**1**

vote

**2**answers

265 views

### May integration spoil real-analyticity?

Is there an example of a function $f:(a,b)\times(c,d)\to\mathbb{R}$, which is real analytic and bounded in its domain, integrable in the second variable, and such that the function
$$ ...

**-1**

votes

**0**answers

13 views

### Dimension of the set of the polynomial growth harmonic function on the hyperbolic plane

We consider the hyperbolic plane and the harmonic function there. Pick any point $p$. Let $H_n, n \in\mathbb N$ be the set of the harmonic functions $f$ such that $|f(x)|\leq c d(x,p)^n$.
What is the ...

**15**

votes

**0**answers

423 views

### A Linear Order from AP Calculus

In teaching my calculus students about limits and function domination, we ran into the class of functions
$$\Theta=\{x^\alpha (\ln{x})^\beta\}_{(\alpha,\beta)\in\mathbb{R}^2}$$
Suppose we say that ...

**-4**

votes

**0**answers

44 views

### Are all derivatives of sinc function bounded on real axis? [on hold]

It seems that all derivatives of sinc function (sinc(x)=sin(x)/x) are bounded on real axis. Is it true or no? Thanks in advance.

**2**

votes

**1**answer

117 views

### Hardy space, Lebesgue space for $p<1$,

We denote $\mathcal D'(\mathbb R^n)$ the space of distributions, and $\mathcal D(\mathbb R^n)$ the space of smooth, compactly supported functions.
Let $\rho\in \mathcal D'(\mathbb R^n)$ such that ...

**35**

votes

**3**answers

958 views

### Why does $d^n \exp(-x-x^{-1})/(dx)^n$ only have $n$ positive real zeroes?

Set $f(x) = \exp(-x-x^{-1})$. An easy induction shows that
$$\frac{d^n}{(dx)^n} f(x) = \phi_n(x^{-1}) f(x)$$
for $\phi_n$ a polynomial of degree $2n$. Clearly, the roots of $\phi_n(x^{-1})$ are the ...

**0**

votes

**1**answer

41 views

### For a given $n$, under what condition(s) there exists (at least) two different $c$ and $c′$ such that $X_n^c=X_n^{c'}$

Let $X_n^c=\{\cos\left((4k-c)\frac{\pi}{2n}+\frac{\pi}{4}\right): k=0, 1, \dots, n-1\}$ where $c\in\{0, 1, \ldots, \lfloor\frac{n}{2}\rfloor\}$ and $n$ is any positive integer greater than 3. I want ...

**1**

vote

**0**answers

170 views

### Density of subspace with nonlocal/Wentzell boundary condition

Given the space $F$ defined by:
$$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$
I want to prove that the subspace $E$ of $F$ defined by ...

**3**

votes

**1**answer

94 views

### Scaling properties of the Hölder estimate for heat equation

Lately, I have been interested in scaling properties of parabolic equations, and this question is related to an earlier one I asked about Harnack constants.
Let $Q(R) := Q(R^2,R) = B(0, R) \times ...

**-3**

votes

**0**answers

94 views

### On the common zeros of $1-x\tan{x}$ and $1-1/x\arctan{1/x}$ [on hold]

Let $f(x)=1-x\tan{x}$.
Let $g(x)=1-1/x\arctan{1/x}$.
Let $r=0.8603335890193797624838\ldots$ be a real root of $f(x)=0$.
High precision numerical computations suggest $f(\pm r)=g(\pm r)=0$.
$g(x)$ ...

**3**

votes

**0**answers

96 views

### Involutions on $[0,1]$ given by power series (related to probability generating functions)

Let $A$ be a function from $[0,1]$ to $[0,1]$. $A$ is an involution if $A(A(x))=x$ for all $x\in[0,1]$.
Which involutions $A$ exist such that $A(x)=\sum_{k=0}^\infty a_k x^k$ with $a_0=1$ and ...

**-4**

votes

**0**answers

23 views

### The derative as the slope of the tangent [closed]

Regarding teaching the derative via tangents, do you have any source material that I can access?

**10**

votes

**5**answers

435 views

### Identities and inequalities in analysis and probability

Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of ...

**1**

vote

**0**answers

35 views

### Can Mumford-Shah functional be adapted to lower $L^1$ space?

The well know Mumford-Shah functional functional
$$
F(u)=\int_\Omega|\nabla u|^2+\mathcal H^{N-1}(S_u) \tag 1
$$
where $u\in SBV(\Omega)$ and $\nabla u$ is the absolutely continuous part of ...

**0**

votes

**0**answers

23 views

### Absolute continuity and the Luzin N-Property for functions of two variables

It is a well known fact that absolutely continuous functions of one real variable have the so-called Luzin N-property. That is, if $E\subset\operatorname{Domain}(f)$ has zero measure, then $f(E)$ has ...

**4**

votes

**1**answer

183 views

### Smoothening a measure, II

There is an almost invisible, but significant difference between the question below and that recently answered by Boris Bukh.
Given a probability measure $\mu$ supported on a finite set ...

**2**

votes

**1**answer

120 views

### Smoothening a probability measure

Given a probability measure $\mu$ supported on a finite set $S\subset{\mathbb R}^2$, define
$$ f(z):=\max\left\{\frac{\mu(x)+\mu(y)}2\colon \frac{x+y}2=z,\ x,y\in S \right\},
\ z\in{\mathbb ...

**1**

vote

**0**answers

43 views

### Monotone version of one-dimensional Whitney extension theorem

Is there a version of the Whitney extension theorem that would extend a monotone $C^\infty$ function on a compact subset of $\mathbb R$ (satisfying the usual Whitney's compatibility conditions) to a ...

**2**

votes

**1**answer

85 views

### Heat equation: impact of the diffusion coefficient on the Harnack constant

Consider the heat equation
$$
u_t - div[a(x,t) \nabla u] =0,\quad (x,t) \in B(r) \times [-r^2, 0] \subset \mathbb R^{d+1}
$$
for a Hölder continuous coefficient $a(x,t)$ satisfying
$$
0<C_o \le ...

**7**

votes

**6**answers

762 views

### Almost-converses to the AM-GM inequality

Let us consider the Arithmetic Mean -- Geometric Mean inequality for nonnegative real numbers:
$$ GM := (a_1 a_2 \ldots a_n)^{1/n} \le \frac{1}{n} \left( a_1 + a_2 + \ldots + a_n \right) =: AM. $$
...

**30**

votes

**1**answer

1k views

### Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$

Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ such that $f(x)=0$ on $\mathbb{R}\setminus (-1,1)$. Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least $n+1$ zeros on $(-1,1)$
...

**1**

vote

**3**answers

386 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 ...

**3**

votes

**1**answer

470 views

### Does integrating with respect to a finitely additive measure respect addition?

Let $X$ be a set and $\mathcal{A} \subseteq P(X)$ a $\sigma$-algebra. Assume $\nu : \mathcal{A} \to [0,\infty]$ is a finitely additive measure. If $f : X \to [0,\infty]$ is a measurable function, we ...

**0**

votes

**0**answers

46 views

### The union of weighted compact supported continuous function

Let $\Omega\subset \mathbb R^N$ be open. Given a weight function $v\geq 1$ such that $v\in L^1_{\text{loc}}(\Omega)$ and $l.s.c$. Also supposethere exists a Lipschitz continuous sequence $v_n$ such ...

**6**

votes

**1**answer

610 views

### Maximal ideals of the rings of Baire-One Functions

A real function $f:X\rightarrow \mathbb{R}$ is called Baire-one function, if there is a sequence $(f_n)_{n=1} ^\infty$ of continuous functions $f_n:X\rightarrow \mathbb{R}$ on $X$ so that for all ...

**2**

votes

**2**answers

104 views

### The convolution between weighted $L^1$ space and normal $L^1$ space

Let $\omega\in L^1_{\text{loc}}(\mathbb R^N$) be given. We assume that $\omega\geq 1$, l.s.c, and satisfies, for a constant $C>0$,
$$
\frac{1}{|B(x,r)|}\int_{B(x,r)}\omega(y)dy\leq C\omega(x)
$$
...

**3**

votes

**1**answer

110 views

### Real analysis on vector-valued spaces, $L^{p}(\mathbb{R}^N,E)$ ,$H^{s}(\mathbb{R}^N,E)$

I am dealing with some vector-valued Sobolev spaces $H^{s}(\mathbb{R}^N,E)$ where $E$ is a Banach space.
I am looking for references about results for the scalar case ...

**0**

votes

**1**answer

91 views

### Discussion for the sign of a specific sum

Given a modular form $f$ of an even weight $k$ for the full modular group. Let $\lambda_f(n)$ the $n$-th normalized Fourier coefficient of $f.$ For a fixed positive integers $a$ and $b,$
I want to ...

**0**

votes

**0**answers

25 views

### The Best Korn's constant for bounded deformation

I am studying the following version of Korn's inequality. For $u\in BD(\Omega)$, $BD$ denotes the bounded deformation space, we have, there exists a $r(u)\in \operatorname{ker}\mathcal E$ such that
...

**4**

votes

**2**answers

187 views

### Dependence of the constant in Korn's inequality on the domain

Let $\Omega \subset \mathbb{R}^N$ be an open, connected set with Lipschitz boundary, $N \geqslant 2$ and
$$ \mathcal{E} ( v) := \int_{\Omega} \sum_{i, j} \varepsilon_{i
j} ( v) \varepsilon_{i j} ...

**2**

votes

**1**answer

695 views

### For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?

I am trying to characterize all measures on $\mathbb{R}$ such that
$$
\sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty,
$$
where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...

**5**

votes

**0**answers

171 views

### Version of Stone Weierstrass for functions not vanishing at infinity

I am trying to see what is known about uniform density of function spaces in $C(\mathbb{R}^n)$ or $C_b(\mathbb{R}^n)$ (bounded continuous functions on $\mathbb{R}^n$). By uniform density, I mean ...

**2**

votes

**1**answer

99 views

### Convolution vanishes on an interval

Fix a "test" function $f(x)=x\exp(-x^2)$, which is nonzero except $x=0$. Suppose that $g$ is a function with some necessary regularity. Consider the convolution.
$$
(f\ast g ...

**0**

votes

**1**answer

117 views

### How can I show that “almost all function” have property P?

The following is cross-posted from
http://math.stackexchange.com/questions/1391293/is-almost-all-function-a-well-defined-concept
since I didn't (yet) get an answer there.
(I hope that's okay?)
...

**1**

vote

**0**answers

11 views

### Is this function $u\in SBV(\Omega)$ also belongs to $L^\infty(\Omega)$?

Some early discussion can be found here.
My question:
Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Also $\mathcal H^{N-1}(\partial\Omega))<\infty$.
Let $u\in ...

**0**

votes

**1**answer

176 views

### Is this set of function belongs to $L^\infty$?

Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $u\in SBV\cap L^\infty(\Omega)$ be given. We write
$$
Du = \nabla u\lfloor \mathcal L^N + (u^+-u^-)\otimes \nu_u\mathcal ...

**4**

votes

**1**answer

665 views

### A result attributed to Whitney

One of the basic results of real analysis says that any closed subset of a smooth ($C^\infty$) manifold $M$ is the set of zeros of some map $\lambda\in C^\infty(M;[0,1])$. This result (or some ...

**5**

votes

**4**answers

1k views

### analysis over non-Archimedean ordered fields

Can anyone suggest any good references for (or any experts on) analysis over non-Archimedean ordered fields, such as the field of rational functions in one variable (ordered at 0, or if you prefer at ...

**0**

votes

**2**answers

58 views

### Smoothness of a power of smooth non-negative function [closed]

Let $f$ be a non-negative infinitely smooth function on the real line. Is it true that for any constant $\alpha$ the function $f^\alpha$ is infinitely smooth?

**0**

votes

**0**answers

33 views

### Is the heat kernel satisfies the heat equation in viscosity sense?

Let us see the heat kernel
\begin{equation}
k(x,t)=
\begin{cases}
(4\pi t)^{-\frac{n}{2}}\exp\{-\frac{1}{4}t^{-1}|x|^{2}\},t>0\\
0,t\leq0.
\end{cases}
\end{equation}
It is easy to see that $k\in ...

**1**

vote

**1**answer

54 views

### Image of a Jordan compact set under a degenerate map

This is crossposted from MSE, I hope this is suitable here, since there is no reaction there. I need this lemma for teaching, and I would appreciate any help.
Briefly:
Is the image of a Jordan ...

**1**

vote

**1**answer

434 views

### Uniform $L_1$ convergence implies uniform convergence pointwise a.e.

Let $\Omega$ be a measure space (which can be assumed to be an interval with Lebesgue measure).
It is well known that for a sequence $(f_n)$ in $L^1(\Omega)$ which converges to zero (in ...

**6**

votes

**1**answer

130 views

### Brownian motion, exists $c < \infty$?

Suppose $B_t$ is a standard Brownian motion. Does there exist $c < \infty$ such that with probability one$$\limsup_{t \to \infty} {{B_t}\over{\sqrt{t \log t}}} \le c?$$I need to know whether or not ...

**0**

votes

**0**answers

28 views

### Uniform convergence problem of the iterative function series

A process $\{\theta_{t}\}_{t=1}^{\infty}$ with finitely continuous state space $\mathcal{S}=[\underline{\theta},\bar{\theta}]$.The transition density is $\phi(\theta_{t},\theta_{t+1})$.I have known ...

**0**

votes

**0**answers

42 views

### The property reservation conditions in the functional iteration process

Given a integral equation:
$$K(y,p)=\int_{a}^{b}f(x,y)K(x,p)dx$$
Using the iteration method,choosing an arbitrary inition function $K^{0}(x,p)$:
$$K^{1}(y,p)=\int_{a}^{b}f(x,y)K^{0}(x,p)dx$$
...

**0**

votes

**1**answer

203 views

### The weighting function for the infinite product of necklaces

Let us consider the limit $\lim_{n\to \infty}\prod_{p=1}^n N(p,a)$ where $N(n,a)$ is the number of fixed necklaces of length $n$ composed of $a$ types of beads.
Let's rewrite the product in a way ...

**1**

vote

**1**answer

189 views

### Two Concepts of Monotonicity

Let $K$ be a closed convex subset in $\mathbb{R}^n$ and $F: K\rightarrow \mathbb{R}^n$. We say that
$F$ is strongly monotone on $K$ if there exists $\gamma>0$ such that
$$
\langle F(y)-F(x), ...

**5**

votes

**1**answer

312 views

### Are piecewise linear curves dense among Hölder curves?

Consider for some $0 < \alpha \leq 1$ the space functions $x:[0,1] \to \mathbb{R}^n$ such that $x(0) = 0$ and
$\sup_{s,t} \frac{\|f(t)-f(s)\|}{|t-s|^{\alpha}}$
is finite.
There are at least two ...

**1**

vote

**0**answers

72 views

### Norm inequalities between difference operators

Assume $(v_i)$ to be a sequence in $\ell_\infty(\mathbb{R})$ for $i=1,\dotsc,N$. Define the difference operator as $\Delta v_i:=v_{i+1}-v_i$ and $\Delta^n v_i:=\Delta(\Delta^{n-1} v_i)$. Then, how can ...

**0**

votes

**2**answers

81 views

### Expected summation of dropped intervals?

For each $n\in\mathbb{N}$, let $I_n$ be an interval of length $1/2^{n}$. We drop each $I_n$ onto the interval $[0,1]$ uniformly at random (so that there is "wraparound" if need be). What is the ...