**-1**

votes

**0**answers

35 views

### Closure property of completely monotone functions [on hold]

A $C^{\infty}$ function $f(x_1, \dots, x_n)$ defined on $(0,\infty)^n$ is said to be completely monotone if
$$
(-1)^{k}\frac{\partial^k f}{\partial x_{i_1} \cdots \partial x_{i_k}} \geq 0
$$
...

**-1**

votes

**0**answers

12 views

### Rigorious formulation of approximation of integral as area of a square and its radius of convergence [migrated]

We know that the taylor expansion of
$$\left(\int_{t}^{t+\Delta t}a(t')dt'\right) = a(t)\Delta t + \frac{1}{2}\frac{da}{dt}(t) \Delta t^2 + \frac{1}{3!}\frac{d^2a}{dt^2}(t) \Delta t^3 + \frac{1}{4!}\...

**1**

vote

**1**answer

36 views

### Polynomial with subset of critical points and values prescribed

Motivated by this question I am motivated to pose the following question:
Given $k$ points $(x_1, y_1), \ldots (x_k, y_k)$ with (WLOG) $x_i < x_{i+1}$, can we find a polynomial $p(x)\in\Bbb R[x]$ ...

**1**

vote

**1**answer

63 views

### Every $W^{1,p}$ has a representative in ACL

Let $\Omega:=(0,1)^n$ and define $ACL_i(\Omega)$ as the set of all Borel functions $u:\Omega\to\mathbb{R}$ such that
$$ t\mapsto u(x_1,\dots,x_{i-1},t,x_{i+1},\dots,x_n) $$
is $AC$ for a.e. $(x_1,\...

**0**

votes

**0**answers

41 views

### Questions about the regularity of the solution of the heat equation in a bounded domain [closed]

I have questions about the proof of the following theorem:
Theorem 8 (Smoothness). Suppose $u\in C^2_1(U_T)$ solves the heat equation in $U_T$. Then $u\in C^\infty(U_T)$
Here is the statement and ...

**0**

votes

**1**answer

98 views

### Doubling metrics, doubling measures, Lebesgue density

As stated in this question,
Lebesgue differentiation theorem holds on locally doubling space?
and proved here,
http://www.math.uiuc.edu/~tyson/595f15lecture2.pdf
the Lebesgue differentiation theorem (...

**0**

votes

**1**answer

80 views

### A uniform Lebesgue density theorem

The Lebesgue density theorem in $\mathbb{R}^n$ may be stated as follows. For a Lebesgue-measurable $A\subseteq\mathbb{R}$ and $r>0, x\in\mathbb{R}^n$, define
$$ \chi_{A,r}(x)=\frac{\mu(A\cap B_r(x))...

**1**

vote

**0**answers

45 views

### BV functions with values in metric space

$
\newcommand{\IR}{\mathbb{R}}
\newcommand{\IN}{\mathbb{N}}
\newcommand{\supp}{\operatorname{supp}}
\newcommand{\divergence}{\operatorname{div}}
\newcommand{\Lip}{\operatorname{Lip}}
$
Let $E$ be a ...

**1**

vote

**1**answer

111 views

### Global Poincaré type estimate

For simplicity let us assume we are considering $\mathbb{R}^3$. Let us define the weighted Sobolev norm $\| u \|^2_{L^2_{\alpha}}= \int_{\mathbb{R}^3} |u|^2 \langle x\rangle^{\alpha}$ where $\langle x ...

**0**

votes

**0**answers

44 views

### Global Harmonic Oscillator

My question essentially is how to find the appropriate functional space to study uniqueness of solutions to a specific pde.
Consider the following pde in three dimensions globally:
$ -\tau^2 \...

**3**

votes

**0**answers

52 views

### Convolution of decaying polynomials [closed]

I conjecture that if the functions $f$, $g$ defined on $\mathbb{R}^n$ satisfying
$$|f(x)| ≤ A(1+|x|)^{−M}, \quad |g(x)| ≤ B(1+|x|)^{−N}$$for some
$M$, $N > n$, then$$|(f * g)(x)| ≤ ABC(1+|x|)^{−L},$...

**0**

votes

**0**answers

92 views

### The eigenfunction of modified $1$-laplace equation?

Let $\Omega\subset \mathbb R^2$ be open bounded with smooth boundary. It is well known that the laplace equation
$$
-\Delta u=0
$$
has a set of eigenvalues $0<\lambda_1<\lambda_2\leq\lambda_3<...

**10**

votes

**0**answers

213 views

### Integrability property of polynomials in several variables

This might be very trivial, or not.
Let $p\colon\mathbb{R}^n\to \mathbb{R}$ be a polynomial of even degree, at most $n-2$. Assume that $p(x)\leq 0$ for any $x\in\mathbb{R}^n$. Assume that there ...

**2**

votes

**0**answers

46 views

### level sets portrait near a critical point

Let $f:\mathbb{R}^{2}\rightarrow\mathbb{R}$ be a smooth ($C^{\infty}$) function and $O$ be an
isolated critical point of $f$. I am looking at the local level sets diagram
near $O$ from topological ...

**2**

votes

**1**answer

307 views

### An irresistible inequality

The following occurred while working on some research project. Since the methods of proof I used were lengthy, I wish to see a skillful or insightful approach (perhaps even conceptual). Anyhow, here ...

**5**

votes

**0**answers

99 views

### For $f$ a polynomial, does strict convexity of $\log f(e^s)$ imply that the second derviative of $\log f(e^s)$ has no zeros?

Let $f(t)$ be a monic real polynomial such that $f(t) > 0$ for all $t \ge 0$. Suppose that $\log f(e^x)$ is strictly convex on $\mathbb{R}$, i.e.
$f(s^2) \cdot f(t^2) > f(st)^2$
for all $s, t \...

**1**

vote

**0**answers

226 views

### Prove this function is increasing

I'm stuck in showing that the following function is increasing over the domain $\left[0,\hat{b}\right]$:
\begin{eqnarray}
\Pi\left(z\right) & = & \int_{0}^{\phi\left(z\right)}\int_{x}^{\bar{x}...

**2**

votes

**0**answers

76 views

### If $f_j\to f$ in $L^1(\Bbb R^n)$ then $Tf_j\to Tf$ in $L^{1,\infty}(\Bbb R^n)$

Let's define $A:=\{f\in L^1(\Bbb R^n)\cap L^2(\Bbb R^n)\;:\;f\;\mbox{has compact support}\}$. So $A$ is dense in $L^1(\Bbb R^n)$.
Given then $f\in L^1(\Bbb R^n)$; by density there exists $\{f_j\}_j\...

**2**

votes

**1**answer

105 views

### estimation of a vector-function

Let $x(t)\in C^1(\mathbb{R}_+,\mathbb{R}^m)$ be a vector-function such that
1) $\|x(t)\|+\|\dot x(t)\|\to 0$ as $t\to\infty$ and
2) for some real $c_1>0$ and all $t>0$ one has $\|x(t)\|\le c_1\...

**2**

votes

**0**answers

72 views

### Imposing boundary conditions and self-similarity on a PDE

This question is an exact duplicate of the question
Imposing boundary conditions AND self-similarity on a PDE
posted by Stan Corey Carter on math.stackexchange.com.
I have a PDE in the ...

**0**

votes

**0**answers

72 views

### Differntiability of Distance to a CLosed Convex Set

Let $A$ be a closed convex set in Banach space $( \mathbb{R}^n, \| \cdot\| )$. For any $\mathbf{x} \in \mathbb{R}^n$, define $$Ｐ_{A}(\mathbf{x}) = \arg\min_{\mathbf{y}\in A} \| \mathbf{x} - \mathbf{y} ...

**0**

votes

**0**answers

20 views

### Splitting the region and estimating fractional Sobolev norms

x-post from math.stackexchange (http://math.stackexchange.com/q/1836766/349671), since the question arose from reading through a scientific paper:
I've been reading the paper "On the Bourgain, Brezis,...

**1**

vote

**0**answers

149 views

### A certain measure on Banach algebras

According to the comments of Nate Eldredge I did revise the question. In particular I change "$C^{*}$ algebras" to "Banach algebras".
Is there a reference who introduce the following measure on ...

**1**

vote

**0**answers

31 views

### The jump set of $SBV$ function for different value of parameter in image denoising problem

The classical Mumford-Shah image denoisng problem study the minimizer of the following functional, for each $\alpha>0$ where $\Omega\subset \mathbb R^2$ is open bounded with sommth boundary,
$$
u_\...

**4**

votes

**2**answers

222 views

### Variation of Radon transform for probability measures on $\mathbb C$

Let $\mu$ be a probability measure on $\mathbb C$. For $z \in \mathbb C$, let $$f^z \colon \mathbb C \to \mathbb R_{\geq 0}$$ be the function $f^z(\lambda) = |\lambda - z|$. Consider now the family $(\...

**4**

votes

**0**answers

66 views

### I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection

I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...

**7**

votes

**1**answer

276 views

### A question about composition of functions

Recently, I heard this question: are there two functions $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ such that $f\circ g$ is strictly increasing on $\mathbb{R}$ and $g\circ f$ is ...

**1**

vote

**0**answers

77 views

### Can we always extract a proper Hausdorff measurable subset from a Hausdorff measurable set?

I also put this question on MSE here
Let $\Gamma\subset \Omega\subset \mathbb R^N$ be such that $\mathcal H^{N-1}(\Gamma)<+\infty$ (this also implise that $\Gamma$ is Hausdorff measurable).
Let $\...

**0**

votes

**0**answers

38 views

### Inner Products vs Discretizations of Functions

Let $f:(0,1)\rightarrow [0,1]$ be a $\mathcal{C}^{\infty}$ function.
We say that a function $D_r^f:\{1,...,r\}\rightarrow [0,1]$ is an $r$-discretization of $f$ if $$D_r^f(j) = \frac{1}{|a(j)-b(j)|}\...

**0**

votes

**0**answers

20 views

### Measure of sub level of a torsion energy

Given a domain $\Omega$ (not necessarily open, but bounded. We can take quasi open domain). And let $u_{\Omega}$ be the minimizer of the torsion energy,
$$
\int_{\Omega}|\nabla u |^2\, -\, \int_{\...

**0**

votes

**1**answer

56 views

### characterization of normality by selection theorem

The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $...

**1**

vote

**1**answer

111 views

### Convergence of Discrete Geodesic

Let $M$ be a Riemmanian manifold, $p\in M$ and $V\in T_p(M)$.
Suppose $f^{-1}:U_p \mapsto U$ is a diffeomorphism of a neighborhood of p to an open subset of $\mathbb{R}$ and define the sequence:
\...

**3**

votes

**0**answers

44 views

### Does there exist $\lambda_{\sigma(1)}$ such that $\mu(A\cap\{\lambda_{\sigma(1)}\neq0\})>0$?

Let $(\mathcal F,\Omega,\mu)$ be a measure space and $A\subseteq\Omega$ such that $\mu(A)>0$. Let $L^0$ be the space of all measurable functions.
We say $X_1,\ldots,X_k\in(L^0)^d=\prod_{k=1}^dL^0$...

**6**

votes

**1**answer

220 views

### Density of convolution

Let $\{X_i\}$ be i.i.d random variables uniform on a measurable, symmetric set $A$ contained in $[-1,1]$. Let $g_{n}$ be density of $X_1+\ldots + X_n$.
Question (general): Is there any non-trivial ...

**2**

votes

**0**answers

34 views

### Well-definedness for a singular integral

Let $T_\alpha$ be a singular integral operator defined by
$$
T_{\alpha}[f](t):=\int_{0}^{t}\frac{f(t)-f(s)}{(t-s)^{\alpha+1}}ds
$$
for continuous functions $f$ on $[0,\infty)$ and $0<\alpha<1$.
...

**2**

votes

**2**answers

90 views

### Continuous upper envelope of upper semicontinuous function

Let $u$ be a upper semicontinuous function on a compact set $K$ in $\mathbb R^d$. Define a space of continuous function dominating $u$ by
$$A = \{\phi \in C(K): \phi \ge u\}.$$
[Q.] Is the following ...

**0**

votes

**1**answer

133 views

### Reproducing Kernel Hilbert Spaces with positive kernels

In my research I'm dealing with the following question.
Let $E$ set, $K:E \times E \to \mathbb R$ a positive type function, and $\mathcal H := \mathcal H(1+K)$ (in the sense of the Moore theorem). ...

**8**

votes

**1**answer

257 views

### Interpolation between $L_1^0$ and $L_2^0$

Let $L_p^0$ be the mean zero functions in $L_p(G)$, where, say, $G$ is an infinite compact group endowed with normalized Haar measure. Suppose that $T$ is a bounded linear operator on $L_1$ that maps $...

**2**

votes

**1**answer

82 views

### Step 2 of The Strichartz's Estimates in Cazenave's Book

My question is from Cazenave's book "Semilinear Schrödinger Equation", page 35. I am stuck with Step 2 of the Strichartz's estimates.
The book says that $||\Phi_f(t)||_{L^2}^2=\left(\int_0^t \...

**1**

vote

**0**answers

33 views

### Compute Mixed Volume with Respect to Some Regular Sets

Let $( \mathbb{R}^n, \mathcal{B}, \gamma)$ be a measure space where $\mathcal{B}$ is the Borel sigma algebra and $\gamma$ is a continuous measure. For $A, B\in \mathcal{B}$ that are convex, the mixed ...

**4**

votes

**1**answer

74 views

### weak convergence in $H_0^1$ and strong convergence in $L^2$

I'm reading a hand-waving argument in a proof of Chapter 7 of the Navier-Stokes Equations by Constantin and Foias. I would like to know if I understand it correctly.
Let $\Omega\subset{\mathbb{R}^n}$ ...

**0**

votes

**0**answers

41 views

### Equivalent Definitions of the Gaussian Surface Measure for Regular Sets

I wonder if the following definitions of the Gaussian surface measure are equivalent.
First, let $\mathbb{R}^n$ be the Euclidean space and $A \subseteq \mathbb{R}^n$ be a sufficiently regular set, e....

**5**

votes

**3**answers

174 views

### Morrey's inequality for Sobolev spaces of fractional order

Let $H^s(\mathbb T)$, where $s\in\mathbb R$, be the space of $2\pi$-periodic functions (or distributions), $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx}$, such that
$$
\|u\|_{H^s}^2=\sum_{k\...

**5**

votes

**0**answers

105 views

### Generalized singular numbers and the Haagerup $L^p$ spaces

Let $M$ be a semi-finite von Neumann algebra with a trace $\tau$.Let $S(M)$ be the algebra of all affiliated operators measurable with respect to $M$.
The $L^p$ norm on $M$ is given by
\begin{...

**0**

votes

**0**answers

110 views

### Smooth Approximation of Indicator Function of Convex Sets in $\mathbb{R}^n$

Let $( \mathbb{R}^n, \| \cdot \|_P)$ be the $n$-dimensional Euclidean space equipped with $\ell_p$-norm $\| \cdot \|_p$ for some $p\in [1, + \infty]$. Let $A$ be a convex set in $\mathbb{R}^n$ and ...

**0**

votes

**1**answer

132 views

### Methods to tackle this series and get to the limit?

Take a look at the averaging sum
$$\frac{\pi}{n}\sum_{k=1}^n\;\exp{(-\sin\theta_k)}\cdot \sin(\theta_k +\cos\theta_k)\, \qquad\text{where }\;\theta_k=(2k-1)\frac{\pi}{2n}$$
depending on $n\in\...

**34**

votes

**1**answer

3k views

### Is the following identity true?

Calculation suggests the following identity:
$$
\lim_{n\to \infty}\sum_{k=1}^{n}\frac{(-1)^k}{k}\sum_{j=1}^k\frac{1}{2j-1}=\frac{1-\sqrt{5}}{2}.
$$
I have verified this identity for $n$ up to $5000$ ...

**6**

votes

**1**answer

216 views

### bounding derivative of a sequence

I've been banging my head against the wall on this one ... define a sequence of polynomials $q_n$ by $q_0 = 0$ and $$q_{n+1} = q_n + .5(t^2 - q_n^2).$$ If $q_n \leq t$ on $[0,1]$ then $$.5(t^2 - q_n^2)...

**10**

votes

**1**answer

257 views

### Sequence of nested sets in $[0, 1]$ with bound on gaps

What is the best possible $\epsilon$ and sequence $(a_n)_{n = 1}^\infty \subset [0, 1]$ we can find such that
$$
d_{N}:=\sup_{x\in [0,1]}\inf_{n=1}^N |x-a_n|\leq \frac{1+\epsilon}{N}
$$
for all $N\in ...

**4**

votes

**0**answers

72 views

### Equivalence between Sobolev norm and Sobolev-Slobodeckij norm for $W^{s,p}(\Omega)$ when $s$ is an integer

Take $W^{1,2} = H^1$ for example. If we still use Slobodeckij norm (which is normally defined for a fractional Sobolev space) as follows for a $u\in H^1(\Omega)$ with the exponent being in integer:
$$
...