Asymptotics in real analysis
Suppose we have a sum $$\sum_{n<x}\frac{f(n)g(n)}{n},$$ and we know asymptotically the value of $\sum_{n<x}\frac{f(n)}{n}\sim h(x)$, and that $g(x)=O(j(x))$ for another function $j$. Is it then...
View ArticleSome properties of averaged modulus of smoothness $\tau_{k}(f, \delta)_{p}$
I am currently reviewing a paper published in the Journal of Approximation Theory and working on proving some unresolved properties. Below are the necessary definitions and the properties...
View Articleapplication to Lusin's Theorem
If $f:\mathbb{R}\rightarrow \mathbb{R}$ is Lebesgue measurable, then there exists a sequence of continuous functions $\{f_n \}$ converges pointwise to $f$ almost everywhere on $\mathbb{R}$.My work:I...
View ArticleProof of integrals being equal to zero
Alright hello, the proof on paper is really simple yet I'm having a lot of trouble with it. The proof is this:Let $f(x)$ be continuous on $[a,b]$, and $\forall x\in [a,b], f(x) \geq 0$. If $\int_a^b...
View ArticleIs my derivation of Taylor's Formula in $\mathbb R$ for the remainder...
Statement from a textbook:Let $n \in \mathbb N$ and let $a,b$ be extended real numbers with $a<b$. If $f:(a,b)\rightarrow \mathbb{R},$ and if $f^{(n+1)}$ exists on $(a,b)$, then for each pair of...
View ArticleCompute $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{\tan(x)}{1+\tan(x)}\ dx$
I have to calculate$$\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{\tan(x)}{1+\tan(x)}\ dx$$This is my attempt.I decide to do it by sostituition, so:$t=\tan(x) \Longrightarrow x=\arctan(t) \Longrightarrow...
View ArticleIs there any counter example that any infinite triangle construction converge?
Given an Initial Triangle $A_1B_1 C_1$ construct the triangle $A_{n+1}B_{n+1}C_{n+1} $ from $A_n B_n C_n$, where $A_{n+1}$ is the $r_1$-th triangle center of $A_n B_n C_n$, $B_{n+1}$ is the $r_2$-th...
View ArticleProve/disprove: $d(x, \phi(x)) \leq f(x) - f(\phi(x))$ implies $\phi$ is a...
According to Approach0, this question seems new.There is a similar question, however in that question asked for a proof about a function, which satisfied the definition (see Problem), has a fix point....
View ArticleLinearity of the Integration of nonnegative simple functions
Let $(X,\Sigma,\mu)$ be a measure space and $\phi$ and $\psi$ nonnegative simple function on $X$. If $\alpha$ and $\beta$ are positive real numbers, then$$ \int_X[\alpha \psi +\beta \phi]d\mu=\alpha...
View ArticleHow can I show that $\sup(AB)\geq\sup A\sup B$ for $A,B\subset\mathbb{R}$...
The question is based on the following exercise in real analysis:Assume that $A,B\subset{\Bbb R}$ are both bounded and $x>0$ for all $x\in A\cup B$. Show that $$\sup(AB)=\sup A\sup B$$ where...
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Bounding $\sum_{k=-N}^{N} |a_k|$ where $a_k=\hat{f}(k)$ and $P_N(x)$ is a...
I need to prove that inequality above:My approach was to write $|a_k| = |a_k| \cdot k \cdot \frac{1}{k}$ for all $k \neq 0$. Then, using the QM-GM inequality, we obtain:$$|a_k| \leq \frac{1}{2}\left(...
View Articlefunctions increasing on nowhere dense set
let f a real function on [0,1]. Assume that A is a closed subset of [0,1] which does not contain intervals and that f is constant on any connected part of [0,1]\A. We know that possibly there are...
View ArticleOpen Set is Equal To Union of Closed Sets
Let $(X,d)$ be a metric space, and $U\subset{X}$ be an open set. Then $U$ may be written as a union of closed sets.I think I have a proof for this but I'm not quite sure, I feel like there must be...
View ArticleSolving a second order separable differential equation but having problems...
Problem:Solve the following differential equation:$$ \frac{ d^2 s }{dt^2} = g(1 -s^2) $$with the following initial condition:\begin{align*}\dfrac{ ds}{dt} \left( 0 \right) &= 0 \\s(0) &=...
View ArticleThe arclength of a rectifiable curve is continuous:
Let $(X,d)$ be a metric space and $\gamma:[a,b]\rightarrow X$ a curve. We define the length of $\gamma$ on $[a,b]$ as follows:\begin{equation} \ell_a^b(\gamma):=\sup\left\{\ell_a^b(\gamma,P)\mid...
View ArticleCharacterization of sine and cosine functions. Uniqueness of $\pi$.
I would like to prove the following theorem:There only exists two functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ which satisfies the following properties:For all $x\in\mathbb{R}$,...
View ArticleKolmogorov–Arnold representation for permutation invariant functions
Kolmogorov Arnold reperesentation theorem states that (the extended version by Lorentz) every multivariate continuous function $f:[0,1]^n\rightarrow \mathbb{R}$ can be written in the folowing form,...
View ArticleProof that the angular measure is a measure.
Let $\mathbb{S}^1=\{x\in \mathbb{R}^2: \lvert x \rvert=1\}$. Consider $\mathcal{B}(\mathbb{S}^1)=\{B\cap\mathbb{S}^1:B \in \mathcal{B}(\mathbb{R}^2)\}$. I want to prove that there exists a measure on...
View ArticleBounding $\sum_{k=-N}^{N} |a_k|$ where $a_k=\hat{P_N}(k)$ and $P_N(x)$ is a...
$$ \sum_{k=-N}^N |a_k| \le \frac1{\sqrt{2\pi}} \int_0^{2\pi} |P_n(x)| \mathrm dx+\frac{\pi}{\sqrt 3} \int_0^{2\pi} |P'_N(x)|^2 \mathrm dx $$I need to prove that inequality above:My approach was to...
View ArticleIntegration and boundedness of integral Proof and Statement
I am not sure if this is a standard theorem, since I can't find a name for it: Suppose that $f\in L^1(\mathbb R, m)$ where $m$ is the standard lebesgue measure. Then for every $\epsilon >0$, there...
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