**0**

votes

**1**answer

135 views

### When can two Cauchy transforms intersect?

Given two polynomials $p$ and $q$ over reals and being guaranteed that both have all roots real I want to know if there is any characterization of the solutions of the equation $\frac{p'}{p} = ...

**2**

votes

**0**answers

39 views

### Dual cone of 'positive' Bochner integrable functions

If we consider the space of integrable functions $L^1([0,1];\mathbb{R})$, it can be ordered by the convex cone of positive integrable functions $L^1([0,1];\mathbb{R}_+)$. It is known that the ...

**4**

votes

**0**answers

113 views

### $H^s$ norm of a solution of a nonlinear Schrödinger equation

I'm reading the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$ by Colliander, Keel, Staffilani, Takaoka and Tao.
They study the ...

**4**

votes

**1**answer

136 views

### Loss of derivative of subelliptic operator

Consider the differential operator $P$ on $\mathbb{R}^2$, given by $P = \frac{\partial^2}{\partial x^2} + x^2\frac{\partial^2}{\partial y^2}$. Clearly it is elliptic everywhere except on the $y$-axis. ...

**1**

vote

**1**answer

214 views

### Condition to obtain a not compact embedding

I have the two spaces $W_0^{1,p}$ with the norme $$||u||^p=||u||^p_{L^p}+||\nabla u||^p_{L^p}$$ and $$L^{p^*}_{\alpha}=\{ u~\text{measurable}, \int_{\Omega} (|x|^{\alpha} u(x)|)^{p^*} dx<\infty\}$$ ...

**4**

votes

**1**answer

146 views

### application of factorization theorem

Young's inequlity tells us that $L^{1}(\mathbb R)\ast L^{p}(\mathbb R) \subset L^{p}(\mathbb R)$ with norm inequality
$$\|f\ast g\|_{L^{p}} \leq \|f\|_{L^1}\|g\|_{L^p};$$
and of course this ...

**5**

votes

**1**answer

117 views

### Simultaneous approximation of arbitrary functions in Hölder space and in $L^2(\mu)$ by a smooth function and its derivative

Let $\mu$ be a probability measure on the circle $S^1=\mathbb{R}/\mathbb{Z}$ which is singular with respect to the Lebesgue measure $\lambda$. Consider the functions spaces $L^2(\mu)$ on the one hand, ...

**0**

votes

**0**answers

119 views

### Problem with operator and Fourier transform

I am currently dealing with a problem in functional analysis where I want to show the following.
Let $\phi_{k+1}(x):=\sqrt{2} \sum_{n \in \mathbb{Z}} h(n) \phi_k(2x-n)$ and $\phi_0= \chi_{[0,1]},$
if ...

**1**

vote

**1**answer

166 views

### Finite trigonometric polynomial

I noticed by numerical and some explicit calculations for a few examples that for real-valued finitely supported functions $\phi \in L^2(\mathbb{R})$ we have that
$T(x):= \sum_{n \in \mathbb{Z}} ...

**7**

votes

**3**answers

277 views

### Do non-normal states exist in the Solovay model?

Let H be an infinite dimensional Hilbert space. Then there exist non-normal states on B(H) in ZFC (i.e. states that are not represented by a density operator).
Is this also true in the Solovay model ...

**7**

votes

**1**answer

285 views

### understanding of rough path

A rough path is defined as an ordered pair
$ (X, \mathbb X)$, where $X$ is a path mapping from $[0,T]$ to some Banach space $V$
and $\mathbb X:[0,T]^2 \mapsto V^2$ is another mapping for additional ...

**3**

votes

**1**answer

96 views

### Associative convolution on p-adic distribution

Let $\mathcal{D}(\mathbb{Q}_p)$ be the space of the locally constant functions with compact support and let $\mathcal{D}'(\mathbb{Q}_p)$ be the space of distributions: linear functionals on ...

**1**

vote

**0**answers

20 views

### Counting variables to look for invariances/range conditions

A while back, I asked this question on m.se. I wasn't terribly happy with the answer, and when someone asked a very similar question which isn't getting any action, it got me thinking again. Let me ...

**3**

votes

**1**answer

162 views

### A question on the Frechet derivative

Suppose the derivative of a functional is given by
\begin{equation*}
\int_{\Omega}(\vec{v}.\nabla u)|\nabla u|^{p-2} \phi=\int_{\Omega}\nabla.(u\vec{v})|\nabla u|^{p-2} \phi,~\phi\in ...

**0**

votes

**0**answers

46 views

### Preimage of singular points of smooth map between vector space and $SU(n)$

(Moved from Math SE as no answer was forthcomming: http://math.stackexchange.com/q/1294521/161684)
Given a smooth ($C^{\infty}$) map $\phi: V \rightarrow SU(n)$ (which is taken to be surjective) ...

**0**

votes

**1**answer

101 views

### solution uniqueness of non-linear Fredholm equations

the equation is
$F(x)=G(\int k(x,y)f(y)dy)$ $(*)$
where $f(x)=dF(x)/dx$ is the unknown and it's required to be non-negative. With integral by parts we'll have the form of a non-linear Fredholm ...

**0**

votes

**1**answer

112 views

### Is the space of smooth maps $C^{\infty}(M,N)$ with the Whitney $C^{\infty}$ topology locally compact, if $M$ is compact

The title says it all:
Let $M$ be a compact manifold and $N$ a (possible non compact) manifold. Equip the space of smooth functions $C^{\infty}(M,N)$ with the Whitney $C^{\infty}$ topology. (The ...

**3**

votes

**2**answers

179 views

### $\mathcal S'(\mathbb R^d)$ is separable [closed]

I Think the statement is true, but I struggle to find a reference for the fact that the space of tempered distributions equipped with the weak-* topology is separable.
Thank you for your help!

**6**

votes

**1**answer

125 views

### Sobolev space for Mixed Dirichlet - Neumann boundary condition

Consider the subset $\Omega\subset \mathbb{R}^N$ with boundary $\partial\Omega$ sufficiently regular and let $\Gamma\subset\partial\Omega$ be a $(N-1)$- dimensional submanifold of $\partial\Omega$. ...

**2**

votes

**0**answers

63 views

### Sobolev embedding on warped product

Consider the warped product $X = M \times \mathbb{R}$, with the metric $g = dr^2 + \varphi(r) g_M$, where $M$ is a compact manifold. Consider the Sobolev space $H^1(X)$ and let $H^1_{rad}(X)$ denote ...

**0**

votes

**0**answers

48 views

### interpolation between Bochner spaces

Is there a reference for the interpolation result stating the existence of an embedding
\begin{equation}
L^2(I;W^{2,p}(\Omega)) \cap H^1(I;L^p(\Omega)) \hookrightarrow ...

**1**

vote

**0**answers

58 views

### Related to derivative of Modified Bessel I function wrt the order

I recently met some problems related to the modified Bessel I funtions. Let $I(\nu,x):=I_\nu(x)$, and $I'_\nu(\nu,x):=\dfrac{\partial}{\partial \nu}I(\nu,x)$.
Using maple, it seems that ...

**2**

votes

**0**answers

69 views

### Resolvent estimate of hyperbolic Laplacian [closed]

Consider the Laplacian $-\Delta$ on the hyperbolic space $\mathbb{H}^n$. For $\lambda \in \mathbb{C} \setminus [0, \infty)$, do we have resolvent estimates of the form
$$\Vert (-\Delta - \lambda ...

**0**

votes

**0**answers

47 views

### Completing Karlin's proof of variation diminishing transformation theorem

In S Karlin's book total positivity there's a theorem
that says if $K(x,y)$ is $TP_r$ (totally positive with degree $r$) and the sign change count of function $h$, $S(h) = n\leq r-1$,
then ...

**1**

vote

**1**answer

91 views

### Is the endpoint map smooth

Given $a,b \in \mathfrak{su}(n)$ and (with $U_0 = I$ taken) the following ODE:
$\frac{d U_t}{dt} = (a + w(t)b)U_t$
consider the "fixed time" endpoint map $V_T: L^2([0,T]) \rightarrow SU(n)$ for ...

**0**

votes

**0**answers

75 views

### Alternating projections and trace preserving maps

Let $\{\Pi_i\}_{i=1}^N$, $\Pi_i\in \mathbb{C}^{n\times n}$ be a set of orthogonal projections. By the Von Neumann alternating projection theorem, it holds that
$$
...

**2**

votes

**0**answers

39 views

### Smooth bivariate functions identifiable under permutations

Consider a "smooth" $f : [0,1]^2 \to [0,1]$ and let $\pi : [n] \to [n]$ be a permutation of $[n] =\{1,\dots,n\}$. Let us define
$$
A^{f,\pi} := \Big( f\Big(\frac{\pi(i)}{n},\frac{\pi(j)}{n}\Big) ...

**3**

votes

**0**answers

84 views

### independent symmetric 3-valued random variables in Lp

Consider the following excerpt from this paper:
Given $1<p<2$, $0<w\leq 1$ and $n\in\mathbb{N}$, we fix once and for all a sequence $f_j^{(n)}=f_{p,w,j}^{(n)}$, $1\leq j\leq n$, of ...

**3**

votes

**2**answers

171 views

### Examples of TVS with no non-trivial open convex subsets

I give here the classical example of the space $E = L^p([0,1])$ which has no open convex subsets apart from $\emptyset$ and $E$. Consequently, there is no non-trivial continuous linear form on $E$.
...

**5**

votes

**2**answers

155 views

### Schwartz space of functions with values in a Frechet space

While reading some papers about $\psi$DOs I found some spaces of vector valued functions which I am not familiar with. I am looking for references about the Schwartz space of functions with values in ...

**6**

votes

**2**answers

371 views

### Is there a nice “synthetic” way for doing differential geometry on infinite dimensional vector spaces?

If $V$ is an infinite dimensional vector space, for example the space of smooth functions on $\mathbb{R}$, we can introduce some differential geometry concepts by choosing a topology on $V$ and doing ...

**2**

votes

**1**answer

232 views

### Interpolation between weighted $L^p$ spaces

Let $K:\mathbb{R}^3\backslash\{0\}\times\mathbb{R}^3\backslash\{0\}\rightarrow\mathbb{C}$, such that $K(x,y)=K(y,x)$ and $K(x,y)=|x|^{-1}|y|^{-1}H(x,y)$, with $H$ locally bounded.
Let $T$ be the ...

**1**

vote

**0**answers

77 views

### Existence of Euler product on critical line for $L(\chi,s) L(\overline{\chi},1-s)$?

Generally there is no Euler product for Dirichlet L-functions $L(\chi,s)$ in the critical strip.(cf Is the Euler product formula always divergent for 0<Re(s)<1?)
But I would like to know if ...

**2**

votes

**0**answers

97 views

### Transformation of kernel

I have the following problem at hand.
Define the kernel
$$K(x_1,x_2) = \int_{-1}^1\int_{-1}^1 \exp(-2\pi\jmath x_1 y_1)R(y_1,y_2)\exp(2\pi\jmath x_2 y_2)\mathrm{d}y_1\mathrm{d_2}.$$
Now, if ...

**9**

votes

**1**answer

258 views

### Sard's Theorem For Banach Spaces

Given a smooth map from $\phi: B \rightarrow M$ where $B$ is a Banach Space and $M$ is a finite dimensional smooth manifold (for example, the end point map for a control system), what is the strongest ...

**1**

vote

**1**answer

114 views

### Uniform convergence of infinite sum with Dirichlet characters

I would like to prove uniform convergence of function series like :
$$\sum\limits_{n=1}^{\infty} \chi(n) f(nx)$$ where $\chi$ is a primitive character and $f(x)$ a function decreasing to zero in ...

**5**

votes

**2**answers

270 views

### Weak solutions for a PDE of fourth order

I deal with two-dimensional Kirchhoff equation with $L^\infty$ coefficient and distributional right hand side:
$$
\Delta\Delta w+u(x,y)\left(\alpha^2\frac{\partial w}{\partial ...

**0**

votes

**0**answers

83 views

### Existing complete function space under suitable norm

This question was asked in math.stackexchange.com but no suitable answer was received, so I am posting it here.
This is a question which came to me due to several previous question: sorry for the all ...

**2**

votes

**1**answer

95 views

### Integral kernels of self-adjoint operators [closed]

If the integral kernel $k(x, y)$ of an operator $T : C^\infty_c(M) \to \mathcal{D}'(M)$ is symmetric ($M$ is a compact manifold), then the operator $T$ is symmetric. Is the converse true? That is, ...

**1**

vote

**0**answers

100 views

### Sobolev trace theorem

Set $Q:=\Omega\times(0,T)\subset\mathbb{R}^{n+1}$,
where $\Omega$ is knows as a bounded domain
with smooth boundary $\partial D$.
We choose any subdomain $D\subset Q$
with smooth boundary $\partial ...

**6**

votes

**1**answer

282 views

### An indicator of a planar subset as an element of a tensor product

Denote $I=(0, 1)$, and let $\mu$ be the Lebesgue measure on $I$. Does there exist a function $f$ on $I\times I$ viewed as an element of the space $L^\infty(\mu\times\mu)$ such that
$$
f^2=f
$$
(that ...

**3**

votes

**1**answer

201 views

### Variations on the Mellin and Dirichlet transforms

There are a number of variations on the Laplace transform that turn up all over math. Some examples:
$\int_{-\infty}^{\infty} f(t)e^{-st} dt$ - The Laplace transform
$\sum_{-\infty}^{\infty} ...

**3**

votes

**2**answers

198 views

### When does a function space allow for point evaluations? [closed]

Consider a space of (generalized) functions $F$ defined on a measure space $\Omega$ and equipped with a topology.
What are necessary and sufficient conditions for point evaluations at arbitrary $x ...

**3**

votes

**1**answer

150 views

### Mixed (anisotropic) Sobolev spaces

Consider real variables $x, y$ and a function $f(x, y) \in H^s(\mathbb{R}^2)$, say for some $s \in (0, 1)$. I am trying to get an understanding of mixed Sobolev spaces of the form $H^s_x(H^s_y)$, ...

**3**

votes

**1**answer

110 views

### Can you pair $H^s(\Omega)$ and $H^{-s}(\Omega)$ on a domain $\Omega$?

Consider the fractional Sobolev spaces on $\mathbb R^n$
$H^s(\mathbb R^n) := \left\{ u \in \mathcal S'(\mathbb R^n) \; : \; ( 1 + |\xi|^2 )^{s/2} \hat u \in L^2(\mathbb R^n) \right\}$.
Let $\Omega$ ...

**6**

votes

**0**answers

170 views

### A “slice-map” type problem for symmetric tensors in the square of a nuclear C*-algebra

Throughout: let $\otimes$ denote the minimal (i.e. spatial) $\newcommand{\Cst}{{\rm C}^*}\Cst$-tensor product of two $\Cst$-algebras.
Let $B$ be a unital, nuclear $\Cst$-algebra and let $A\subset B$ ...

**1**

vote

**0**answers

92 views

### Asymptotics of “heat” semigroup

Consider a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary. Consider a second order elliptic operator $L$ on $L^2(\Omega)$, defined by either the Dirichlet or Neumann boundary ...

**-1**

votes

**1**answer

256 views

### Which operators other than self-adjoint operators have no purely imaginary eigenvalues? [closed]

Given an operator mapping between suitable spaces, what is the condition that guarantees all eigenvalues have nonzero real part? Obviously self-adjointness implies all eigenvalues are real, but how ...

**4**

votes

**1**answer

112 views

### Relation between dual of nuclear space $(\substack{\text{lim} \\ \leftarrow i} H_i)'$ and $\substack{\text{colim} \\ i \rightarrow } H_i$

Let $\substack{\text{lim} \\ \leftarrow i} H_i$ be a nuclear space, considered as the limit of the codirected diagram $$... \to H_2 \to H_1 \to H_0,$$
with $f_{ji}:H_i \to H_j$ being the trace class ...

**0**

votes

**0**answers

65 views

### prove that a function is approximatively three dimensional

Let $D_n(x)$ be a diagonal matrix of size $N\times N$ where the $k$th element is $\exp(2\pi\jmath x(n+(k-1)/N)$.
Let $P_n$ be a random diagonal $N\times N$ matrix where each diagonal element is a ...