**0**

votes

**0**answers

62 views

### Functions $f \in S(\mathbb{R})$ with slow decay rate?

I came across this question by thinking about whether there are functions $f \in S(\mathbb{R})$ that satisfy for all(!) $a>0$ that $f e^{a|.|^a} \notin L^{\infty},$ as somebody on math....

**14**

votes

**2**answers

523 views

### Is there a $C_c^{\infty}( \mathbb{R}^d)$ function whose Fourier transform we can explicitly write down?

I noticed that although $C_c^{\infty}$-functions are dense in some quite large spaces and well understood (especially their Fourier transform) I have never encountered an explicit example of a ...

**1**

vote

**0**answers

18 views

### Concavity of maxima [closed]

Suppose we have the following optimization problem : $\min\limits_x kf(x) + g(x)$ where $f$ is a decreasing convex function in $x$ and $g$ is an increasing convex function. Can we say that $x^*$ is ...

**2**

votes

**0**answers

118 views

### Conditions for continuity of non-simple eigenvectors

Here, http://math.stackexchange.com/a/1146455, it is noted that eigenprojections are continuous, but eigenvectors are not. Are there any conditions where the eigenvalues are not simple, but the ...

**1**

vote

**0**answers

58 views

### A problem on Markov chains and Dirichlet forms

Let $X$ be a countable set. Let $c:X\times X\to[0,+\infty)$ satisfy
$$c(x,y)=c(y,x)\text{ for all }x,y\in X,$$
$$m(x)=\sum_{y\in X}c(x,y)\in (0,+\infty)\text{ for all }x\in X,$$
$$c(x,x)=0\text{ for ...

**0**

votes

**1**answer

58 views

### Long term behavior of a certain discrete time dynamical system on graphs

Consider the graph $(V,E)$ with vertex set $V=\{v_1,...,v_n\}$ and edge set $E\subset V\times V$. Further, assume that $\forall v_i\in V, (v_i,v_i)\in E$.
Assume that each vertex has an $\textit{...

**2**

votes

**0**answers

58 views

### Harmonic functions in tempered distribution sense

Suppose $g$ is a metric on $\mathbb{R}^3$ and $\Omega \subset\subset \mathbb{R}^3$. We assume that $g$ is euclidean outside $\Omega$.
My question concerns solutions to $\triangle_g u =0$ that are say ...

**4**

votes

**1**answer

270 views

### Two similar integrals

Let $n$ be a given even positive integer. We have the following integral
\begin{align}
\int_0^{\infty}\cdots\int_0^{\infty}e^{-(x_1+\cdots+x_n+y_1+\cdots+y_n)}\prod\limits_{i=1}^n\prod\limits_{j=1}^n(...

**2**

votes

**1**answer

150 views

### Simplify proof for rapidly decaying functions

I want to show the following theorem in a lecture:
Let $F \in C^{\infty}(\mathbb{C}^{k}, \mathbb{C})$ such that $F(0)=0.$
Let $G: \mathbb{R}^n \rightarrow \mathbb{C}^{k}$, $x \mapsto (f_1(x),..,f_k(...

**4**

votes

**1**answer

175 views

### Is the number of intervals you get by slicing the closed region under a nice graph continuous from below?

Let $I$ be a closed interval in $\mathbb{R}$, and let $f:I \to \mathbb{R}$ be a bounded function, smooth except at finitely many points (piece-wise smooth). There are also only finitely many critical ...

**1**

vote

**1**answer

71 views

### Partition of Real Number [closed]

Can the set of Real numbers be partioned into two parts such that both are uncountable,dense and have empty interior and any closed interval intersects both at uncountably many points?

**0**

votes

**0**answers

89 views

### Does this set of (structured) equations always have a solution?

Let $r_1,\ldots,r_K$ be arbitrary positive numbers.
Does
$$|\mathcal{A}|\log\left(1+\frac{1}{|\mathcal{A}|}\left(\sum_{n\in \mathcal{A}} \sqrt{x_n(\exp(r_n)-1)}\right)^2\right)\leq \sum_{n\in \...

**5**

votes

**1**answer

620 views

### Is the following integral nonzero?

Recently I met an integral as follow:
$$\int_0^{2\pi}\cdots\int_0^{2\pi}\left(\prod\limits_{1\leq i<j\leq9}\sin\frac{\theta_i-\theta_j}{2}\right)\left(\prod\limits_{i=1}^9(1+\cos(\theta_i-\theta_{i+...

**43**

votes

**1**answer

1k views

### Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...

**4**

votes

**0**answers

116 views

### Classifying countable sets of weighted dots on a real line

Each dot is located on the real line and assigned a weight that can be positive or negative. A dot is equivalent to two(or more) dots located at the same place whose weights sum is equal to that of ...

**50**

votes

**1**answer

2k views

### Square root of dirac delta function

Is there a measurable function $ f:\mathbb{R}\to \mathbb{R}^+ $ so that $ f*f(x)=1 $ for all $ x\in \mathbb{R} $, i.e $$\int\limits_{-\infty}^{\infty} f(t)f(x-t) dt=1 $$ for all $ x\in \mathbb{R} $.

**1**

vote

**0**answers

78 views

### Convolution Integral involving an unknown function

I've got the following problem I'm working on which is related to some of my research.
I am trying to solve the following equation for the function $f$.
$$t^{-\alpha} \exp{ \left(- \beta x^2 t^{-2 \...

**7**

votes

**0**answers

105 views

### Well-definedness on $C_{0}^{\infty}(\mathbb{R}^{n})$

Let $T$ be a Calderon-Zygmund operator associated to a Calderon-Zygmund kernel $K\in CZK_{\alpha}$ of order $\alpha>0$ and $b\in BMO(\mathbb{R}^{n})$. Then for $f\in C_{0}^{\infty}(\mathbb{R}^{n})$ ...

**3**

votes

**1**answer

158 views

### Uniqueness from orthogonality relation?

This question was posted yesterday on MathOverflow by Michael Smith and received a number of upvotes. I too think the question was interesting. However, for some unknown to me reasons, it has been ...

**0**

votes

**0**answers

38 views

### On the continuity of Riemann-Liouville integral

For a function $f:(0,1)\to\mathbb{R}$, the Riemann-Liouville integral of $f$ is defined by
$$
(I^{1-\nu}f)(t):=\frac{1}{\Gamma(1-\nu)}\int_{0}^{t}\frac{f(s)}{(t-s)^{\nu}}ds,
$$
where $\nu\in(0,1)$ is ...

**1**

vote

**1**answer

126 views

### Integration against Borel measures on compact Hausdorff spaces

I am studying the properties of integration against Borel measures and Baire measures. And I am not sure whether the following proposition is correct and I tried to give a proof.
Suppose that $X$ ...

**2**

votes

**1**answer

96 views

### A singular integral of several functions

While playing with some PDE I came across a singular integral that looks something like
$$T(f_1,f_2,\ldots,f_n)(x)=p.v.\int_{-\infty}^\infty\frac{(f_1(x)-f_1(y))(f_2(x)-f_2(y))\cdots(f_n(x)-f_n(y))}{(...

**5**

votes

**1**answer

138 views

### Multidimensional integrals that diverge by oscillation

It's not hard to extend the theory of integration over ${\bf R}$ so that the integral of any compactly supported function is its usual value, while the integral of $f(t) = \cos (at+b)$ (with $a \neq 0$...

**9**

votes

**1**answer

261 views

### Is the regularization of a Fourier transform unique?

The Fourier transform of the Coulomb potential $1/\vert \mathbf r \vert$ of an electric charge doesn't converge because one obtains
$$F(k)=\frac {4\pi}{k} \int_0^\infty \sin(kr) dr.$$
The standard ...

**0**

votes

**2**answers

78 views

### Root of a special rational function with positive coefficients

During my research I came across the following problem:
I need to find a root of the following function:
$$\Gamma_{N}(x) = \sum\limits_{i=0}^{M}\left(\frac{\sum\limits_{n=0}^{n_F}n\ \alpha_{i,n} x^n}{...

**1**

vote

**1**answer

66 views

### Inverting two paraboloid relations

Let's suppose we have two variables $(\alpha, \beta)$ on two functions $k_1, k_2$ that can be defined in terms of matrix relations:
$$
k_1 = \left| \xi^T \mathcal{F}^T \mathcal{F} \xi \right|
$$
$$
...

**0**

votes

**0**answers

89 views

### The norm of the operator in the Calderon-Marcinkiewicz interpolation theorem

I read a general Marcinkiewicz interpolation theorem (the Calderon-Marcinkiewicz theorem) in J.Bergh's book "Interpolation Spaces - An Introduction".(Page 113-114, Theorem 5.3.2). If $T:L_{p_ir_i}\to ...

**0**

votes

**0**answers

34 views

### Interpolation functional for BV spaces?

Recently, while trying to tackle a problem, I found that it would be convenient that I could find some sort of 'interpolation theorem' for Bounded Variation Spaces. More specifically, let's define, ...

**3**

votes

**2**answers

74 views

### Sequence of subharmonic functions on shrinking domains

Set $G_\eta:=\{(x,y)\in \mathbb{R}^2|-\eta<x<\eta, 0<y<1\}$. If $u_\eta\geq 0$ is a sequence of subharmonic functions defined on $G_\eta$ such that
$$
\int_{G_\eta}|u_\eta|^2dx\wedge dy\...

**-4**

votes

**1**answer

284 views

### Proof of formula for $\pi$ [closed]

The number $\pi$ can be expressed as $\pi=\lim_{n\to\infty} \frac{n\sqrt[n]{-1}-n}{\sqrt{-1}}$ or more poetically $\pi=\frac{\infty\sqrt[\infty]{-1}-\infty}{\sqrt{-1}}$. Here we choose the principal ...

**6**

votes

**0**answers

190 views

### Density of polynomials in $C^k(\overline\Omega)$

Let $\Omega$ be an open and bounded subset of $\mathbb{R}^2$ and let $C^k(\Omega)$, $1\leq k<\infty$, be the space of functions $f$ with continuous derivatives of order $\leq k$ in $\Omega$, ...

**1**

vote

**0**answers

29 views

### Pointwise convergence of a sequence of approximate limits of BV functions

So, let $\Omega\subset \mathbb R^2$ bounded and consider a sequence of functions $\{u_k\}_{k\in\mathbb N}\subset BV(\Omega)$ and $u\in BV(\Omega)$ such that $u_k\rightarrow u$ weakly* in $BV(\Omega)$. ...

**2**

votes

**0**answers

67 views

### Nonlinear smooth bijection from $\mathbb Q$ to itself [duplicate]

Is there a bijection $\phi: \mathbb Q \to \mathbb Q$ such that
$\phi$ is nonlinear: different from $ax+b$,
$\phi$ is smooth: the extension $\hat{\phi}$ of $\phi$ over $\mathbb R$ is $\mathcal C^2$ ?
...

**2**

votes

**0**answers

165 views

### Metric analogues of bounded variation

A function $f:[a,b]\to\mathbb{R}$ is said to be of bounded variation if
$$ \sup_I \sum_{i=1}^n |f(x_i)-f(x_{i-1})| \le V $$
for some finite $V>0$, where the supremum is over all finite partitions
$...

**1**

vote

**1**answer

70 views

### Lower Matuszewska index of positive increasing $O$-regular functions

I am not sure if this question is too specific on notations (I think the question is intuitive, but basically the only reference I know with this kind of notations is Bingham, Goldie & Teugels ...

**0**

votes

**0**answers

50 views

### stochastically decreasing sequence converges in distribution

Let $(X_i)_{i=1}^\infty$ be independent nonnegative integer valued random variables. Suppose that $X_n \succeq X_{n+1}$ (in the stochastic dominance sense). Does it follow that $X_n \overset{d}\to X$ ...

**1**

vote

**0**answers

57 views

### Norm-averaging reference request

(Apology in advance for the broadness of this question) I recently came across a relatively simple application where I needed to "balance" the "spreaded-out-ness" of a function with the "peaked-ness" ...

**3**

votes

**1**answer

97 views

### Restrictions on spectral measure

Given any Borel measure $\mu$ on $\mathbb{R}$, define a map that sends any $f\in C_c(\mathbb{R})$ to $$T_\mu(f)(y)=\int \langle\exp(-i x \lambda),f(x)\rangle\exp(iy\lambda)d\mu(\lambda).$$
Here $\...

**0**

votes

**1**answer

47 views

### On 1-iso maps and subsets of the unit circle

Let $S$ be the unit circle and for any $x,y \in S$ let $d(x,y)$ be the lenght of the smallest arc between $x$ and $y$. A bijective map $\phi : S\longrightarrow S$ is called 1-iso if the following ...

**1**

vote

**0**answers

70 views

### Roots of generalized homogeneous polynomials

A polynomial $P(\xi)$ of $n$ real or complex variables is homogeneous of order $m$ with respect to $\lambda \in \mathbb{Z}_{+}^n$ if $$P(\xi) = \sum\limits_{\substack{(\alpha, \lambda) = m \\ \alpha \...

**8**

votes

**1**answer

139 views

### Does infinitesimal variance imply continuity?

Let $u:[0,1]\to\mathbb{R}^n$ be a bounded Borel function.
It is well-known that if, for any compact interval $I\subseteq [0,1]$,
$$ \int_I|u-u_I|^2\le C|I|^{1+\alpha} $$
for some $C,\alpha>0$ (here ...

**1**

vote

**0**answers

71 views

### Differentiability criterion in the Zygmund class

Let $ f: \mathbf{R}^{m} \rightarrow \mathbf{R} $ be a continuous
function, $ \omega $ be a modulus of continuity and assume
$$ | f(x+h) +f(x-h) -2f(x) | \leq \omega(|h|)|h| $$
whenever $ x,h \in \...

**0**

votes

**0**answers

55 views

### limit multiple integral

I want to know if $\lim_{T-> \infty}$ of this integral
$$ \frac{\sigma^{4}C_{H,K}^{2}}{4 T^{4HK}e^{2\theta T }}\\
\times \int\limits_{[0,T]^{4}}e^{\theta(t_{1}-s_{1})}e^{\theta(t_{2}-s_{2})}\left\...

**1**

vote

**1**answer

109 views

### What are the spaces for which the Fourier transform is an automorphism? [closed]

this is well-known that the Fourier transform is an automorphism of $L^2(\mathbb R)$ and also of $\mathcal S(\mathbb R)$ (Schwartz space). Is there any other spaces of functions of one real variable ...

**1**

vote

**0**answers

60 views

### An inequality for integral on spheres

I have a question concerning to the integral on sphere. It's maybe true and simple but I don't know how to prove it. Could anyone have some suggestions? Thanks.
Denote $S^{n-1}$ the unit sphere in $R^...

**2**

votes

**1**answer

117 views

### $L^p$-bounding inequality [closed]

Do we have that$$\|Du\|_{L^{2p}} \le C\|u\|_{L^\infty}^{1\over2} \|D^2u\|_{L^p}^{1\over2}$$for $1 \le p < \infty$ and all $u \in C_c^\infty(U)$? Here, $U$ denotes an open subset of $\mathbb{R}^n$.

**3**

votes

**1**answer

88 views

### Sobolev inequality involving summing from $j = 0$ to $m - 2$, exists constant

Let $I = (0, 1)$ and $1 \le q < \infty$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, q)$ such that$$\|D^{(m - 1)}u\|_{L^q(I)} + \sum_{j = 0}^{m - 2} \|D^ju\|_{L^\infty(I)} \le \...

**1**

vote

**1**answer

128 views

### A problem about the quotient space of an extended Dirichlet space

Let $(\mathscr{E},\mathscr{F})$ be a recurrent Dirichlet form on $L^2(X;m)$ and $\mathscr{F}_e$ the corresponding extended Dirichlet space, then $1\in\mathscr{F}_e$ and $\mathscr{E}(1,1)=0$. Let ${\...

**0**

votes

**1**answer

77 views

### a sum of ratios of quadratic forms

I have the following function that I would like to optimize over the value A
$$f(A)=\sum_k \frac{\mathbf{y}_k^H\left[\begin{array}{cc} 1&0\\ 0& A \end{array} \right]\mathbf{x}_k\mathbf{x}_k^H\...

**9**

votes

**2**answers

229 views

### Two elementary inequalities for real-valued polynomials

I am looking for references discussing two inequalities that come up in the study of the dynamics of Newton's method on real-valued polynomials (in one variable). The inequalities are fairly different,...