Distributional Fourier expansion of $\cot$ or something else?
Cody mentions in his answer here thatA handy formula when integrating a polynomial times cot or csc.It can be shown...
View Articleequivalent bases implies that the sequence spaces coincide
From Topics in Banach space theory bookIf we select a basis in a finite-dimensional vector space, then we are, in effect,selecting a system of coordinates. Bases in infinite-dimensional Banach spaces...
View ArticleShow that for all $\varepsilon>0$,...
I'm having trouble succeeding in doing this exercise, where $d(n)$ is the number of divisors of $n$. My first instinct was to use the following line of reasoning: for all $|z|<1$,...
View ArticlePreimage of a closed 2-to-1 continuous function related to Alexandroff...
Consider the Alexandroff duplicate $X × \{0,1\}$, the space $X × \{0,1\}$ where the points of the form $(x,1)$ are isolated and for each open set $U$ in $X, (U×\{0,1\}) ∖ (x,1)$. If $X$ is having...
View ArticleSequence with infinite number of zeros
My question reads:Let’s call a sequence $(x_n)$ zero-heavy if there exists $M\in\mathbb{N}$ such that for all $N\in\mathbb{N}$ there exists $n$ satisfying $N\leq\ n\leq\ N + M$ where $x_n = 0$.If a...
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Find all roots of a Lipschitz function
I'm looking for advice to pick a numerical method to find all roots of a univariate function $f$ on a finite interval:$f$ is Lipschitz-continuous with unknown $L$ (without that assumption, it is...
View ArticleWhy is the probability $0$ that random vectors all lie perpendicular to some...
I'm trying to solve Gilbert Strange's 18.065 problem 3 from first problem set and struggling with this probability argument:Given:Matrix $A$ is $1000×1000$ with $\operatorname{rank}(A) < 10$Matrix...
View ArticleWhen is a matrix function the Jacobian matrix of another mapping
Suppose $J(x)$ is a continuous matrix function $\mathbb{R}^D \to \mathbb{R}^D \times \mathbb{R}^D$. Do there always exist a mapping $f: \mathbb{R}^D \to \mathbb{R}^D$ so that $J = \nabla f$. If not,...
View ArticleOn the Definition of Gateaux Derivative
My question is about two different definitions Gateaux derivative. I have seen the following two definitions but whether they are equivalent or which one is better to use I am not sure about:Definition...
View ArticleIsomorphism in Banach Spaces
Let $E$ and $F$ be Banach spaces. Let $T: E \rightarrow F$ be an isomorphism (i.e., a continuous vector space isomorphism with a continuous inverse). Let $J_E$ and $J_F$ be the canonical injections of...
View ArticleThe R-L fractional derivative interpretation
The Riemann-Liouville fractional derivative of order $\alpha$ of a continuous function $f:(0,\infty)\rightarrow \mathbb R$ is defined as:$$D^{\alpha}=\frac{1}{\Gamma(n-\alpha)}(\frac{\partial...
View ArticleSwapping the order of summation
I want to swap the order of summation in the following sum: $$\sum_{0<a\le X^k}\sum_{b\le(\frac{a}{2\pi})^{\ell}}.$$ Here $0<\ell<1$.I want to get this into two sums, of the form $$\sum_{b\le...
View ArticleProving that $|x^*(x)| \leq \rho(x,Y)$
Exercise :Let $(X,\|\cdot\|)$ be a normed space, $Y$ a subspace of $X$ and $x^* \in X$ with $\|x^*\| \leq 1$ such that $x^*|_Y = 0$. Show that $\forall x \in X \setminus Y$, it is : $|x^*(x)| \leq...
View ArticleStrtictly convex functional has unique minimizer on a convex set
On page 8 of the book Regularity of Free Boundaries in Obstacle-type Problems, it considered a functional$$J(u):=\int_D (|\nabla u|^2+2fu)\,dx \qquad D=B(0,1)$$where $f\in L^\infty(D)$ is fixed. It is...
View ArticleDerivative of concave function with growth condition in L^2? [closed]
Let $f: [0,\infty) \to [0,\infty)$ be a concave $C^1$-function with $f(x) = O(x^\alpha)$ for some $\alpha < 1/2$ as $x \to \infty$. Does this imply $f' \in L^2$?
View ArticleIs this better or worse than Gronwall inequality?
Let $a, b, u: \Bbb R\to\Bbb R$ be continuous functions and $r, c\in \Bbb R$ such that\begin{align*}u(t)\leq a(t)+ r \int_c^t u(s)d s\end{align*}It is well known that, Gronwall inequality...
View ArticleProve $\sup (\{x \in \mathbb Q | x^2 < 2\}) \geq \sup(\{x \in \mathbb R | x^2...
I'm learning real analysis recently by reading Basic Anlysis I by Jiří Lebl, and I encountered this question in exercise 1.2.14Let $A = \{x \in \mathbb Q \mid x^2 < 2\}$ and $B = \{x \in \mathbb R...
View ArticleWhy is the angle $\arccos\left(\frac{1}{3}\right)$ always used in...
I am currently studying Stan Wagon's book about the proof of the Banach-Tarski Paradox. In his proof and in many others I've seen on the internet, one of the most important steps is that there is a...
View ArticleDoes the series $\sum_{k=3}^{\infty} \frac{(-1)^k}{\ln k + \sin k} $ converge?
QuestionDoes the series$$ \sum_{k=3}^{\infty} \frac{(-1)^k}{\ln k + \sin k} $$converge?My workI checked absolute convergence and saw$$\frac1{\ln k+\sin k}\;\ge\;\frac1{\ln k+1},$$and $\sum1/(\ln k+1)$...
View ArticleFunction of class $C^1$ implies locally Lipschitz
Let $A$ be open in $\mathbf{R}^{m} ;$ let $g: A \rightarrow \mathbf{R}^{n} .$ If $S$ is a subset of $A,$ we say that $g$ satisfies the Lipschitz condition on $S$ if the function$$\lambda(\mathbf{x},...
View ArticleWhy can we treat differential operators as if they behave like algebraic...
In college, I've come across many instances where we multiply a derivative by a function, and the result somehow becomes the derivative of the function i.e $\frac{d}{dx}\times f=\frac{df}{dx}$— as if...
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is $ \sum_{n \in A}\frac{1}{n}=\infty$? [closed]
We know that$$\sum_{n=1}^\infty \frac{1}{n}=\infty$$If we take an infinite set $A \subseteq \mathbb N$ can we say that$$ \sum_{n \in A}\frac{1}{n}=\infty$$I see that in the article: A classical...
View ArticleFind all real function $f(x)$ satisfy the functional equation [closed]
Find all real function $f(x)$ satisfy this functional equation $|f(x) - f(y) | \leq |\sin(x-y) -x+y| $ for all $x, y \in \mathbb{R}$?My ideas: I showed that the constant functions are a class of such...
View Articlerelationship between functional derivative and simple derivative.
Consider a function $f(x)$. The derivative of $f$ with respect to $x$ is defined as $$f'(x)=\lim_{\epsilon\rightarrow 0} \frac{f(x+\epsilon)-f(x)}{\epsilon}.$$Sometimes, we define a functional...
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What is the use of assumption of concavity in this result?
I am trying to understand the following Proposition $8.5$ of this book:Here, $R,~R_n$ are functions on $\Theta$ and $R$ has a unique maximizer $\theta_0\in\Theta.$ Authors assumed $R_n(\theta) $ is...
View ArticleConvergence of $ \sum_{k=1}^\infty \Bigl(\tfrac{a_k}{A_k}\Bigr)^{\alpha} $...
Let $\{a_k\}$ be a non‑negative, non‑decreasing sequence satisfying$$0 < a_k \le a_{k+1} \le k+1,$$and set$$A_n = a_1 + a_2 + \cdots + a_n.$$It’s known that$$\sum_{k=1}^\infty...
View ArticleProof of that every interval is connected
I don't understand the following proof of that every interval is connected in $\mathbb{R}$.Let $Y$ be an interval in $\mathbb{R}$ and suppose that $Y$ is not connected.Then $Y=A\cup B$, where...
View ArticleA Quantified Definition of Real Analytic Functions [closed]
I have recently been reading Krantz's Primer on Real Analytic Functions. I have only one issue with the book. I find its description of real analytic functions ambiguous. Here is what I mean. Usually...
View ArticleVan der Vaart's notation on convolution between measures
On page 113 in Van der Vaart Asymptotic Statistics, the statement of Anderson's lemma, he used the following notation:$$\int \ell \,d [ N(0, \Sigma) \ast M],$$where $N(0, \Sigma)$ is a standard normal...
View ArticleThe inequality for the Bessel functions $J_\nu(x)^2 \leq J_{\nu-1/2}(x)^2 +...
Let $J_\nu$ be the Bessel function of the first kind of order $\nu$.Does the inequality\begin{equation} \label{eq:1} \tag{1}J_{\nu}(x)^2 \leq J_{\nu-1/2}(x)^2 + J_{\nu+1/2}(x)^2\end{equation}hold for...
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