When convolution of $L^1$ function with $C_0$ function is $L^1$
For simplicity, let's only consider "signals", functions $f(t)$ on $\mathbb R$ that are identically zero on when $t<0$.We know that it true that $L^1*C_0\subseteq C_0$ and $L^1*C_0\not\subseteq L^1$...
View ArticleCountability of rationals
We worked on a proof for countability of rational numbers in the analyis class. Even though an explanation is provided, I hit a dead end trying to understand it, and can't seem to find an appropriate...
View ArticleDiscontinuity of Dirichlet function
Define $$f(x)=\begin{cases}1, & \text{if }x\in\mathbb{Q}, \\0, & \text{if }x\in\mathbb{R}\setminus\mathbb{Q}.\end{cases}$$Then $f$ has a discontinuity of the second kind at every point $x$,...
View ArticleConfusing result of an infinite sum from Taylor series
my motivation is to find a "closed form" (may contain integrals) of the general infinite sum $$\sum_{n=0}^\infty\frac{1}{(2n+2k+1)(4n+2k+3)}$$, where k is an arbitrary natural number. The sum can be...
View ArticleQuestion about behavior of integral of a function in $L^1$ with respect to...
Let $f:(0,\infty)\to \mathbb{R}$ function such that $f\geq0$ and$$\int_{0}^{\infty}f(x)\mathrm{d}x=1.$$Consider $0<t_1<t_2$. My question is: Is it possible find constants $A,B>0$ with $B<1$...
View ArticleInequality when fixing a variable in a function of $L^1(\mathbb{R}^N)$
I need to find if the following is true: Let $N>1$, given a function $f \in L^1(\mathbb{R}^N), f > 0$,I want to know if it is true that\begin{equation} \int_{\mathbb{R}^{N-1}} f(x, y)dx \leq...
View ArticleIsometry Between Parameterized Manifolds
I'm struggling with the following problem. I have some $Y_\alpha = \alpha(A)$ parameterized manifold in $\mathbb{R}^n$ with $A \subset \mathbb{R}^k$ open, $k \leq n$, and $\alpha \in C^r(A)$. Given an...
View ArticlePartitioning ℝ into sets $A$ and $B$, such that the measures of $A$ and $B$...
Motivation: I want to partition $\mathbb{R}$ into sets $A$ and $B$, where the measures of $A$ and $B$ in each non-empty open interval have positive ratios with an upper and lower bound, such that the...
View ArticleGriffiths and Harris Regularity Lemma II
I'm trying to make my way through Griffiths and Harris's proof of the Hodge Theorem in Principles of Algebraic Geometry. I've gotten through most of it just fine, but I've been stuck on the estimates...
View ArticleProving $f \equiv 0$ on $[a,b]$ given $\left|\int_\alpha^\beta...
So this problem is from my analysis exam, it states as follows: assume that the real function $f$ is continuous on a finite interval $[a,b]$, and for any subinterval $[\alpha,\beta]\subset[a,b]$, we...
View ArticleDetermining a derivation on the unit sphere of the $\mathbb{R}^3$
Let $S^2 \subseteq \mathbb{R}^3$ be the unit sphere in $\mathbb{R}^3$ (which is a smooth manifold of dimension $2$). Let $\phi = (x_1, x_2): S^2 \backslash \{N\} \to \mathbb{R}^2 $ be the stereographic...
View ArticlePartial derivatives of polynomial in two variables
Let $k \in \mathbb N$, $a_{ij} \in \mathbb R$ for $i,j \in \mathbb N$, $i+j \le k$. A function $f:\mathbb R^2 \to \mathbb R$$$f(x,y) = \sum_{i+j \le k} a_{ij}x^iy^j$$is called polynomial of degree $k$...
View ArticleHow to evaluate $\int_{-\infty}^\infty \frac{x-2}{(x^2+x+4)\sqrt{x^2+2x+4}}dx $
I would like to study how to evaluate the definite integral$$\int_{-\infty}^\infty \frac{x-2}{(x^2+x+4)\sqrt{x^2+2x+4}}dx$$Based on the integration techniques that I know, I proceed by completing the...
View ArticleProve that $\prod_1^{\infty}(1-a_i) > 0$ iff $\sum_1^{\infty}{a_i} < \infty$
Suppose $\{a_i\}_1^{\infty} \subset (0,1)$a) $\prod_1^{\infty}(1-a_i) > 0$ iff $\sum_1^{\infty}{a_i} < \infty$b) Given $\beta \in (0,1)$, exhibit a sequence $\{a_i\}$ such that...
View ArticleIs the zero set of an analytic function closed nowhere dense?
Given a non-constant analytic function $f(x)$ on a domain $D \subseteq \mathbb{R}^n$. I want to prove that$\mathcal{Z} = \{x \in \mathcal{D} |f(x) = 0 \}$is closed nowhere dense.I originally wanted to...
View ArticleWhat conditions must a sequence of functions satisfy to represent any “nice”...
What conditions must a sequence of functions$ \{f_n\}_{n=1}^{\infty} $ must have in order to generate any "nice" function $F(x)$ as $F(x)=\sum\limits_{-\infty}^\infty a_n f_n(x)$ .For example:The...
View ArticleIf linear operator is bounded on $L^p(\mathbb{T})$ and $L^q(\mathbb{T})$,...
Let $1 \leq p < r < q \leq \infty$, which implies $L^q(\mathbb{T}) \subset L^r(\mathbb{T}) \subset L^p(\mathbb{T})$. I have a linear operator $T \colon L^p(\mathbb{T}) \to L^p(\mathbb{T})$ such...
View ArticleFourier tricks for $f(x)=\sin(x)+x\cos(x)$
Let $f(x)=\sin(x)+x\cos(x)$, $x\in(-\pi,\pi)$.Find the Fourier series of $f$I know all the formulas for the coefficients $a_0,b_0,c_0$ and $a_n,b_n,c_n$ and the property for odd functions that then...
View ArticleFind the limit of $\lim_{n \to \infty} \frac{\sqrt[n]{(n+2025)!}}{n - 2025}$
my calculus-1 professor gave us this as homework to find the limit above when n goes to infinity. I tried to put the n square in the bottom but that didn't solve anything. I tried a few...
View ArticleShowing that $\lim_{n\to\infty}\left(\sum_{i=0}^N\frac{x^i}{n^ii!}\right)^n=e^x$
It is well known that $e^x$ can be expressed as a limit of a sequence:$$e^x=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n.$$I can't help but to notice that the terms in parenthesis are just the first...
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