# Tagged Questions

**27**

votes

**1**answer

543 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 ...

**0**

votes

**0**answers

38 views

### minimum distance between sets and relation with functions [on hold]

Let $A, B$ be two convex and closed subsets of $\mathbb{R}^n$. We would like to the minimum distance between these two sets. i.e., we want to find a solution for the following problem.
$$ \min ...

**1**

vote

**1**answer

129 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 ...

**1**

vote

**0**answers

84 views

### Relationship between weak Lp and strong Lq topologies for q<p

Specificaly:
Does convergence in $L^{\frac{1}{2}}$ imply weak $L^2$ convergence?
Having a limit in $L^{\frac{1}{2}}$ topology and a limit in weak $L^2$ topology whether these are always equal? If ...

**3**

votes

**1**answer

155 views

### Characterization of a set in $\mathbb{R}^d$

Let $X= (X_1,\dots, X_d)$ be a fixed vector of random variables on the space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the following set.
\begin{equation}\label{main12}
C= \{x\in \mathbb{R}^d ~|~ ...

**1**

vote

**2**answers

140 views

### Smooth but non-analytic kernel functions

Does there exist a (stationary) covariance kernel function which is $C^\infty$-smooth but not real analytic? If so, could you please provide an example?

**0**

votes

**0**answers

346 views

### Given an even function how to obtain the most close odd function and vise versa?

Given an even function $f(x)$, how to obtain the most close to it continuous odd function $g(x)$?
By most close I mean that $\int_0^\infty |f(x)-g(x)| dx$ be the minimum possible and the difference ...

**5**

votes

**3**answers

143 views

### Method to compute fundamental solutions which are distributions

The Malgrange-Ehrenpreis theorem tells us that there is a fundamental solution for any linear differential operator of constants coefficients. The original proof was not constructive (it was based on ...

**3**

votes

**1**answer

152 views

### When is the bound in Riesz-Thorin Interpolation Theorem attained?

Let me recall the statement of Riesz-Thorin theorem (see also http://en.wikipedia.org/wiki/Riesz%E2%80%93Thorin_theorem).
Theorem (Riesz-Thorin): Let $(X,\mu)$ and $(Y,\nu)$ be $\sigma$-finite ...

**19**

votes

**2**answers

447 views

### Which smooth compactly supported functions are convolutions?

If $f,g$ are smooth functions with support in the interval $[-r,r]$ for some $r>0$, then their convolution $f*g$ is smooth with support in $[-2r,2r]$. My question is about the converse: Given ...

**2**

votes

**1**answer

148 views

### Equivalence of negative Sobolev norm of derivative to $L^2$-norm

Let $S:=(0,1)^2$ be the unit square in $\mathbb{R}^2$, and let $M:=\{u\in L^2(S)\mid \int_S u=0\}$ be the space of (real-valued) $L^2$-functions with mean value zero. On $M$ we can consider the ...

**-1**

votes

**1**answer

110 views

### Is the countably infinite product of locally convex topological vector spaces locally convex?

Let $(X,\tau)$ be a locally convex topological vector space and denote the product space
$$X^{\infty}=X\times X\times X\cdots:=\big\{x=(x_i)_{i\geq 1}:~ x_i\in X\big\}$$
If we endow $X^{\infty}$ ...

**1**

vote

**1**answer

162 views

### Inequality in the Sobolev space $H^1$

I've found the following inequality
$$\int_{B_r}\vert u\vert^q\leq C \bigg(\int_{B_r}\vert\nabla u\vert^2\bigg)^{a}\bigg(\int_{B_r}\vert u\vert ...

**4**

votes

**1**answer

392 views

### A generalization of a theorem of Grothendieck

In this question the norm of $L^{P}[0,1]$ is denoted by $\parallel . \parallel _{p}$.
Let $p$ and $q$ be two arbitrary real numbers with $2<p<q$.
Assume that $S$ is a subvector space ...

**3**

votes

**2**answers

107 views

### series representation of bivariate functions

Given a bivariate function $f(x, y)$ with $x \in [-a,a]$ and $y \in [-b, b]$, what is the necessary and sufficient condition under which we can write $f(x, y) = \sum g_k(x)h_k(y)$ for all $(x,y)$ in ...

**1**

vote

**1**answer

184 views

### Is the space of test functions separable? [closed]

Consider the space $\mathcal D(\mathbb{R}^n)$ of smooth functions (in the sense of having continuous derivatives of all orders) which are compactly supported. Endow it with its usual topology, i.e., ...

**0**

votes

**3**answers

187 views

### dual space of a subspace of the space of bounded measures

Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Equipped with the weak convergence, the dual space of $\mathcal{M}$ is $\mathcal{C}_b(\mathbb{R})$ consisting of continuous ...

**0**

votes

**0**answers

107 views

### Dyadic approach to Marcinkiewicz interpolation for Lorentz Spaces

In Exercise 21, in a note by professor Terrence Tao on his own blog
http://terrytao.wordpress.com/2009/03/30/245c-notes-1-interpolation-of-lp-spaces/
Exercise 21 Suppose we are in the situation ...

**0**

votes

**0**answers

36 views

### compactness related to some distance defined on the space of increasing functions2

Let $I=[0,1]$ and denote by $C^{+}(I)$ the space of continuous increasing functions. Can we find a distance $d$ for $C^+(I)$ such that the set of the form
$$d(f,g)\rightarrow 0\Longrightarrow ...

**2**

votes

**0**answers

44 views

### A fixed point problem in infinite dimensions for monotonic algebras

Let $A$ be an algebra over $[0,1]$, whose operations are all unary monotone (increasing or decreasing) bijections, except that $A$ also includes the infimum operation over finite or countably many ...

**0**

votes

**2**answers

153 views

### Let f:J→R be an absolutely continuous and f'\in…?

Let $f:J\rightarrow \mathbb{R}$ be an absolutely continuous.
Under what kind of extra condition for $f'$, (not $C$) holds the following relation?
$$
\Big | \frac{1}{|I_{1}|}\int_{I_{1}}f'(x)dx- ...

**0**

votes

**0**answers

113 views

### Dual of the space of vector valued Borel measures

What is the dual of the space of all vector valued Borel measures?

**2**

votes

**0**answers

81 views

### Weak relative compactness in $L^1_{loc}$.

In my work I stumbled upon a proposition (without proof, alas), which I can't really prove.
Suppose we have a family of functions $\left\{\phi_\epsilon (t,x,v)\right\}_{\epsilon\in(0,1]}$, and $M(v)$ ...

**0**

votes

**1**answer

138 views

### An Integral Functional Equation

Let $f$ be a non-negative function supported and integrable on the positive real axis, such that
$$\int_0^\infty f(x+y)p(y) dy = c[p] f(x), $$
where $c[p]$ a number (functional) dependent on function ...

**1**

vote

**1**answer

129 views

### Lebesgue's integrability condition in several variables

The well known Lebesgue's condition of Riemann integrability says that a bounded function in one variable
$f\colon [a,b] \to \mathbb{R}$ is Riemann integrable if and only if it is continuous almost ...

**2**

votes

**0**answers

78 views

### Is Modulation space is close under absolute?

It is clear that, for $1\leq p \leq \infty$, $f\in L^{p}$ iff $| f| \in L^{p}.$
By the usual modulation space $M^{p, q}(\mathbb R^{d})$, $1\leq p, q \leq \infty, d\in \mathbb N $, we mean the ...

**1**

vote

**0**answers

57 views

### Fixed point theorm that does not require the hemi-continuity of the set valued map?

All of the fixed point theorem I have seen (like Kakutani and Brower, Browder ) required the set valued map to be hemi-continuous (lower). Is any fixed point theorem that can assure the existence of ...

**2**

votes

**0**answers

100 views

### Is reflexive Banach space valued scalarwise Lebesgue space isomorphic to the Bochner space?

I first specify the setting and then formulate the question precisely. (A very long post follows.)
Definitions 1. For $E$ a (real Hausdorff) locally convex space, say that $E$ is suitable iff there ...

**2**

votes

**2**answers

182 views

### specific improper integral involving erf

I have encountered an integral, and kindly ask for help with a solution. It is beyond my own capabilities, and neither Maple nor Mathematica were of any help:
$$
\int_{1}^{\infty} ...

**1**

vote

**1**answer

182 views

### Continuous and dense embeddings and the density of sets in Hilbert space

Suppose $H$ is a Hilbert space of functions $f:\Omega\to \mathbb{R}^n$ with $\Omega\subset \mathbb{R}^n$ open, bounded and with Lipschitz boundary (take for example $H=H_0^1(\Omega)^n$) and suppose ...

**3**

votes

**1**answer

117 views

### Weak continuity of Lebesgue decomposition

Let $X$ be a space with its $\sigma$-algebra $\mathcal{B}$; we are given a finite measure $\mu$ and a sequence of finite measures $\nu_n$ such that, for every bounded continuous function ...

**3**

votes

**0**answers

66 views

### Fourier multiplier with a singularity on a convex curve

Let $h$ be a strictly convex function such that $h(0) = h'(0)=0$. Let $\Phi: \mathbb{R}^2 \to \mathbb{R}$ be a $C^{\infty}$-function with compact support (say, $\Phi$ is supported on ...

**4**

votes

**1**answer

220 views

### Approximation of an integral over the unit ball of L_1

For every $\varepsilon>0$ find a piecewise continuous function $q:[0,1]\rightarrow \mathbb{R}$ such that $\int_0^1 q(x)dx=1$ and
$$\int_{0}^1 \int_{0}^{s} \left|\frac{q(s)q(t/s)}{s}- ...

**1**

vote

**1**answer

158 views

### Does a particular iteration produce a weak solution to a non linear pde?

Consider the following non linear pde in the unknown $v(x,y)$:
$$ \frac{\partial v(x,y)}{\partial x} +
\Big(\frac{\partial v(x,y)}{\partial x} \Big)^2 = e^{2 ty}-1 $$
where $t$ is some fixed small ...

**4**

votes

**0**answers

130 views

### Is there an appropriate weighted Sobolev space to include exponential map and projection map?

Observe that given a non negative function
$\omega: \mathbb{R^2} \rightarrow [0, \infty)$, we can define the weighted
$L^{p}(\mathbb{R}^2, \omega) $ spaces. They are measurable functions
$f: ...

**2**

votes

**1**answer

157 views

### Limit involving regularlized gamma function and its inverse

Let
$$L(x)=Q\left(\frac{x}{2},\frac{a}{a+f(x)/\sqrt{x}}Q^{-1}\left(\frac{x}{2},1-b^{1/g(x)}\right)\right)$$
where $Q(s,x)=\frac{\Gamma(s,x)}{\Gamma(s)}$ is the upper incomplete gamma function ...

**0**

votes

**0**answers

84 views

### Two distribution spaces ${\mathcal S}'/{\mathcal P}$ and ${\mathcal S}_\infty'$

Let ${\mathcal S}'$ be the set of all distributions.
Denote by ${\mathcal P}$ the set of all polynomials,
which is embedded into ${\mathcal S}'$ as a closed subspace.
Equip ${\mathcal S'}/{\mathcal ...

**0**

votes

**2**answers

409 views

### Is there a probability density function satisfying the following conditions?

I find myself in need of the solution of this problem in finding a probability density function. I had asked this question in Math Stack Exchange but I did not get an answer so I am posting it here.
...

**2**

votes

**1**answer

133 views

### AM-GM interpolation in the limit

Given a sequence of positive numbers $X = (x_1, x_2, \dots)$, define the $k$th elementary symmetric mean of the first $n$ entries to be
$$ S(X, n, k) := \frac{\displaystyle\sum_{1 \leq i_1 < ...

**4**

votes

**0**answers

163 views

### decreasing rearrangements: why the asymmetry of measure-preserving maps?

Ryff proved in 1970 that the decreasing rearrangement $f^*$ of a, say, continuous function $f:[0,1]\to\mathbb{R}$ admits a measure preserving map $\phi$ such that $f=f^*\circ\phi$. In general it is ...

**1**

vote

**2**answers

220 views

### Is there a reference for compact imbedding theory of Hölder space?

This question is posted and unanswered from math.stackexchange.
Suppose $0 < \alpha < \beta$ and $\Omega$ is bounded. Then, the Hölder space $C^\beta(\Omega)$ is compactly imbedded to ...

**10**

votes

**0**answers

334 views

### Classes of (non-continuous) functions with the fixed point property

Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$.
...

**4**

votes

**1**answer

214 views

### Norms for complex measures

I'm searching for a state of the art introduction to norms on the space of complex measures (on $\mathbb R^n $, for example, or some compact subset thereof). I'd be interested in inequalities of the ...

**1**

vote

**1**answer

189 views

### When can we “displace” an ultrafilter limit with another limit?

Let $\cal A$ be a Banach algebra, $\cal U$ be a free ultrafilter, and $\phi$ be a character. Let ${(w_{\alpha})}_{\alpha}$ be a net in $(\cal A)_{\cal U}$, and suppose that for every $(a_i)\in (\cal ...

**3**

votes

**1**answer

99 views

### A recurrent sequence related to the Brouwer fixed-point theorem

Let $K$ be a non-empty compact convex subset of a Banach space $E$, and let $f : K \longmapsto K$ be a continuous function. Fix $u_0 \in K$, and define by recurrence $u_{n+1} = \frac{1}{n+1} ...

**1**

vote

**0**answers

211 views

### A strange Weakly Compactness in $L^1 ( \Omega, \mathcal{F}, \mathbb{P})$

Hi to everyone,
The ingredients of my problem are the following:
I have a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, a set (continuum cardinality) $\mathcal{Q}$ of probability measures on ...

**0**

votes

**0**answers

123 views

### Does this sequence of H\"older functions have a limit?

Let $\left\{\alpha_{n}\right\}_{n\in \mathbb{N}}$ a sequence of positive real numbers with
$$\alpha_{n}\in (0,1)\quad \textrm{and}\quad \alpha_{n}>\alpha_{n+1}$$
Moreover suppose
...

**7**

votes

**1**answer

438 views

### complete metric space

Hallo, I have the following question:
Let $(X,d)$ be a complete metric space. Is then $(X,\operatorname{dist})$ also complete? Here by $\operatorname{dist}$ I mean the metric induced by $d$ by: ...

**0**

votes

**0**answers

42 views

### Question on weak Lorentz spaces

I have the following question. Let $\Omega$ be a bounded domain in $\mathbb{R}^d$. Suppose we have a function $f$ such that for $h>0$ it holds that
$$\|f\|_{H^1}\lesssim h^k.$$
By the Sobolev ...

**3**

votes

**1**answer

335 views

### Is a Cauchy principal value invariant under a “change of variables”?

Let $f \in C^{\gamma}_c(\mathbb{R}^n) $. Let $K:\mathbb{R}^n \backslash \{\vec{0}\} \rightarrow \mathbb{R}^n$ be a singular integral kernel with the following properties:
1) K smooth everywhere ...