**3**

votes

**2**answers

76 views

### Abstract ODE; PDE; uniqueness of solution

I have a somewhat vague question regarding an abstract ODE in a Banach space.
Suppose $A:D(A) \subset X \rightarrow X$ is some linear operator (let's assume it's closed) and maybe add some other ...

**2**

votes

**1**answer

50 views

### Coercivity for functional and complete orthonormal system

Consider with $\rho \in W^{1,2}([0,\pi])$ the following functional
$$J(\rho)=\frac{1}{2}\int_{0}^{\pi}{\rho^2\,dx}$$
I know that in the $L^{2}([0,\pi])$ the coercivity condition is satisfied, but i'm ...

**-3**

votes

**0**answers

31 views

### Can you give me an example of signed distance function [on hold]

Can you give me an example of signed distance function？
Thank you！

**0**

votes

**0**answers

117 views

### About the proof of the Morse lemma

In the Chang's book "Infinite dimensional Morse theory and multiple solution problems"
the Morse lemma is a special case of the spliting lemma but i dont understand in the proof why ...

**2**

votes

**0**answers

39 views

### Almost-Monotone Kernels - Examples and/or Covering Theorems

I am looking for examples (or, if it exists, a theory) of almost-monotone kernels. First, a bit of notation.
Recall that if $(\leq, \Omega)$ is a partially ordered set, then the set of measures ...

**14**

votes

**2**answers

361 views

### What can be said about the Fourier transforms of characteristic functions?

What can be said about the Fourier transform of the characteristic function $1_A$, where $A\subset \mathbb{R}^n$ is of finite Lebesgue measure? In particular,
What properties are common to ...

**-1**

votes

**0**answers

52 views

### Measure generated by Semigroup $\exp[-t|p|]$

I am studying Ingrid Daubechies' paper 'An Uncertainty Principle for Fermions with Generalized
Kinetic Energy'.
On page 514, the measure $\mu_{x,y;t}$ is introduced as generated by the semigroup ...

**0**

votes

**0**answers

47 views

### The proximality of low rank function approximation

The paper "Best $n$-Dimensional approximation to sets of functions" by A. L. Brown in 1964 gave a negative answer to the following question:
Q1: Is there for a given integer $n$ always a best ...

**0**

votes

**0**answers

43 views

### Find an analytic function [migrated]

Is it possible to find an analytic expression for a smooth, continuous, single-variable function $y=f(x)$ such that:
1) $f(0) = 0$
2) $f(x_1) = y_1$
3) $f(x) > 0, \forall x>0$
4) ...

**0**

votes

**1**answer

179 views

### Theorem with an example [on hold]

i have this theorem
in the paper they gives an example:
but here $H_1$ is not satisfied !
How to correct it please?

**7**

votes

**1**answer

275 views

### About the convergence rate for an approximation to the heat kernel

Let $G(t,x)$ be the heat kernel
$$
G(t,x)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}, \quad t>0, \:x\in\mathbb{R}.
$$
Here is one approximation to $G(t,x)$:
$$
G_\epsilon(t,x)=e^{-t/\epsilon} ...

**3**

votes

**1**answer

115 views

### Characterization(?) of coersive(?) elements in the special linear group

Take your favorite matrix norm $\|\bullet\|$ (my favorite is the Frobenius norm $\|A\| = \sqrt{\operatorname{tr} A A^t}$). Now consider the set $S_x$ of matrices $A,$ such that
$\|A\| < x$ and ...

**1**

vote

**1**answer

103 views

### A comparison principle for parabolic equation

(Crossposted from http://math.stackexchange.com/questions/757672/how-to-prove-comparison-principle-for-parabolic-pde-nonlinear)
Suppose $F:\mathbb{R} \to \mathbb{R}$ is smooth with $F(x) > 0$ for ...

**3**

votes

**2**answers

156 views

### Structure of an intersection of $L^p$-spaces

In what follows, $L^p$ denotes the space of functions from $\mathbb{R}$ to $\mathbb{R}$ such that $\int_{\mathbb{R}} |f(x)|^p\mathrm{d}x < \infty$.
I am interested to understand the structure we ...

**2**

votes

**0**answers

69 views

### $f, \hat{f} \in L^{p}\cap L^{\infty} \implies f\in B(\mathbb R)$ (algebra of Fourier- Stieltjes transforms )?

For a bounded complex Borel measure $\mu$ on $\mathbb R$, we define, its Fourier-Stieltjes transform, $\hat{\mu}(y)= \int_{\mathbb R} e^{-2\pi ix\cdot y} d\mu(x); (y\in \mathbb R).$
Let $1\leq p \leq ...

**2**

votes

**1**answer

117 views

### Estimate infinity norm with Lp and W1p norm

Let $p \in [1,\infty)$. Does there exist $C>0$ such that for every $f \in W^{1,p}([0,1],\mathbb{R})$ we have
$$\|f\|_{L^\infty}\leq C\|f\|_{L^p}^{1-\frac{1}{p}}\|f\|_{W^{1,p}}^{\frac{1}{p}}?$$
My ...

**0**

votes

**1**answer

65 views

### reference request: trace/lifting operator for $L^{\infty}$ data in bounded $\Omega\subset R^d$

I know the answer to my question must be somewhere in the literature. Probably in [Adams-Fournier], [Dautray-Lions] or something alike, but I don't have access to a library right now so I'll ask ...

**-1**

votes

**0**answers

56 views

### Rising Sun Inequality (Dunford-Schwartz maximal inequality) [migrated]

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function
$$f^*(x) := ...

**1**

vote

**0**answers

34 views

### Bounding Rayleigh quotioent for stochastic matrix

Suppose you have an irreducible, stochastic matrix $A$ with left Perron-Frobenius eigenvector $v$ (corresponding to the eigenvalue $1$), and suppose the next largest eigenvalue for $A$ is $\lambda$. ...

**5**

votes

**0**answers

197 views

### Products of spaces containing no copies of $\ell_2(\Gamma)$

Given an infinite set $\Gamma$, I would like to know if the class of Banach spaces containing no copies of $\ell_2(\Gamma)$ is stable under finite products.
When $\Gamma$ is countable the answer is ...

**0**

votes

**0**answers

38 views

### Is $\partial^\alpha$ a map $H^{s,p}(\mathbb R^N,F)\to H^{s-|\alpha|,p}(\mathbb R^N,F)$?

More precisely, the question is formulated as follows. Let $F$ be an arbitrary Banach space, especially not having the UMD−property. Let $N\in\mathbb N$ and $s\in\mathbb R$ and $1\le p\le +\infty$ . ...

**2**

votes

**0**answers

36 views

### Closure of pseudodifferential operators of order 0 on compact manifolds

Let $M$ be a compact manifold. If we have a pseudodifferential operator $P$ of order $0$ on $M$, then $P$ is pseudolocal, i.e., every commutator $[f,P]$ with a continuous function $f \in C(M)$ is a ...

**29**

votes

**1**answer

579 views

### For which maps $S^1\to S^1$ is the winding number defined?

There are two classes of maps $S^1\to S^1$ for which I know how to define the winding number:
• Continuous maps:
Using the unique path lifting property of the universal covering map $\mathbb R\to ...

**1**

vote

**1**answer

160 views

### When are completely positive maps monic/epic?

In the category of C*-algebras and $*$-homomorphisms, a morphism is monic precisely when it is injective, and epic precisely when it is surjective (see Mono- and epi-morphisms for C*-algebras). Is ...

**0**

votes

**0**answers

64 views

### $C^\infty$ approximations of $f(r) = |r|^{m-1}r$ [migrated]

Consider $f(r) = |r|^{m-1}r$ where $m \geq 1$.
Is it possible to find $C^\infty$ functions $f_n$, such that
$f_n \to f$ uniformly on compact subsets of $\mathbb{R}$,
$f_n' \to f'$ uniformly on ...

**6**

votes

**2**answers

181 views

### Literature on “real” $C^*$-algebras

I am trying to get a better understanding of "real" $C^*$-algebras. I encountered them in the paper
D. Voiculescu, Dual algebraic structures, J. Operator Theory 17(1987), 85-98,
which cites
G.G. ...

**1**

vote

**0**answers

74 views

### Tauberian theorem from generalized Gelfand transform

Wiener's theorem gives the necessary and sufficient conditions for the set of translates of a set of functions to be dense in $L^1(\mathbb{R}^n)$, which translates algebraically into a statement about ...

**-1**

votes

**1**answer

80 views

### Adjoint operator of a Convex operator is convex [closed]

Let $B(X, Y )$ be the collection of all continuous linear operators from the ordered normed space $X$ to the ordered normed space $Y$ . Given $A\in B(X, Y )$, the adjoint operator $A^\ast : ...

**0**

votes

**1**answer

82 views

### Getting an a priori bound on a nonlinear gradient term in PDE; how to adapt trick from $L^2$ case to $H^{-1}$ case?

I have the PDE
$$u_t(t) - \Delta f(u(t)) = 0$$
in $H^{-1}(\Omega)$ where $f$ is a nonlinear function.
Define $F(s) = \int_0^s f(s)$. Note that if $u_t(t) \in L^2(\Omega)$,
$$\frac{d}{dt}F(u(t)) = ...

**1**

vote

**0**answers

130 views

### Cotangent bundle of symmetric space is symmetric space?

Let $G$ be a connected Lie group. Then a symmetric space for $G$ is a homogeneous space $G/H$ where the stabilizer $H$ of a typical point is an open subgroup of the fixed point set of an involution ...

**2**

votes

**0**answers

109 views

### Equivalence of Gaussian measures on Hilbert space

Suppose we have 2 nondegenerate Gaussian measures given by N(0,T) and N(0,S) supported on a separable Hilbert space H. T and S are such that eigenbasis of S lies in the cameron martin space of ...

**1**

vote

**3**answers

140 views

### tranforms that lowers the number of variables of a function

Is there any linear map that lowers the number of variables of functions, namely a map that maps a function of several variables to functions of one variable and at the same time the original ...

**5**

votes

**0**answers

88 views

### Special elements in $L^{\infty}(G)^*$

Let $G$ be a locally compact group. Let $M(G)$ denote the measure algebra and $L^1(G)$ denote the group algebra on $G$. Then $M(G)$ acts on $L^1(G)$ by convolution. So by duality $M(G)$ acts on ...

**1**

vote

**0**answers

91 views

### on high order Laplacian

Roughly speaking, we have good understanding of the solution to heat equation $u_t-\Delta u=0$, on bounded or unbounded domain. For example, we know the decay rate, we know it generates analytic ...

**5**

votes

**0**answers

102 views

### Reference Request: Elliptic differential operators in the Fréchet setting

Normally the theory of (elliptic) differential operators between vector bundles (or $\mathbb{R}^n$) is presented in the language of Sobolev spaces. I'm searching for a book (or something similar) ...

**2**

votes

**1**answer

156 views

### Existence of Minimizer of $h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|} $

given the following functional
$h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|} $ with $\rho>0$ , $\|\rho\|_1 = 1$ and obviously $\rho\in L^1(\mathbb R^3)$.
Can I see ...

**5**

votes

**0**answers

76 views

### Is the space of $C^r$ vector fields inducing locally uniformly bounded trajectories Baire?

Let $\mathcal{V}$ be the space $C^r$ vector fields on a non-compact (smooth) manifold $M$. Being a subspace of $C^r(M, T M)$, it inherits the natural $C^r$ topology (i.e. the strong topology) of that ...

**4**

votes

**0**answers

97 views

### Is there a tensor norm that preserves Rosenthal Banach spaces?

By a Rosenthal Banach space I mean one that does not contain an isomorphic copy of $\ell_1$. Is there a tensor norm $\alpha$ such that the Banach tensor product $E\otimes_\alpha F$ is Rosenthal if $E$ ...

**1**

vote

**1**answer

76 views

### Techniques to show existence for a PDE with dynamic boundary condition

Let $\Omega$ be a bounded domain. I am looking for techniques to show existence of solutions to dynamic boundary problems of the form
$$\Delta u = 0 \quad\text{on}\quad \Omega \times (0,T)\\
...

**0**

votes

**0**answers

53 views

### Fredholm Integral Involving Stochastic Process

I wish to solve an integral equation of the form $$g(X) = c\int_0^1 K(X,t)f(t) \ dt $$ where $f\in L^1([0,1])$ and $g$ is some function on finite sequences of random variables. So, $X$ is a stochastic ...

**9**

votes

**1**answer

277 views

### A version of von Neumann inequality

Assume that $X,Y,Z$ are three commuting operators acting in a Hilbert space $H$. Assume also that they satisfy following properties:
1) $\|Z\| \le 1$, i.e. $Z$ is a contraction;
2) For any complex ...

**0**

votes

**0**answers

76 views

### Application of Schauder fixed point theorem

Let $\Omega$ an open bounded in $\mathbb{R}^n$, and let $A(x,u)$ an Caratheodory function, bounded and coercive, i.e.,
$$|A(x,u)|\leq \beta$$
$$A(x,u) \geq \alpha I,\quad \alpha > 0$$
and let
...

**-2**

votes

**1**answer

58 views

### Approximation of locally Lipschitz function by globally Lipschitz functions? [closed]

Let $f(x)=|x|^mx$ for $m \geq 0.$
$f$ is differentiable and locally Lipschitz. Is it possible to approximate $f$ by a sequence of globally Lipschitz functions $f_n$? Can we write down $f_n$ ...

**0**

votes

**1**answer

46 views

### Finiteness of “novel variance” from a kernel on a compact space [closed]

Let $c(i,i')$ be a kernel function on a reasonable index space $I$. Choose a dense sequence of points $\{i_1, i_2, \cdots \} \subseteq I$, and define the one-point kernel functions $k_n := c(\cdot, ...

**1**

vote

**1**answer

155 views

### Final topology of surjective linear map on Banach space

Let $L:X\rightarrow Y$ be a surjective linear map from Banach space $(X,||\cdot||_X)$ to vector space $Y$ and denote with $\tau_L$ the final topology on $Y$ induced by $T$.
Is $\tau_L$ equivalent ...

**1**

vote

**1**answer

130 views

### Question on Morse inequalities

I want to understand why: From K.C Chang's book "Infinite Dimensional
Morse Theory and Multiple Solution Problems":
if i have
then $(4.1)$ is formal : it means that
EDIT1: $(4.1)$ tel us that ...

**3**

votes

**2**answers

114 views

### Existence for ODE in Banach space (accretive operators and Crandall-Liggett)

There is a theory of mild solutions $u \in C^0(0,T;X)$ where $X$ is a Banach space for equations of the form
$$\frac{du}{dt} + Au = f$$
where $A$ is an accretive nonlinear operator under some ...

**5**

votes

**1**answer

163 views

### Measures which exhibit the “uncorrelated implies independent” property

Let $X$ be a topological linear space, and let $X^*$ be its dual space. Suppose that $X$ is complete and Hausdorff, and $X^*$ separates points. Let $Y$ be another such space, and let $f : X \to Y$ be ...

**6**

votes

**2**answers

469 views

### Is this Hankel matrix in trace class

Let A be the infinite Hankel matrix with the coefficient
$$A_{kj}=e^{(-t(k+j)^2)}-e^{(-t(k+j+2)^2)},$$ with $t$ a nonnegative real number.
Is $A$ in trace class with a norm bounded by an absolute ...

**2**

votes

**1**answer

199 views

### Modulus of of continuity of a convolution operator with respect to Wasserstein metric

For a (discrete) measure $G$ on some reasonable metric space $\Theta$, consider the map $G \mapsto f_G$ defined as
$$
f_G := f*G(dx) := \int f(dx|\theta) G(d\theta)
$$
for some nice kernel function ...