The real-analysis tag has no wiki summary.

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48 views

### In Dedekind' construction of real numbers, what's wrong with this understanding [on hold]

in this prove, every cut corresponds to a real number. and a cut is a subset of Q.and cut have these three properties.1.is not empty 2.if p belong to this cut,any qp
so in my understanding, every cut ...

**-1**

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**0**answers

43 views

### Particular case of every sequence has a Cauchy subsequence? [on hold]

A metric space (X,d) has the following property:
Given $\epsilon >0$ and non-empty finite subset $X_\epsilon \subset X$
$$ \inf \{ d(x,p) : p \in X_\epsilon \} < \epsilon$$
I would like to ...

**0**

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**0**answers

45 views

### Sequence of cosine converges? [on hold]

Does the following sequence
$$(\cos (\pi \sqrt{ n^2 + n}))_{n=1}^\infty $$
converge?
Can I use the ratio or root test?

**0**

votes

**0**answers

55 views

### Bolzano-Weierstrass application? [on hold]

I am having problems proving the following claim:
Given a bounded set $A \subset R^n$, I want to prove the existence of $a_1, \dots, a_N \in R^n$ and numbers $r_1, \dots, r_N \in [0, +\infty)$ such ...

**0**

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**3**answers

65 views

### On the existence of compactly supported functions whose its Fourier transform satisfies a given condition

My question is concerned with the existence of compactly supported functions whose its Fourier transform satisfies a given condition: For $\gamma\ge 1$, one can prove that there is no compactly ...

**0**

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**0**answers

10 views

### Bounding a function pointwise from bound on the expectation [migrated]

Suppose we have a Lipschitz continuous positive function with bounded expectation Eg < a. What can be said about the point-wise bound ?

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62 views

### Uniqueness of the maximum derivative of a rational function

This may seem like an elementary question, but bear with me; you'll find that it is actually quite hard. Consider the function
$$f(x)=\frac{a_nx^n}{\sum_{i=0}^n a_ix^i}$$
with all $a_i\geq 0$ and ...

**0**

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**0**answers

43 views

### Which of the following is true? [on hold]

$f(x),g(x)$ are defined on $[-1,1]$, $f'(0),g'(0)$ exist, $f(0)=g(0)$, and $f(x)\ge g(x)$ holds for an open interval containing $0$. Then which of the following is correct:
I, $f(x)$ and $g(x)$ have ...

**3**

votes

**1**answer

155 views

### Chances for a cosine polynomial to be positive at a point

Let $k_1,\ldots,k_n$ be distinct integers. Let $s_n(t)=\cos (k_1t)+\cdots+\cos (k_nt)$ be a trigonometric sum. Consider any interval $I\subset [-\pi,\pi)$ of length $\delta=\delta(n)$. Let $\,U$ be a ...

**0**

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**0**answers

51 views

### How to prove $\lim _{ \delta\rightarrow {0}^{+}}\int_{a}^{b} F_{\delta}(x)dx=0 ? $ [on hold]

This question comes from http://math.stackexchange.com/questions/982231/a-function-fx-that-riemann-integrable-on-a-b
Define a function $f(x)$ that Riemann integrable on $[a,b]$.
Let
...

**2**

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**1**answer

117 views

### Quantitative stability: Hausdorff distance between subdifferentials

Suppose I have two convex functions $f$ and $g$ mapping $\mathbb{R}^{n}\to\mathbb{R}$ (so they are 'more' than proper). Suppose $\|f-g\|_{\infty}<\epsilon$, the sup-norm. In particular, the ...

**1**

vote

**1**answer

168 views

### Can (how) one distinguish germs of continuous functions by a countable set of params?

Continuous functions can be distinguished by their values at say rational points of [0 1].
Germs of analytic functions can be distinguished by derivatives at a point.
So in both cases we see ...

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**0**answers

44 views

### Existence and Uniqueness of Volterra integral equations of the first kind

$$
\int_0^t k(s,t)f(s)ds=g(t)
$$
To prove the existence and uniqueness of solutions of Volterra integral equation(VIE) of the first kind, we usually differentiate it and convert to the VIE of the ...

**3**

votes

**1**answer

207 views

### “Nice” functions on infinite-dimensional space of germs of continuous functions at a point

Consider set of all germs of continuous functions at some point.
Question: What are some functions ("any/nice/constructive/whatever") from this set to R (reals) ? (Except evalution at point and made ...

**-2**

votes

**1**answer

156 views

### A calculus question [closed]

Fix $q>1$. Define the function
$$
f_q(c):=\int_e^\infty \frac{e^{-c r^2}r}{\log(r)^q}d r.
$$
The problem is whether the following is true,
$$
\lim_{c\rightarrow 0} c \log(1/c)^q f_q(c) = C \in ...

**0**

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**0**answers

47 views

### Guess A Property Of The Integral Average Value Function [closed]

Let $f$ be a function that is defined on $[a,b]$ and Riemann integrable on $[a,b]$.
Def 1.
$$\hat f(x)=\begin{cases}
f(x),& \text{if }x\in[a,b], \\
f(a),& \text{if }x<a, \\
...

**4**

votes

**1**answer

256 views

### Strong Law of Large Numbers for arrays of partly dependent random variables

Suppose $X_1$, $X_2$ are two independent real-valued random variables. Let $F$ be a continuous (unbounded) function from $\mathbb{R^2}$ to $\mathbb{R}$. Assume that the necessary measurability and ...

**-1**

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**2**answers

61 views

### Inner Product of Given Sum Positive Sequence

Let $$A = \Big\{(a_1,a_2,\dots)\ \Big|\ a_i\ge 0, \sum_{i=1}^\infty a_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum_{i=1}^\infty a_ib_i\ \bigg|\ (a_i)_{i=1}^\infty,\, (b_i)_{i=1}^\infty \in ...

**0**

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**0**answers

98 views

### Can we define log-convex operators?

Let $I\subset\mathbb{R}$. A function $f:I\rightarrow\mathbb{R}$, is said to be log-convex if $\log f$ is convex or equivalently for all $x,y\in I$ and $\alpha\in [0,1]$
$$f(\alpha x+(1-\alpha)y)\leq ...

**0**

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**1**answer

46 views

### Analytic extension of the exterior Newtonian potential into the domain

I just want to know for a domain with smooth boundary whether exterior Newtonian potential has singular analytic extension into the domain or into a part of the domain.
Definition of Newtonian ...

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**0**answers

56 views

### How is this transformation related to the Legendre transform?

I stumbled over the following transform in a statistical mechanics paper:
Unfortunately, no mathematical details were given there, which is why I wanted to ask here about this transform.
Let $s : ...

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**0**answers

143 views

### Establishing an upper bound for a dyadic average of a function in $ {L^{p}}([0,1)) $

Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1)) $, where $ p > 1 $. Let
$$
(D_{n})_{n \in \mathbb{N}_{0}} =
\left( \left\{
I^{n}_{j} \stackrel{\text{df}}{=} \left[ ...

**27**

votes

**1**answer

2k views

### What did Rolle prove when he proved Rolle's theorem?

Rolle published what we today call Rolle's theorem about 150 years before the arithmetization of the reals. Unfortunately this proof seems to have been buried in a long book [Rolle 1691] that I can't ...

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**0**answers

30 views

### Stability of simple conditions on functions under convolution and/or mixture

We consider families of smooth probability densities defined on $\mathbb{R}^+$, $p=(x\in \mathbb{R}^+ \mapsto p_n(x))_{n\in\mathbb{N}}=(p_n)_{n\in\mathbb{N}}$ satisfying
(i) $\int_{\mathbb{R}^+} ...

**1**

vote

**1**answer

124 views

### Estimates on evolution operator

Let's consider the following evolution operator in $\mathbb{R}^3$
$$S(t)=e^{(i+\delta)t\Delta }$$
How to get the following estimate
$$\Vert S(t)f\Vert_2\leq C_\varepsilon t^{-\frac{1}{4}}\Vert ...

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**1**answer

104 views

### Relationship between LlogL and Hardy spaces

I think that for positive, one-dimensional, periodic functions, the following statement is true:
$$
f\in L log L(\mathbb{T})\Leftrightarrow f\in H^1(\mathbb{T}),
$$
where
$$
LlogL=\{f\in ...

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59 views

### Equivalence of ensembles and $\sum_{N=0}^{\infty} \int_{-Bn}^{\infty} 1_{\text{{N>K or u>A}}}(u)e^{-\beta |\lambda_m|(\frac{u}{2}+n)}du $

I am thinking about a step in the proof of the equivalence of ensembles in StatMech on page 25. step from 3.19 to 3.20.
It seems to be argued there that for $A$ and $K$ large enough, the term
...

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**2**answers

246 views

### Non-smooth function with all differences of translates smooth?

Suppose $f:\mathbb{R} \to \mathbb{R}$ has the property that for every fixed $t\in\mathbb{R}$ the function
$$
g_t : x \mapsto f(x) - f(x-t)
$$
is $C^\infty(\mathbb{R})$. Does it follow that $f$ is ...

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240 views

### Partitions of ${\rm Sym}(\mathbb{N})$ induced by convergent, but not absolutely convergent series

Let $(a_n) \subset \mathbb{R}$ be a sequence such that the series
$\sum_{n=1}^\infty a_n$ converges, but does not converge absolutely.
Then there is a partition of the symmetric group ${\rm ...

**1**

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**1**answer

93 views

### Evaluation of the multiple integral [closed]

Would you give me any suggestions or comments on evaluating the following $n$-dimensional
integral? $$ \int_{[0,t]^n} h(x) dx $$
where
$ x=(x_1 ,x_2 , \cdots, x_n ), h(x)= \prod_{k=1}^n min( ...

**0**

votes

**1**answer

117 views

### Most natural smooth interpolation of 1,4=2^2,3^27,4^4^4^4, [closed]

Is there a functional equation for extending this to a smooth real function?

**1**

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**1**answer

30 views

### Can we implicitly fit a system of linear ODEs by reduced information?

Assume we are given several initial vectors $x^{(1)},\ldots,x^{(r)} \in \mathbb{R}^n$, where the dimension $n$ is in the range of 50 to 100, and the number of initial vectors $r$ is in the range of ...

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**1**answer

92 views

### Extending derivations to the superposition closure

Let $X$ be a set and $\mathcal{F}\subseteq {\mathbb{R}^X}$ an arbitrary family of functions.
The superposition closure of $\mathcal{F}$ is defined as
$$
\overline{\mathcal{F}}=\{ ...

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**0**answers

80 views

### An integration limit

Given $z\geq 0$, denote
$$A_m(z) = \{x\in \mathbf R^{m-1}\, :\, \min_{1\leq i\leq m-1} x_i > z\},$$
and
$$F_m(z) = \int_{A_m(z)} (1+|x|^2)^{1-m} dx.$$
Does the following limit
...

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**0**answers

46 views

### question about a genralized Skorokhod topology

Let $D:=D([0,1], R)$ be the space of all cadlag functions defined on $[0,1]$. Now we have the known Skorokhod topology defined by: $\forall f, g\in D$
...

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83 views

### Convergence in energy of bounded (semi)subharmonic functions

Consider a sequence $(f_n)$ of functions in the flat torus $T^d$ converging Lebesgue-a.e. to a limit function $f$.
Assume that:
1) $|f_n|(x)\leq 1$ for every $n,x$
2) $\Delta f_n\geq -1$ in the ...

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71 views

### Is every supersmooth function a local polynomial?

This question is a follow up question to this question that I recently asked.
A $C^{\infty}$ function $f:(c,d)\rightarrow\mathbb{R}$ shall be called a local polynomial if whenever ...

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188 views

### Does there exist a supersmooth non-polynomial function?

Let's call a $C^{\infty}$-function $f:\mathbb{R}\rightarrow\mathbb{R}$ Lebesgue supersmooth if whenever $a_{n}\in\mathbb{R}$ for all $n$, then $\lim_{n\rightarrow\infty}a_{n}f^{(n)}(x)\rightarrow 0$ ...

**2**

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**2**answers

88 views

### First order pde with characteristics [closed]

Consider a first order pde of the type $$u_y+b(x)u_x=0$$ and suppose that the coefficient $b$ is not necessairly continuous (for instance with a jump in some point).
Is it still possible to apply in ...

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**0**answers

29 views

### computational question concerning singular integral theory

Let $m\in C^\infty(\mathbb{R}^d-\{0\})$ be homogeneous of degree zero and has mean zero on the sphere$(S^{d-1})$. Then $m$ defines a tempered distribution and $\partial_j^dm\in S'(\mathbb{R}^d)$ is ...

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114 views

### extension of function in an abstract metric space

my question is the following.(Maybe my title is not quite proper for this question):
Let $(E,d)$ be a Polish space (or a separable metric space), let $\xi: E\to R_+$ be a Lipschitz function. Now set ...

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**1**answer

93 views

### Characterization of a subset of [0,1] $III$

I have a question related to the previous one.
Characterization of a subset of [0,1] $II$
Let $T\subseteq [0,1]$ be some subset closed under lower limit topology, i.e.
$t_n$ is said to converge to ...

**-1**

votes

**1**answer

151 views

### Does the divergence of the sum of reciprocals of a set of integers imply this density statement about the set?

Suppose $A \subseteq \mathbb{N}$ is such that $\displaystyle{\sum_{n \in A} n^{-1}} = \infty$. Suppose $B \subseteq \mathbb{N}$ is infinite.
Is there a set $X \subseteq [1,\infty)$ and a increasing ...

**2**

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**1**answer

69 views

### Does directional limits along any given direction, always exist for a function of bounded variation?

If a function $f:\mathbb{R}^N\to\mathbb{R}$ is of bounded variation (in modern sense or in Tonelli sense or according to any of the existing definitions), then can we say that, given any point $x\in ...

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86 views

### Density of function spaces

Let $\Omega$ be a subset of either $\mathbb{R}^n$ ($n\geq 3$, if it matters) or of a compact manifold. In either case, we'll call the manifold $M$. Let $V_i\subset V_{i+1} \subset \Omega$ be an ...

**3**

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**1**answer

153 views

### Generalized Hardy-Littlewood-Sobolev Inequality

The Hardy-Littlewood-Sobolev Inequality says that
$$\text{for $p,q,r\in (1,+\infty)$ such that }\quad
1-\frac1p+1-\frac1q=1-\frac1r,\tag {$\sharp$}
$$
$$
\exists C, \forall u\in L^p(\mathbb ...

**1**

vote

**1**answer

201 views

### is $x_{n}\ll \overline{x}_{n}^{2}$?

This question is a cross-post from MSE, cause I didn't get any answer there. I hope it is well suited for MO:
Let $(x_{n})_{n\ge 1}$ be an increasing sequence of positive integers and ...

**7**

votes

**1**answer

234 views

### Counting norms on an infinite dimensional vector space

It is known that whenever E is a finite dimensional real vector space, there is only one norm on E up to equivalence (actually one non discrete vector space topology).
Is it known what happens when E ...

**3**

votes

**1**answer

223 views

### Characterization of a subset of [0,1] $II$

My question follows the previous one
Characterization of a subset of $[0,1]$
But I don't know whether it is correct to ask again with a new title.
Thanks a lot for pointing the mistake and I ...

**2**

votes

**1**answer

111 views

### Differentiability of Nemytskii operator on Sobolev space

I am trying to consider hypothesis on $g$ such that the operator
$$ H_0^1 (\Omega) \to L^2(\Omega), \qquad v \mapsto g(v) $$
is $\mathcal C^1$. As additional hypothesis $\Omega$ is bounded and $g(0) = ...