# Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

3,096
questions

**3**

votes

**0**answers

224 views

### Mesh for 3d dungeons game. [closed]

Hallo, I look for some F: R^2->R height function which would generate the Speleothem ceiling http://en.wikipedia.org/wiki/Speleothem for 3d game taking place in dungeons/caves.
The function might be ...

**0**

votes

**1**answer

2k views

### Is the sum sin(n) bounded? [closed]

I wonder whether the sequence $s_{n} = \sum_{k = 1}^{n} \sin k$ is bounded.
The answer seems no, but I have no idea how to prove this from the irrationality of $\pi$.

**2**

votes

**2**answers

384 views

### Higher order partial derivatives and global regularity.

Let $f$ be a function of two variables $x$ and $y$. Assume that $f$ is $C^1$. Assume that $f_{xx}$ exists and continuous.
Is it true that $f_{xy}$ exists and continuous?
Is it true that $f_{yx}$ ...

**3**

votes

**1**answer

1k views

### Can one represent a generalized hypergeometric function 1F2 as a product of two confluent hypergeometric functions?

I am trying to somewhat simplify a series, whose coefficients feature generalised hypergeometric functions ${}_1F_2(1;a,a+1;z)$. I was unable to find useful functional relations for this specific ...

**12**

votes

**2**answers

2k views

### Asymptotics of the n-th prime using the gamma function

In the paper http://rgmia.org/papers/v8n2/eepnt.pdf, the author proves that proves an explicit inequality on prime numbers using the gamma function and as a corollary, he showed that.
$$
p_n = n \...

**6**

votes

**3**answers

618 views

### On the inequality $\sum_{i=1}^nx_i^4\sum_{i=1}^nx_i^2 -\sum_{i=1}^nx_i^6 \leq c\left(\sum_{i=1}^nx_i^3\right)^2$

I'm have some difficulties in bounding the following inequality:
I want to find a c as small as possible s.t.
$$\sum_{i=1}^nx_i^4\sum_{i=1}^nx_i^2 -\sum_{i=1}^nx_i^6 \leq c\left(\sum_{i=1}^nx_i^3\...

**1**

vote

**0**answers

737 views

### Question about Riemann integral and total variation [closed]

Let $g$ be Riemann integrable on $[a,b]$, $f(x)=\int_a^xg(t)dt$ for $x∈[a,b]$.
How to show that the total variation of $f$ is equal to $∫_a^b|g(x)|dx$?

**3**

votes

**1**answer

304 views

### Integral Equation with “convolution”

I've got the following problem I'm working on which is related to some of my research:
Solve:
$f(x) = \int_{-\infty}^x G(x,y)f(y)f(x-y)dy$
for f, given $G$ which has whatever smoothness ...

**0**

votes

**1**answer

619 views

### Hölder continuity of uniform limit of piecewise constant functions

Consider a piecewise constant function $v: [a,b] \rightarrow \mathbb{R}$ defined by a finite partition $a=t_0 < t_1 < t_2 < ... < t_s=b$ of the interval $[a,b]$, and constants $m_1,m_2,...,...

**7**

votes

**3**answers

705 views

### On the set of divergence to infinity for sequences of positive continuous functions

Hi,
I have asked this question on math.stackexchange but it has not received much attention, so I ask it here.
This question is partly motivated by this one, which contains an example of a sequence $...

**0**

votes

**1**answer

721 views

### Asymptotic equivalence for functions with zeros

I am considering the relative asymptotic behavior of a pair of real functions on the positive real axis, say $f$ and $g$. There is no $x_0$ such that $f$ and $g$ are non-zero for all $x>x_0$.
...

**0**

votes

**0**answers

90 views

### Lower bound for double sums with power law decay terms.

This question is related to a work in progress about Ballistic-Diffusive phase transition for some random polymers with long range self-repulsion.
The motivation to ask here if the inequality below ...

**4**

votes

**1**answer

715 views

### What is the domain of the “average operator”?

I can try to define an averaging operator for functions, namely let
$$A: D \subset L^\infty([0,\infty]) \to \mathbb{R}$$
by
$$Af = \lim_{N\to\infty} \frac{1}{N}\int_0^N f(x)dx$$
whenever the limit ...

**0**

votes

**3**answers

331 views

### Some Questions about zero-dimensional subsets of the unit interval related to cantor set

Let $\mathbb{P}$ denote the set of all irrational numbers in the open segment$(0 , 1)$. let $K$ be the intersection of $\mathbb{P}$ and the standard cantor set and $H=\mathbb{P}-K$. as you know these ...

**1**

vote

**1**answer

147 views

### represented as a series of periodic function

Is there any necessary and sufficient condition for function $f$ such that:
$f(x)=\sum_{k=1}^{\infty} f_k(x)$ for all $x \in \mathbb{R}$,where $(f_n )_{n=1}^{\infty}$ is a sequence of periodic ...

**0**

votes

**1**answer

684 views

### Is Jordan outer measure finitely additive on positively separated sets in $\mathbb{R^n}$?

I am trying to argue that exterior measure has nice properties that Jordan outer measure doesn't have. One of them is finite additivity, but I can't find a simple way to show Jordan outer measure is ...

**1**

vote

**1**answer

299 views

### Strong convergence in reflecxive Banach space

Let $(X, \|\cdot\|)$ be an Banach space. Assume that a sequence $f_n \rightarrow f$ weakly in $X$, and $\|f_n\| \rightarrow \|f\|$ as $n \rightarrow \infty$. It's known that if $X$ is a uniformly ...

**45**

votes

**4**answers

6k views

### Why could Mertens not prove the prime number theorem?

We know that
$$
\sum_{n \le x}\frac{1}{n\ln n} = \ln\ln x + c_1 + O(1/x)
$$
where $c_1$ is a constant. Again Mertens' theorem says that the primes $p$ satisfy
$$
\sum_{p \le x}\frac{1}{p} = \ln\ln ...

**12**

votes

**1**answer

782 views

### Does every strictly increasing, unbounded sequence of positive real numbers contain arbitrarily long, finite subsequences which are “sort of increasing” or “sort of decreasing” (as defined below)?

Is the following true?
If $(x_0, x_1, \dots)$ is a strictly increasing, unbounded sequence of positive real numbers, then there exist fixed $M,N \geq 1$ such that the sequence $(x_0, x_1, \dots)$ ...

**1**

vote

**1**answer

217 views

### f_n(x,p) converge uniformly to nice f(x,p); do zeros of f_n(.,p) converge uniformly to zeros of f(.,p)?

Fix compact intervals $X, P \subseteq \mathbb{R}$.
Let $f_n : X \times P \to \mathbb{R}$ be a sequence of $C^2$ functions converging uniformly to a $C^2$ function $f$. The first and second ...

**3**

votes

**0**answers

281 views

### How well do continuously differentiable functions behave from R^2 to R^2 ?

The behaviour of complex smooth vs 1-dimensional real smooth functions is discussed in a previous question.
In "Complex Analysis as Catalyst" by Steven G. Krantz, the Cauchy integral formula is ...

**8**

votes

**2**answers

2k views

### Divergent series expansion in Apéry's proof of the irrationality of $\zeta(2)$ and $\zeta(3)$

UPDATE. I am now making this a CW in the hope someone can improve the content of this question and/or correct the text.
This is a concise version of this math.SE question of mine. I've got an answer ...

**2**

votes

**0**answers

125 views

### Bounding an integral with a small parameter by log

I have been working through Erdos & Yau's `Linear Boltzmann equation as the weak coupling limit of a random Schrodinger Equation,'
(arXiv link: http://arxiv.org/abs/math-ph/9901020), and for an ...

**4**

votes

**2**answers

908 views

### Reducing system of equations involving Erf, Error Function

I have a system of equations:
$$1/2 + {\rm Erf}(x) - {\rm Erf}(\frac{x+y}{2})=0$$
$$-1/2 + {\rm Erf}(y) - {\rm Erf}(\frac{x+y}{2})=0,$$
Where $x \le y$ and ${\rm Erf}$ is the Error Function.
By ...

**1**

vote

**2**answers

446 views

### Chebyshev's Theorem

Hi,
I´m looking for Chebyshev´s theorem which says that the inequality $|x(k)-y|<3/k$ has infinitely many solutions, where $x(k)=x_0+k\alpha \pmod 1$, $\alpha$ is an irrational number, and $x_0,y\...

**8**

votes

**2**answers

4k views

### Who was the first to formulate the inverse function theorem?

Let $U\subset \mathbb{R}^n$ and let $F:U\to \mathbb{R}^n$. The 'classical' inverse function theorem gives a sufficient condition for the existence and differentiability of the inverse function of $F$.
...

**2**

votes

**2**answers

706 views

### The extension of smooth function

If $U$ is a bounded domain in $\mathbb R^n$ whose boundary is smooth, and $f$ is a smooth function on $U$ whose partial derivatives of all orders have a continuous extensions to $\bar U$. For an ...

**2**

votes

**1**answer

531 views

### Partitions of an interval

This question asks about properties of functions which are "piecewise" polynomials. I would like to ask a specific question about the meaning of "piecewise" there.
Specifically, consider "partitions" ...

**29**

votes

**4**answers

2k views

### is f a polynomial provided that it is “partially” smooth?

Let $f$ be a $C^\infty$ function on $(c,d)$ ,and
let $O=\cup_{n\in \mathbb{Z}^+} (a_n,b_n)$ where $(a_n,b_n)$ are disjoint open interval in $(c,d)$ and $O$ is dense in $(c,d)$.
Suppose for each $n\in ...

**1**

vote

**1**answer

468 views

### A question about the tail of an absolutely integrable function

Assume $X$ is a measure space and $f : X \to [0,\infty]$ is an absolutely integrable function (that is $\int_X f \; d \mu < \infty$). This question is about the asymptotic behaviour of the function ...

**4**

votes

**2**answers

2k views

### Chain rule for fractional laplacian

Does anyone know a formula of chain rule for fractional laplacian?
say we take the fractional laplacian of order a on function $g(U(x))$ $x\in \mathbb{R}^2$, $U \in \mathbb{R}$, $g \colon \mathbb{R} \...

**4**

votes

**2**answers

673 views

### Articles with examples of Darboux functions without fixed points

A function $f: I \to J$ ($I,J$ intervals) has the Darboux property or the Intermediate value property if for every $a < b \in I$ and for every $\lambda$ between $f(a)$ and $f(b)$ there exists $c \...

**4**

votes

**2**answers

456 views

### subsets of $\mathbb{R}^+$ closed under addition

No one's answered this question so here's a simpler question: Has anyone described or catalogued all sets of non-negative real numbers that are closed under addition? In particular, how about those ...

**0**

votes

**1**answer

232 views

### H\"older spaces

In Gilbarg and Trudinger, they have an example where a function is in $C^1(\bar\Omega)$ but not in $C^\alpha(\bar\Omega)$ where $\alpha<1$. $\Omega$ is bounded and is defined as follows
$\Omega:= ...

**2**

votes

**3**answers

347 views

### Non continous representations of $SL_2(\mathbf{R})$

Q: How does one construct a non continuous representation $\rho:SL_2(\mathbf{R})\rightarrow G$ for some connected (finite dimensional) Lie group $G$?

**2**

votes

**0**answers

111 views

### Searching for inequalities relating a convolution-type integral of functions of modulus less than but close to one.

Suppose $f(x,y)$ and $g(x,y)$ are both measurable functions from $[0,1]\times[0,1]\to \mathbb{C}$ with $|f|,|g|<1$, and let $h(x,y)=\int_{0}^1 f(x,z)g(z,y) \ dz$. (So $|h(x,y)|<1$ also.)
...

**2**

votes

**0**answers

280 views

### Continuous function sort

If you have a real-valued function f(x), positive, continuous and bounded on some interval, then what kind of transform would convert this to a monotonic function g(x) on that interval analogously to ...

**1**

vote

**1**answer

2k 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 $L^1(\Omega)$,...

**5**

votes

**1**answer

505 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 ...

**22**

votes

**1**answer

482 views

### Asymptotics of a Selberg-type integral

Let $\Delta(s_1,s_2,\ldots,s_n) := \prod_{i<j}(s_i-s_j)^2$. Is there a standard way to estimate the decay of the Selberg-type integral
$$ I_n:= \frac{1}{n!^2}\int_0^1 \int_0^1\cdots\int_0^1 \...

**2**

votes

**0**answers

508 views

### Young inequality in weighted spaces

Let $U$ be a bounded open set in $\mathbb{R}^2$, $g\in L^1_{\mathrm{loc}}(\mathbb{R}^2)$.
Let moreover $w$ be a weight (i.e. a non vanishing locally integrable function) on $U$ and $p\geq2$.
Does ...

**7**

votes

**1**answer

825 views

### A curious definite integral

I was playing around with $\mathcal{I}=\int_0^1\text{frac}({\frac{1}{x^n}}) dx$, where $\text{frac(.)}$ is the fractional part function, and I discovered that
$$
\mathcal{I} =
\begin{cases}
\frac{1}{...

**5**

votes

**0**answers

666 views

### two versions of the nested interval property

There appear to be two different nested interval properties for the reals with the punchline "... then the intersection of the intervals is non-empty", and I'd like to know their respective histories (...

**7**

votes

**3**answers

3k views

### Is a semicontinuous real function Borel measurable?

Let $f(x,u): [0,1]^2 \mapsto \mathbb{R}$ be a continuous
function.
[Q] Is $g(x) = \inf_{u\in [0,1]} f(x,u)$ always Borel measurable?
If not, can one find a counter-example?
Note that, for any $c$,
...

**1**

vote

**0**answers

159 views

### On explicit eigenfunctions

Given an algebraic surface $S$ defined by an algebraic equation such
as $x^{4}+2y^{4}+3z^{4}=1$, how would one find the third smallest
eigenvalue $\mu_{3}$ for the differential equation $\Delta f\left(...

**2**

votes

**3**answers

2k views

### Extension of pointwise convergence of a sequence of uniformly continuous functions that converges on a dense set

It is known that a sequence of continuous functions on a metric space that converges pointwise on a dense subset need not converge pointwise on the full space. But what about if one assumes uniform ...

**4**

votes

**0**answers

201 views

### The ring generated by measures

Suppose $X$ is a space equipped with a $\sigma$-algebra $\mathcal{M}_X$. Then the set of measures on $X$ is closed under addition and scalar multiplication by elements of ${\mathbb R}$. Formally ...

**7**

votes

**1**answer

554 views

### Mean value property with fixed radius

Let $f$ be a continuous function defined on $\mathbb{R^n}$. It is well known that both the spherical mean value property (MVP) of $f$, i.e.
$$f(x)=\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}f,\ ...

**4**

votes

**0**answers

254 views

### Real Analytic Function and nth Prime

It is trivial that there are no polynomial function $P$ with integer coefficients that has the property $P(n)=p_n$ where $p_n$ is the $n$th prime.While it is true that can always construct a smooth ...

**1**

vote

**1**answer

320 views

### Singular conformally-Euclidean metrics

Suppose $W : \Bbb{R}^n \to \Bbb{R}_+$ is a continuous, positive function, with exactly $n$ zeros $\alpha_1,...,\alpha_n$. Define the following 'distance':
$$ d(\alpha_i,\alpha_j)=\inf\{\int_0^1 \sqrt{...