# Questions tagged [real-analysis]

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

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### 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 ...
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### 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 $\pi$.
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### 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 $f_{yx}$ ...
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### 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 ...
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### 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 $∫_a^b|g(x)|dx$?
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### 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 ...
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### 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 $x>x_0$. ...
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 ...
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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 $$\mathcal{I} = \begin{cases} \frac{1}{... 0answers 666 views ### two versions of the nested interval property There appear to be two different nested interval properties for the reals with the punchline "... then the intersection of the intervals is non-empty", and I'd like to know their respective histories (... 3answers 3k views ### Is a semicontinuous real function Borel measurable? Let f(x,u): [0,1]^2 \mapsto \mathbb{R} be a continuous function. [Q] Is g(x) = \inf_{u\in [0,1]} f(x,u) always Borel measurable? If not, can one find a counter-example? Note that, for any c, ... 0answers 159 views ### On explicit eigenfunctions Given an algebraic surface S defined by an algebraic equation such as x^{4}+2y^{4}+3z^{4}=1, how would one find the third smallest eigenvalue \mu_{3} for the differential equation \Delta f\left(... 3answers 2k views ### Extension of pointwise convergence of a sequence of uniformly continuous functions that converges on a dense set It is known that a sequence of continuous functions on a metric space that converges pointwise on a dense subset need not converge pointwise on the full space. But what about if one assumes uniform ... 0answers 201 views ### The ring generated by measures Suppose X is a space equipped with a \sigma-algebra \mathcal{M}_X. Then the set of measures on X is closed under addition and scalar multiplication by elements of {\mathbb R}. Formally ... 1answer 554 views ### Mean value property with fixed radius Let f be a continuous function defined on \mathbb{R^n}. It is well known that both the spherical mean value property (MVP) of f, i.e.$$f(x)=\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}f,\ ...
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 ...
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{...