Does this sequence of polynomials converge to the square root function?
Taken from Lang's R & F Analysis (p.60). For some reason I can't see why, for $t \in [0,1]$, the following is true for all natural numbers $n$ (by an inductive argument):$$0 \leq \sqrt{t} - P_n(t)...
View ArticleSimplification of inner product
From https://arxiv.org/pdf/2101.12317 how is the simplification of equation (2.16) obtained?Let $g(x),p(x)$ be real functions and $\lambda$ complex. Consider the expression $\langle g,...
View ArticleSolution to a First-Order PDE Using the Lagrange method and division by zero
I need to find a solution for the following PDE:$$ \begin{cases} y u_x - x u_y = 0 \\ u(x, 0) = x^2 & ,x \in \mathbb{R} \\ \end{cases}$$My attempt:I started by using the Lagrange...
View ArticleDoes the integral converge $\int\limits_{0}^{1}\frac{\cos\left ( t^{-2}...
Investigate the convergence and absolute convergence of the integral at $a\in \mathbb{R}$$$I=\int\limits_{0}^{1}\frac{\cos\left ( t^{-2} \right )}{\left ( 2-t^2\cos\left ( t^{-2} \right ) \right...
View ArticleMinimize function two variables II.
Let $x,y,a,b$ be the column vectors $(n,1)$ , $C(n,n)$ be the matrix, and$$\phi(x,y)=\left\Vert x-a\right\Vert _{2}^{2}+\left\Vert y-b\right\Vert _{2}^{2}+\left\Vert x^{T}Cy\right\Vert _{2}^{2}$$be the...
View Article$f:\mathbb R\to [0,\infty)$ is a 3 times differentiable and...
I have a problem in Analysis:Let $f:(-\infty,\infty)\to [0,\infty)$ be a three times differentiable and satisfy:$$\max_{x\in\mathbb R}|f'''(x)|\le 1$$Prove...
View ArticleHamming distance in real analysis
Given two binary strings $x, y\in (0,1)^*$ such that $|x|=|y|$, then the set $$\delta{(x,y)}=\frac{|\{i\in[|x|]:x_i\neq y_i\}|}{|x|}$$ is called relative hamming distance.Given $x\in (0,1)^*$ and $S:$=...
View ArticleWhat is the optimal definition of the convolution of two measurable functions?
Assume that we are dealing with the usual Lebesgue measure on $\mathbb R^n$. In the book "Real Analysis: Modern Techniques and their Applications" written by Folland, the convolution of two measurable...
View ArticleImplicit non linear formula to explicit polynomial formula.
Suppose that $\{P_n\}$ are sequential functions of polynomials of degree $n$, and $x$ is between $[-1,1]$.Put $P_0 = 0$, and define, for $n = 0, 1, 2, \ldots ,$$$P_{n+1} = P_n + \frac{ (x^2 - P_n^2)...
View ArticleComputing an integral using differential under the integral sign
The following integral is in question.$$I(x) =\int_0^x \frac{\ln(1+tx)}{1+t^2}\,dt$$My attempt is finding $I’(x)$ which is$$I’(x) = \int_0^x \frac{t}{(1+t^2)(1+tx)}\,dt + \frac{\ln(1+x^2)}{1+x^2} $$Now...
View ArticleConvergence of the derivative of a BV function in sense of measures
Suppose I have a smooth sequence $f_{n} : \mathbb{R} \to \mathbb{R}$ with $f_{n} \to f$ strongly in $L^{1}_{loc}(\mathbb{R})$ and $f \in BV_{loc}(\mathbb{R})$. Is there any way to justify that...
View ArticleRecursively defined sequence - Convergence and boundedness
I’m stuck on the following task and need your help.Namely, a recursively defined sequence is defined by: \begin{equation}x_0 = 2, \quad x_{n+1} = -x_n + (-1)^{n} \frac{1}{2^{n+1}}\end{equation} I am...
View ArticleWhy discrete set must be countable?
I want to show a set, which every point of it is an isolated point. Then this set must be countable. How to show it?I find this from Wikipedia's article Isolated point, but I don't understand:A set...
View ArticleProblem 30 in the Exercises following Chapter 2 in Baby Rudin: How to...
Here's problem 30, the very last one, in the Exercises after Chapter 2 in Walter Rudin's Principles of Mathematical Analysis, 3rd edition: Imitate the proof of Theorem 2.43 to obtain the following...
View ArticleVolume of solid generated by rotation around y axis of a region between two...
I have to find the volume of the solid generated by the rotation around y axis of the following region: the bounded region between $f(x)=\tan^2{(x)}$ and the line passing through...
View ArticleApplication of fix point theorem for functions $\epsilon,\delta:[0,1]\to[0,1]$
I have two functions $\epsilon,\delta:[0,1]\to[0,1]$ such that the following hold for all $x,y\in[0,1]$:$$\text{(i) $x+\epsilon(x)\in[0,1]$ and $x-\delta(x)\in[0,1]$}$$$$\text{(ii) if $y\in...
View ArticleIs $\sum_{k=1}^\infty\frac{\sin{kx}}{k}$ uniformly convergent in $(0,2\pi)$?
I'm pretty sure this series is divergent because $|\frac{\sin{kx}}{k}| \leq \frac{1}{k}$, which is a divergent harmonic series. I'm not sure on how to formally prove this, i thought about using the...
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Prove that : $\rho(x,y)= \sqrt{(x_1-y_1)^2+4(x_2-y_2)^2} $ is a metric
let $X= R^2$. If $x$ and $y$ are points in the plane and their cordinates are $x=(x_1,x_2), y=(y_1,y_2)$ prove that the given function is a metric.I am currently stuck on proving the triangular...
View ArticleIntegrability of $2\pi$-periodic function
I WTS that if $f$ is periodic and Riemann integrable on $[-\pi, \pi]$, then $f$ is integrable on any closed interval and for every real number $x,$ we have $\int_{x-\pi}^{x+\pi}f = \int_{-\pi}^{\pi}f$....
View ArticleProof of the generalized additive interval property of definite integral
I was trying to prove that for any real number c this equality$\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$holds true.So I started by assuming that a primitive of the function f(x) exists and I...
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