Lp functions regularity
Let $g(x,\theta) = \int_{-a}^0 f(x+t\theta)dt$. If $f$ is in $L^p$ then $g$ is in $L^p$.Proof.$||g(x,\theta)||_{L^p(\Omega \times S^1)}^p = \displaystyle\int_{\Omega \times S^1}|g(x,\theta)|^p...
View ArticleIs $\mathbb{R}$ dense in $\mathbb{R}$?
I am trying to solve a question from my real analysis book.Is this statement correct? For every pair x < y of real numbers, there is a real number z such that x < z < yI was thinking that the...
View ArticleProve that $f(x)=x^2$ is uniformly continuous on any bounded interval.
May I please ask how to prove that $f(x)=x^2$ is uniformly continuous on any bounded interval? I know that there is a theorem saying that every continuous function on a compact set is uniformly...
View ArticleA smooth extension with an assigned regular value.
Let $f\colon S^{n-1} \to \Bbb R^n$ be a smooth function from the unit sphere into $\Bbb R^n$ (not necessarily injective) and $y_0 \in \Bbb R^n\backslash f(S^{n-1})$ be a given point. Can we always...
View ArticleSum of scalar map in $\mathbb{R}^3$
Given the following function defined in $\mathbb{R}^3$, with the restriction $0\leq p\leq 1$:$$f(x,y,z)=p^{| x+y-z| +| x-y+z| +| -x+y+z| +| x+y+z| +| x-y| +| x+y| +| x-z| +| x+z| +| x| +| y-z| +| y+z|...
View ArticleProblems when proving surjectivity.
I have some question when proving surjectivity of functions in some type of propositions such as this one from Terrence Tao Analysis book (Proposition 3.6.14, (c)):Let $X$ be a finite set, and let $Y$...
View ArticleDistance to a set and normal cone
We consider a nonempty set $S\subset\mathbb{R}^n$ and $x\notin S$. We define the set$$P_{S}(x) =\{z\in S : \lVert z - x\rVert =d(x,S)\}$$I would like to prove that for all $z\in P_{S}(x)$ we have...
View ArticlePicard's Theorem for Linear First Order Differential Equations
Problem: Let $I$ be an interval and $x_0\in{I}$. Let $p:I\to\mathbb{R}$, $q:I\to\mathbb{R}$ be two continuous functions. Show there exists a unique differentiable function $f:I\to\mathbb{R}$ such that...
View ArticleMultidimensional Fourier Transform derived from the one dimensional case
Many authors (Rudin, Axler,...) study the Fourier Transform primarily in $\mathbb{R}$, which raises the question: Is this because of pedagogical reasons or because the multidimensional case can be...
View ArticleMultivariate integral of the Dirac delta distribution...
I am confused about the multivariate integral of the Dirac delta distribution $\int_\mathbb{S}\int_\mathbb{S}f(x,y)\delta(x-y-a)dxdy$, where $\mathbb{S}\subset\mathbb{R}^2$ is a unit circle. For any...
View ArticleReference for global implicit function theorem in any dimensions
In the book Advanced Calculus 2E by Patrick M. Fitzpatrick, I found the global implicit function theorem in 1 dimension:Suppose that the function $f: \mathbb{R}^2 \to \mathbb{R}$ is continuously...
View ArticleThe Bernoulli measure is back shift map invariant.
I was reading the solution of the following question here Bernoulli shift is a measure-preserving transformation :Let (for simplicity) $(\{0,1\}^\mathbb{N},\mathcal{B},\mu,T)$ be a Bernoulli scheme,...
View ArticleProve That $\int_{1}^{e}\frac{1}{t}dt=1$ Without the Use of Logarithms
I've been studying Real Analysis by Jay Cummings, and am working through the exercises on integration. The question is as such:Define a function $L:(0, \infty)\rightarrow\mathbb{R}$ by...
View ArticleShow the left shift $T$ and $T^{-1} $ are measurable.
Let $X=\{0,1 \}^{\mathbb{Z}}$. So an element of $X$ is given by a sequence $(x_i)_{i\in \mathbb{Z}}$, where $x_i \in \{0,1 \}$.Let $\mu_i$ be a probability measure on $(\{0,1 \}, \mathcal{P}(\{0,1...
View ArticleIf $\alpha$ is irrational, $\lim_{n \to \infty} \sin(n \alpha \pi)$ DNE
I've read an answer to this on another post here.According to the green-checked answer there, let $y=x/2$, and we first know that $a_n = \text{ny} \mod{1}$ (i.e., the fractional part of $ny$) is dense...
View ArticleA tricky series [duplicate]
I am pretty sure I have seen this somewhere before. The problem states$$\sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1}\sum_{k=0}^{2n}\frac{1}{2n+4k+3}=\frac{3\pi}{8}\log \left(\frac{1+\sqrt5}{2}...
View ArticleRiemann sum (Is this justified)?
Let $f_2$ be integrable functions. I am trying to sum$$\lim_{n\rightarrow\infty}\sum_{j=1}^{\infty}e^{ij}f_{2}\left(\frac{j}{n}\right)\frac{1}{n}$$Under what condition can I write this...
View ArticleFixed points of $\tan\sqrt{x}$
This question came in my class test in an MCQ format.$\DeclareMathOperator{\N}{\mathbb N}$ Let $X=\{x\in\mathbb R^+: \tan(\sqrt{x})= x\}$. Consider the sequence $(b_n)_{n\in\N}$ of real numbers defined...
View ArticleProve $f(a)[g(b)h^{\prime}(c) - h(b)g^{\prime}(c)]+h(a)[f(b)g^{\prime}(c) -...
The question is this: If $f$, $g$, $h$ are continuous functions on $[a,b]$ which are differentiable on $(a,b)$ then prove that there exists $c \in (a,b)$ such that $f(a)[g(b)h^{\prime}(c) -...
View ArticleProve $\prod (1+a_i^2) \le \frac {2^n}{n^{2n-2}}\left (\sum...
Problem. Let $n \in \mathbb{N}_{\ge 3}$. Let $a_1, a_2, \cdots, a_n > 0$ such that $\prod_{i=1}^n a_i = 1$. Prove that$$\prod^n_{i=1} (1+a_i^2) \le \frac {2^n}{n^{2n-2}}\left (\sum^n_{i=1}...
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