a sequence of characteristic functions converge uniformly near t=0, then they...
I encountered a problem when reading A course in probability theory by Kailai Chung:If the sequence of ch.f.'s $\{f_n\}$ converges uniformly in a neighborhood of the origin, then $\{f_n\}$ is...
View ArticleIs Schwarz's Lemma true for squares?
In a recent complex analysis exam, we were asked which step(s) in the proof of the Riemann Mapping Theorem fail, when you replace every instance of $\mathbb{E}$ with the square$$\mathbb{S} =...
View ArticleLinking Fourier Coefficients of periodic functions
Let $\tau\in (0,1)$ and assume that we have a $\tau$-periodic function $$f_1(t) = \sum\limits_{k\in\mathbb{Z}} a^{(1)}_k e^{\frac{2\pi i k}{\tau}t},$$a $(1-\tau)$-periodic function$$f_2(t) =...
View ArticleProving that 1/2 is the least upper bound for $A = \{\frac{(-1)^n}{n}: n \in...
I'm trying to solve Problem 3(a) from https://pi.math.cornell.edu/~zbnorwood/3110-s19/3110-hw2.pdf (Spring 2019 class). We want to find the supremum of $A$ and to prove it as well.(a) $A =...
View ArticleShow that a function admits at most one maximizer or provide a counter example.
Let $g$ be a strictly concave, increasing, and differentiable function defined on the real line and consider the map $f$ defined on the non-negative reals by $ f(x) = (x/(1+x))g(a-x) + (1/(1+x))g(-x)$,...
View ArticleThe Fourier transform of $e^{-i/x}$
Does anyone know what is the Fourier transform of$$ f(x)=e^{-i/x} $$on the real line? I would like to compute it explicitly, or to establish some properties to have a good feeling of “how it looks...
View ArticleCalculate $\sum_{n=1}^\infty (1-\alpha) \alpha^nP^{*n}(x)$
$$\mbox{Let}\quadP'(x)=\sum_{j=1}^n a_j\frac{a^jx^{j-1}}{(j-1)!}e^{-ax},x\geq 0$$be the density function of a mixture of Erlangs and let $\alpha\in(0,1)$:Is is possible to determine an analytic...
View ArticleDetermine all the values of $x\in \mathbb{R}$ for which the series below...
Determine all the values of $x\in \mathbb{R}$ for which the series below converges$$\sum_{n=1}^\infty \frac{x^n}{1+n|x|^n} $$I don't know how to approach these questions in general.
View ArticleUse the definition of convergence to prove that the sequence...
The definition for convergence: The sequence $\{x_n\}$ converges to $L$ where $L\in \mathbb{R}$ provided that for every $\epsilon > 0$ there exists a corresponding integer $N\in \mathbb{N}$ such...
View ArticleFinding a non-affine function satisfying symmetry properties
I am looking for an example of a continuous, non-affine function $u\colon X\to \mathbb{R}$ and a continuous, non-negative function $\epsilon\colon X\to\mathbb{R}_{\geq 0}$ such that the following hold...
View ArticleClik here to view.
Rudin PMA 9.28. Why is $\phi(y)' =(g(y)' k, k)$?
I am revising The implicit function theorem from Rudin's PMA as I forgot the entire proof of this theorem and I couldn't remember or understand why $\phi(y)' =(g(y)' k, k)$ I didn't encounter this...
View ArticleInequality in hamming distance
Let two functions $f,g :[n]\to \{0,1\},$ then we define $$\delta{(f,g)}=\frac{|\{i\in[n]:f(i)\neq g(i) \}|}{n}$$ is called normalized hamming distance(n.h.d).For example, $f=1011, g=0101$, I clearly...
View ArticleApproximation the sum $\sum\limits_{n=0}^\infty (c+n)^{k-1}...
I would like to find lower and upper bounds on the following sum\begin{align}\sum_{n=0}^\infty (c+n)^{k-1} e^{-\frac{(c+n)^k}{2*a}}\end{align}where $c,a>0$ and $k>1$. Note that if we could...
View ArticleAbout the converse of below theorem
My book states thatif a sequence $a_n$converges to $l$, then every subsequence also converges to $l$.Now it states at the end that its converse is not true. but as we know that if every subsequence...
View ArticleClik here to view.
Rudin PMA 9.28. Why is $\phi(y)'k =(g(y)' k, k)$?
I am revising The implicit function theorem from Rudin's PMA as I forgot the entire proof of this theorem and I couldn't remember or understand why $\phi(y)'k =(g(y)' k, k)$ I didn't encounter this...
View ArticleClik here to view.
Papa Rudin $7.24$ Theorem,
There are some necessary definitions for the theorem:There is the theorem:If$(a)$$V$ is open in $R^{k}$.$(b)$$T : V \to R^{k}$ is continuous, and$(c)$$T$ is differentiable at some point $x \in V$, then...
View ArticleTrigonometric polynomials and curve reparametrizations
Let $F\colon \mathbb R \to \mathbb C$ be a complex-valued smooth function of period $2 \pi$ that we consider as a function on the unit circle.I have the feel that the following statement is true but my...
View ArticleProve the minimum $m(t,x)$ of $F(t,x,y)$ is a continuous function
$$F(t,x,y)=\frac{(x-y)^2}{2t}+\int_{0}^{y} u_0(\eta) d\eta,$$$$m(t,x)=\min_{y\in\Bbb R} F(t,x,y),$$where $(t,x,y)\in\Bbb R^3$, $u_0$ is a measurable function, and $|u_0(\eta)|\le M$.There is a lemma:If...
View ArticleAzimuthal moment of indicator function
Suppose $\Omega$ is a smoothly bounded domain in $\mathbb{R}^{2n}$. Then what kind of properties does the function $h_{\alpha \bar{\beta}} (r)= \int_{S^{2n-1}}\chi_{\Omega}(\omega r) \omega^{\alpha}...
View ArticleAlternating series test to prove the convergence of the series
I want to apply the alternating series test to prove the convergence of the series $\sum_{n\geq 1} (-1)^n u_n$ where $u_1=1$ and $\forall n\in \mathbb{N},\quad u_{n+1}=\frac{\cos(u_n)}{n^\alpha}$ where...
View Article