A Bernstein inequality in Littlewood–Paley theory
In Appendix A of Tao's book Nonlinear Dispersive Equations, a Bernstein inequality has puzzled me a long time:Suppose that $P_{\geq N}$ is smoothed out projection to the region $|\xi|\geq N$, where...
View ArticleEstimation of a sequence
Let $(n_1, n_2, \cdots )$ be a sequence of natural numbers. Now we define the sequence:$$a_k := \Big(\frac{1}{2}\Big)^{n_1} + \Big(\frac{1}{2}\Big)^{n_1+n_2}+ \cdots + \Big(\frac{1}{2}\Big)^{n_1 +...
View ArticleShow that $2xy+\frac{1}{x}+\frac{1}{y}$ attains global minimum
Let be $f:]0,\infty[\times]0,\infty[\to\mathbb{R}$ where $f(x,y):=2xy+\frac{1}{x}+\frac{1}{y}$. We already know that $f$ has only one local minimum at...
View ArticleHow is the epsilon neighbourhood defined.What define how we should this...
How do you find epsilon for the neighbourhood point of any point.
View ArticleDoes $y\in[0,\:1]\implies y\in f((-1,\:1])$?
Suppose that $f(x)=x^2$. Find $f((-1,\:1])$.$y\in f((-1,\:1])$ iff $y=f(x)$ for some $x\in(-1\:1]$iff $y=x^2$ for some $x\in(-1\:1]$iff $y=x^2$ for some $x$ such that $-1<x≤1$iff $y=x^2$ for some...
View ArticleDoes this book cover the first topics for undergraduate analysis? What book...
I found a book that will be my first contact, however I do not know if it covers the introductory topics. I would like to know if it is enough for a first contact. Its topics are:Ordered fields and...
View ArticleSobolev spaces and ridge approximations
I am new to approximation theory and do not know much about Sobolev spaces.I aim to approximate a Sobolev function $f$ using linear combinations of ridge functions of the form...
View ArticleShowing that $\nu(E) := \int_E f \,d\mu$ is a signed measure when $f$ is...
Let $(X,\mathcal{M},\mu)$ be a measure space and let $f:X \to [-\infty,\infty]$ be an extended $\mu$-integrable function (i.e. $f$ is measurable and at least one of $\int f^{+} \,d\mu$ and $\int f^{-}...
View ArticleAre observed values for random realized variables random themselves? [closed]
Regarding question What is a realization of random variable? and similarly Probability: are realizations of random variables what is actually observed?:I have a random variable $X$, such that $X$ is in...
View ArticleShow that $\lim\limits_{n\rightarrow\infty}(nb^n)=0$ for $0
Show that something converges to something
View Articlefunctional definition of tuples (Terrence Tao Analysis, execise 3.5.2)
I came across this particular exercise in T. Tao Analysis book:Suppose we define an ordered $n$-tuple to be a surjective function $x : \{i ∈ N : 1 ≤ i \leq n\} \to X$ whose range is some arbitrary set...
View ArticleTheorem 3.17 in Rudin's Analysis
This question refers to Rudin's "Principles of Mathematical Analysis", Theorem 3.17, p.56. In particular let $\left\{s_n\right\}$ be a real sequence and let $E$ be the set of all sub-sequential limits...
View ArticleShow $u=\sup(S)$ if $u+\frac{1}{n}$ is an upper bound and $u-\frac{1}{n}$ is...
Suppose $S\subset R$ is nonempty. Prove $u=\sup(S)$ if $\forall n\in\mathbb{N}$, $u-\frac{1}{n}$ is not an upper bound of $S$ and $u+\frac{1}{n}$ is an upper bound of $S$.I was hoping someone could...
View ArticleReal Analysis, Folland Proposition 2.13 Integration of Nonnegative Functions
Question:Proposition 2.13 - Let $\phi$ and $\psi$ be simple functions in $L^+$.a.) If $c\geq 0$, $\int c\phi = c\int \phi$.b.) $\int(\phi + \psi) = \int \phi + \int \psi$.c.) If $\phi\leq \psi$, then...
View ArticleWhich Analysis books did you learn from and how many years/textbooks did it...
It seems that nearly every graduate program in the U.S. requires an extensive education in Analysis clearly demonstrating its importance not only as a subfield itself, but as a foundation for other...
View ArticleProve or disprove that the sequence $x_n$ satisfying $|x_{n+1} - x_n| =...
I'm given a sequence of real nos. satisfying the condition$$|x_{n+1} - x_n| = \frac{1}{\sqrt{n}},\,\,\forall n \in \mathbb N$$I'm trying to prove that a sequence satisfying this condition will never be...
View ArticleIntegration Inequality for $C^{1}$ functions
Let $f,g$ be $C^{1}(R)$ which are nonnegative. $f'(x) < g'(x) \hspace{3mm} \forall{x} \in (0,s) $, does that mean $$\int_{0}^{s} f'(x) < \int_{0}^{s} g'(x)$$I know that $a_{n} < b_{n}...
View ArticleOscillatory integrals
Is there any result about oscillatory integrals of the form$$I=\int_{0}^{\infty}e^{inx}f(x)dx$$I know by Van der Corput Lemma that $|I|\leq c\frac{1}{n}$ for some constant $c>0$.Can one get...
View ArticleA Divergence Criteria to Series of variables between $1$ and $0$
I want to prove the equivalence of the two conditions below. If someone could help me I would appreciate very much!$$p_1,p_2,.... \in (0,1)$$$$\sum_{i = 1}^{\infty}(1-p_i)=\infty\iff \lim_{n\to\infty}...
View ArticleHow to calculate the diameter of a set in $\ell^{2}$
How can I calculate the diameter of $A$ given by:\begin{equation*}A:=\left\{x\in\ell^{2} : \sum_{n\in\mathbb{N}} \left(1+\frac{1}{n}\right)^{2}|x_{n}|^{2} \leq 1\right\}.\end{equation*}Regards!
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