Deriving an inequality for the integral of maximum indicator functions under...
Let's denote the measure space by $(X, \mathcal{B}, \mu)$ and the measure-preserving transformation by $T: X \to X$. Let $A \in \mathcal{B}$ be a measurable set with $0 < \mu(A) < \infty$. Let $0...
View ArticleContinous function with infinitely many zeroes
Let's say I have a function $f(x)$, that is continous everywhere on the real line. Suppose I have a sequence of real numbers $a_k$ such that $a_0$ = $C$ and as $k$ tends to infinity, $a_k$ tends to a...
View ArticleConfusion about little-o and sums
I was trying to do the following exercise from Apostol, Introduction to analytic number theory. Given two-real-valued functions S(x) and T(x) such that $ T(x) = \sum_{n \leq x}S(\frac{x}{n}) $ If S(x)...
View ArticleAny function with a modulus of continuity proportional to any preassigned...
The following is from Courant & John's Introduction to Calculus and Analysis Volume I, p. 43.Lipschitz-continuity means that the "difference quotient"$$ \frac{f(x_2)-f(x_1)}{x_2-x_1} $$formed for...
View ArticleExtension of a metric defined on a closed subset
If $X$ is any metrizable space, $A$ is a closed subset of $X$. Let $d$ be a compatible metric on $A$ then $d$ can be extended to a compatible metric on $X$.
View ArticleDoes U(f,P,[a,b]) = L(f,P,[a,b]) really imply f is constant
I am currently reading Measure, Integration & Real Analysis by Sheldon Axler, and am working through the practice problems. In particular, I am on this problem right now:Suppose...
View ArticleShifting Index of Recursive Sequence
If I have a recursive sequence defined by:$a_0 = 7$$a_n = a_{n-1} + 3 + 2(n-1),$ for $n \geq 1$How is this recursive sequence the same as the one above. Isn’t $n+1 \geq 2$?$a_0 = 7$$a_{n+1} = a_{n} + 3...
View ArticleShowing that a family of holomorphic functions is normal
Let $\mathcal{F}$ be a family of holomorphic functions on $\mathbb{D}=\{|z|<1\}$ so that for any $f \in \mathcal{F}$,$$\left|f^{\prime}(z)\right|\left(1-|z|^2\right)+|f(0)| \leq 1 \quad \text { for...
View ArticleBound of an integral function
Let $$f(r):=\int_{\mathbb{R}^d}\left|\int_{\mathbb{R}^d}e^{2\pi i \langle x,y\rangle}e^{-|y|^4+r^{1/2}|y|^2}dy\right|dx,$$for $r\geq 0$.Is it possible to uniformly bound $f$ on $\mathbb{R}_+$?...
View ArticleSequential criteria of Continuity [closed]
Using the de nition, prove that the functions f: R -> R and g : [0, infinity ) -> R defined by f(x) = x^2 and g(x) = (x)^(1/2) are continuous on given domains.
View ArticleConfusion about $\lim\sup$ and its definition as the greatest limit point
I posted a question a few days ago and the most voted answer uses $\lim \sup$, a concept I was not familiar with. I decided to jump ahead and read about $\lim\sup$ to understand the answer, but one...
View ArticleAbout a calculation in Grafakos' Classical Fourier Analysis.
I'm reading Grafakos' book on Fourier Analysis and at some point he says "There is an analogous calculation when $g$ is the characteristic function of the unit disk $B(0, 1)$ in $\mathbb{R}^2$. A...
View Articleexponentially decaying weighted integral lower bound
For $f\in L^2$ for example, can we achieve sharpest lower/upper bound for the following integral for small $\epsilon>0$$$\int_{0}^{T}e^{-\dfrac{t}{\epsilon}} |f(t)|dt $$ andunder what conditions on...
View ArticleIs Kolmogorov-Arnold (representation) neural network dense?
The Kolmogorov-Arnold neural networks (KAN), Ziming Liu et al, KAN: Kolmogorov-Arnold Networks draws inspiration from the Kolmogorov-Arnold representation theorem(KA theorem). However, the former, as...
View Articleintegration by parts in Hilbert space
I want to prove that $u,v∈H^1 (R^n )$ we have : $∫_{R^n} \frac{∂u}{∂x_i} vdx=-∫_{R^n} \frac{∂v}{∂x_i} udx$We know that $C_c^∞$$(R^n )$ is dense in $H^1$$(R^n)$, this means that $∃ u_n∈C_c^∞ (R^n )$...
View ArticleFunctions maintain the property that sets are measurable
Let $f$ be a function with continuous derivatives on the real axis $\mathbb R$ which satisfies$\forall x\in\mathbb R\forall y\in\mathbb R(|f(x)-f(y)|\geq m|x-y|)$where $m > 0$ and is a constant....
View ArticleCalculating the derivative of $ I(x) = \int_{x}^{\pi} \bigg( \frac{1 -...
Consider the function ${I}\left(x\right)$ defined by the integral$${I}\left(x\right) =\int_{x}^{\pi}\left[\frac{1 - \cos\left(y\right)}{\cos\left(x\right) - \cos\left(y\right)}\right]^{1/2}{\rm...
View ArticleIf $\lim_{n\to\infty}\int_0^1f(x+n)dx=2$, prove that...
It is given $f:[0,\infty)\to\mathbb{R}$ is a continuous function and $\lim_{n\to \infty}\int_{0}^{1} f(x+n)dx=2$ then prove that $\lim_{n\to \infty}\int_{0}^{1} f(nx)dx=2$.I can prove that $\lim_{x\to...
View ArticleDerivate problem
Let $f$ be continuous on $[a, b]$ and differentiable on $(a, b)$ with $f(c) = 0$ for some $c \in [a, b]$. If there exists $M \in \mathbb{R}$ such that $\vert f'(x) \vert \leq M \vert f(x) \vert$,...
View ArticleGeneralizing a logarithmic inequality
Let $\{x_i\}_{i=1}^N$ and $\{y_i\}_{i=1}^N$ be real numbers in the interval (0,1). Define for each $i$$$\alpha_i = x_i (1-x_i) \log^2 \frac{x_i (1-y_i)}{y_i (1-x_i)}$$and$$\beta_i = x_i \log...
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