is onto
Let us consider the operator $T$ defined by$$\eqalign{
& T:{L^2}((a,b) \times (c,d)) \to {L^2}((c,d)) \cr
& Tf(s,x) \mapsto \int\limits_{q(x)}^{p(x)} {f(\alpha (s,x),s)ds} \cr} $$
where ...
- 617
2
votes
0
answers
55
views
Approximate a one-form with nowhere vanishing one-forms having bounded Laplacians
This is a follow-up question of this one.
Let $\mathbb{D}^2$ be the closed two-dimensional unit disk (endowed with some smooth Riemannian metric), and let $\sigma \in \Omega^1(\mathbb{D}^2)$ be a ...
- 6,741
2
votes
0
answers
146
views
Prove the equicontinuity of a maximizing sequence
Let $X$ be a compact subset of $\mathbb{R}$ and $c(x_1,x_2,x_3,x_4)$ be a fixed bounded continuous functions on $X^4$. Assume $\mu,\nu$ are probability measures on $X^2$, and $\mu\otimes\nu$ is the ...
- 323
2
votes
0
answers
77
views
Homomorphism of composition to additive structure
Consider the following topological groups
$\operatorname{Homeo}(\mathbb{R}^d)$ be the topological group of all homeomorphism from $\mathbb{R}^d$ onto itself; equipped with the compact-open topology (...
- 5,405
2
votes
0
answers
68
views
Sequence of harmonic measure
There is a well-known result stating that if $\mu_{n}$ is a sequence of uniformly bounded measures on a compact set $E$ of $\mathbb{R}^{m}$, then there is a subsequence $\mu_{n_{j}}$ that converges ...
- 411
2
votes
0
answers
108
views
How to evaluate an interpolation method, in terms of converging to the underlying function, as data points go to infinity?
I have an interpolation method, which takes function $f$ values at any given finite number $N$ of points in the domain and interpolate to get a function $f_{int}$.
I want to do some analysis on how ...
- 698
2
votes
0
answers
1k
views
bounds on derivatives of mollifiers/mollified functions
Consider the standard mollifier
$$
\phi(x) = C\exp\left(-\frac{1}{1-x^2}\right), \quad -1<x<1.
$$
such that $\int\phi(x) = 1$.
Let $f(x) = |x|$ and consider the convolution $f\ast \phi$. I am ...
- 640
2
votes
0
answers
171
views
How to use Stein-Tomas theorem to check to following inequality?
Recently, I am reading Rodnianski & Schlag
Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. In lemma 3.2, R&S said that by using Stein-Tomas theorem ...
- 429
2
votes
0
answers
218
views
On a special continuous extension of Sylvester's Theorem
Notations. Denote by $S^{-}\left\{a_0,a_1,\ldots,a_m\right\}$ the number of strict sign changes in the indicated sequence $\left\{a_j\right\}_{j=0}^{m}$ of real numbers (i.e., when counting the sign ...
- 31
2
votes
0
answers
123
views
GUE, tau-function of Painlevé II, and an article of Forrester-Witte
Let $ \mu $ be the Gaussian measure $ d\mu(x) = e^{-x^2/2} \frac{dx}{\sqrt{2\pi} } $. I am interested in the following random matrix integral defined for all $ s \in \mathbb{R} $, $ N \geq 1 $ and $ a ...
- 593
2
votes
0
answers
216
views
Is $f$ defined by $f(x) = t\mapsto G(t , x(t))$ differentiable?
Let us consider $X = AC([0 , 1] , \mathbb{R}^n)$, and $Y=L^{1} ([0,1] , \mathbb{R}^n )$ as Banach spaces with their usual norms. Let $G: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ be a ...
- 369
2
votes
0
answers
232
views
Is an orthogonal projection of a Lipschitz domain still a Lipschitz domain?
Let $\mathcal{X}\subseteq\mathbf{R}^n$ be a Lipschitz domain, i.e., for each $x\in\partial\mathcal{X}$, there exists a radius $r_x>0$ and a Lipschitz continuous function $F^x:\mathbf{R}^{n-1}\to\...
- 21
2
votes
0
answers
103
views
Buridan's principle in computable analysis
In (Lamport, 2012), Lamport proposes the principle
A discrete decision based upon an input
having a continuous range of values cannot be made within a bounded length of time.
I think it could be ...
2
votes
0
answers
241
views
Monotone approximation of continuous function by Lipschitz function
Let $X$ be a normed space, and $f\in C_b(X,\mathbb{R})$. How can I construct a sequence $\varphi_k,\,k\geq 1$ of bounded Lipschitz functions with $\varphi_k\downarrow f$ pointwise? Thanks
- 139
2
votes
0
answers
208
views
On the difference of conditional differential entropy of two correlated random variables
Problem Definition
Let $\mathbf{G}$ and $\mathbf{S}$ be jointly distributed random variables where
$\mathbf{S}$ is continuous and is related to $\mathbf{G}$ through a conditional pdf $f(s|g)$ defined ...
- 31
2
votes
0
answers
249
views
Calculating $\int_1^{\infty}\frac{\operatorname{ali}(x)}{x^3}dx$, where $\operatorname{ali}(x)$ is the inverse function of the logarithmic integral
It is well-known that we can compute the closed-form of the integrals $$\int_1^{\infty}\frac{\log x}{x^2}dx$$
and $$\int_1^{\infty}\frac{\operatorname{li} (x)}{x^3}dx,$$
where $\operatorname{li} (x)$ ...
2
votes
0
answers
90
views
Invariance under diffeomorphisms of the Hajlasz-Sobolev spaces
In this post it was shown that if $\Omega$ and $\Omega'$ are diffeomorphic non-empty open domains in some Euclidean space then the corresponding local Sobolev spaces are diffeomorphic with ...
- 5,405
2
votes
0
answers
337
views
fractional compact Sobolev embedding on lipschitz domain
Let $\Omega$ be a bounded Lipschitz domain. It is well known that $H^1(\Omega)$ can be compactly embedded into $L^2(\Omega)$. I also found references for the compact embedding $H^\delta(\Omega)\...
- 21
2
votes
0
answers
69
views
Extension of a $\delta$-subharmonic function that is subharmonic on a reduced domain
Suppose $B$ is a ball in $\mathbb{R}^{m}$ and $u$ and $s$ are subharmonic on $B$. Suppose there is a closed subset $F$ of the closure of $B$ with no interior such that $v=u-s$ is subharmonic on $B\...
- 411
2
votes
0
answers
109
views
Average number of pieces of a random piecewise-linear function
Let $I$ be a (nonempty) compact interval in $\mathbb R$ and $a_1,b_1,\ldots,a_L,b_L \in \mathbb R$. Let $\varphi$ be a piecewise function with $T \ge 2$ pieces(for example $T=2$ for the choice $\...
- 6,853
2
votes
0
answers
197
views
Orthogonality relation in $L^2$ implying periodicity
Let $\theta(t)$ and $\phi(t)$ be two real $C^1$ functions $[0,2\pi]\rightarrow \mathbb{R}$. Let us assume $\theta$ has the properties
$$
\int_0^{2\pi} e^{i\theta(t)} dt=0.
$$
Geometrically this means ...
- 405
2
votes
0
answers
488
views
Karamata's proof of Hardy-Littlewood Tauberian theorem
I understand Karamata's proof of the Hardy-Littlewood Tauberian theorem as in http://individual.utoronto.ca/jordanbell/notes/karamata.pdf, but what on earth is the motivation behind Lemma 4 - i.e, ...
- 353
2
votes
0
answers
74
views
Can we approximate this matrix field with an invertible matrix field?
Let $\mathbb{D}^2=\{ x \in \mathbb{R}^2 \, | \, |x| \le 1\}$ be the closed unit disk, and set
$$\begin{equation*}
A(x,y)=\left(
\begin{array}{cc}
x & -y \\
y & x
\end{array} \right)
\end{...
- 6,741
2
votes
0
answers
70
views
Essentially anti-Cauchy functions
Call a function $f: \mathbb R+ \to \mathbb R$ essentially $C^\infty$ if there exists a sequence $f_n$ $(n \geq 0)$ such that each $f_n$ is differentiable a.e., $f_0 = f$ a.e., and $f_n’$ is equal to $...
- 2,069
2
votes
0
answers
90
views
A question about how to use the convexity condition?
At page 5 (125), seven line after the Proof of Theorem 2.2(i), of the following article.
THE HEAT EQUATION WITH A SINGULAR POTENTIAL
the authors say that since $p$ is convex, we can deduce that
$$ \...
- 371
2
votes
0
answers
240
views
Discrete Sobolev embedding
It is true in one dimension that $H^1$ is continuously embedded in $L^{\infty}.$
Now, consider a compact interval $[0,1]$ with a partition $I_n:=([m/n,(m+1)/n])_{m \in \left\{0,...,n-1 \right\}}$ and ...
2
votes
0
answers
77
views
How we can do the derivative for this equation w.r.t.to time t>0
Let $x\in[0,L]$ and consider the following equation,
$$\varepsilon \left( t \right)=\frac{1}{2}\int_{0}^{L}{({{\rho }_{1}}{{\left| {{\varphi }_{t}} \right|}^{2}}+{{\rho }_{2}}{{\left| {{\varphi }_{t}} ...
- 199
2
votes
0
answers
269
views
Extending Green's theorem from very special regions to more general regions
Green's theorem
Let $C$ be a positively oriented and consists of a finite union of disjoint,piecewise smooth simple closed curve in a plane, and let $D$ be the region bounded by $C$. If $P$ and $Q$ ...
- 151
2
votes
0
answers
384
views
Is the Fourier transform of a measurable function as a tempered distribution necessarily a complex Borel measure?
Let $u\in\mathcal{S}'(\mathbb{R}^n)$. Suppose that $u$ is also a measurable function on $\mathbb{R}^n$. Is it true that the Fourier transform $\hat{u}$ as a tempered distribution is always a complex ...
- 183