Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.

learn more… | top users | synonyms (1)

0
votes
0answers
23 views

Show that f is measurable

Let $U$ be a open Set of $\mathbb{R} \times [0,\infty]$ and let f be defined as $$f: \mathbb{R}\mapsto [0,\infty], \quad f(x) := \max\{0,\sup\{y| (x,y) \in U\}\} $$ How can I show that $f$ is ...
1
vote
0answers
24 views

A direct application of inverse function theorem

Let $f:U\longrightarrow \mathbb{R}^n$ a function with $U\subset \mathbb{R}^n$ open, $f$ injective of class $C^1$ (i.e. continuous with the first derivate continuous) such that $\forall x\in U$ the ...
2
votes
3answers
60 views

Intuitive Numerical Analysis Texts

Steven Strogatz has a great informal textbook on Nonlinear Dynamics and Chaos. I have found it to be incredibly helpful to get an intuitive sense of what is going on and has been a great supplement ...
4
votes
2answers
126 views

Show a function is monotonically decreasing.

Show that $f(x)=\dfrac{\sin x}{x}$ is monotonically decreasing on $[0,\frac{\pi}{2}]$ I'm trying to show that $f'(x)\leq0$ to show it's monotonically decreasing. So $f'(x)=\dfrac{x\cos x-\sin ...
2
votes
0answers
34 views

a complicated question about double improper integral

how to evaluate $$\iint_{y\ge x^2+1}{dx\,dy\over{x^4+y^2}}$$ My solution: the initial intergral $$ =2\int_0^\infty \left(\int_{x^2+1}^\infty {dy\over {x^4+y^2}}\right)\,dx = \int_0^\infty ...
1
vote
1answer
33 views

Questions about Proof of Lusin's Theorem

I am reviewing my analysis notes, and having trouble understanding certain parts of the proof to Lusin's theorem. $\textbf{Lusin's Theorem}$: Let $F: [0,1] \rightarrow [0,\infty]$ be a nonnegative, ...
1
vote
0answers
25 views

$ d(x,y)^2 \le d(x,z)^2 - d(y,z)^2\color{}{+\varphi\big(x,y,z,d(x,y),d(y,z) \big)}? $

In a metric space $(M,d)$ the triangle inequality $d (x, z) \le d(x, y) + d (y, z)$ gives us's the inequalitie $$ \quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}} $$ ...
0
votes
1answer
18 views

Does convergence of $S_{n!}=\sum{1}^{n!}a_{k}$ implies $a_{k}$ approaches to zero?

**Let $\{{a_n}\}$ be a sequence . Define $ S_{n}=\sum_{1}^{n}a_{k}$. Does convergence of $\{{S_{n!}}\}$ implies $\lim{a_{n}}=0$ as $n\rightarrow\infty$. **
1
vote
0answers
13 views

Real Hardy space question

Please see this related question for the definition of the grand maximal function and the class of normalised test functions $\mathcal{T}$. I will refer to them in this question. Let $f\in ...
4
votes
5answers
64 views

Constructive proof of Euler's formula

In most textbooks on the subject I have seen, Euler's formula (by which I mean $e^{ix}=\cos(x)+i\sin(x)$) is proved by applying either differential equations or the power series of sine and cosine. ...
2
votes
2answers
69 views

How find the function $f(x)$ such $\int_{0}^{\pi}f(x)\cos{(nx)}dx=0$

let $f(x)$ is Continuous function on $[0,\pi]$,and for infinite positive integer $n$ such $$\int_{0}^{\pi}f(x)\cos{(nx)}dx=0$$ Find the $f(x)$? I think the answer is $f(x)=c$?,But maybe have ...
0
votes
1answer
43 views

A problem with the density of sin (N) [duplicate]

Actually I can prove the fact that $\sin(\mathbb{Z})$ is dense in $[-1,1]$ using the result that "any non trivial subgroup of the additive group of $\mathbb{R}$ is either cyclic or is dense in ...
1
vote
1answer
52 views

Prove that A is both open and closed. [on hold]

Let $X = \{ z : | z | \leq 1 \} \cup \{ z : | z - 3 | < 1 \} $ be a subset of $\mathbb{C}$. Let the metric be the usual metric $d(x,y) = | x-y |$. Prove that the set A = $\{ z : | z | \leq 1 \}$ ...
1
vote
3answers
66 views

How to get a function if you have the Fourier coefficients

So I have $$H(e^{i\omega})=\sum_{n=-\infty}^\infty C_ne^{i\omega n}$$ and I know that: $$C_n = \frac{2}{\pi n}\sin^2\left(\frac{\pi n}{2}\right)$$ How can I work out the function that this makes? I ...
2
votes
3answers
78 views

Find $x > 0$ for which $\int_{0}^{x} [t]^2 \ dt = 2 (x-1)$.

What are all possible $x > 0$ for which the following equation is satisfied? $$\int_{0}^{x} [t]^2 \ dt = 2 (x-1),$$ where $[.]$ denotes the bracket (or floor) function. I guess we will have to ...
3
votes
2answers
44 views

Every interval in real numbers has a rational and irrational

How can you prove the following: where $a\not= b$, show every $[a,b]$ of R has a rational and irrational number. The context for my question is as follows. In my intro calculus class, we were showing ...
2
votes
0answers
28 views

Stochastic inequality, true?

Consider two stochastic processes $X$ and $Y$ satisfying the following SDEs (with the same drift!): $$X_t = x + \int_0^t b(X_s)ds + B_t$$ $$Y_t = y + \int_0^t b(Y_s)ds + B_t.$$ If $0<x<y$, is ...
2
votes
3answers
80 views

If $I_{n+1}\subset I_n$, show that $\bigcap_{n=1}^\infty I_n$ is nonempty

Question: If $I_n$ is closed and bounded, $I_{n+1}\subset I_n$, and $I_n\neq\emptyset$, show that $\bigcap_{n=1}^\infty I_n$ is nonempty. This is not a homework help question. I'm actually looking ...
3
votes
2answers
41 views

Test for convergence $\sum_{n = 2}^\infty \frac{1}{(n+1)\ln^2(n+1)}$ [duplicate]

Test for convergence $$\sum_{n = 2}^\infty \frac{1}{(n+1)\ln^2(n+1)}$$ Here's my attempt! I decided to use the integral test for this. $$\frac{1}{(n+1)\ln^2(n+1)} > \frac{1}{(n+1)^2\ln^2(n+1)}$$ ...
1
vote
2answers
45 views

Test for convergence $\sum_{n=1}^\infty \frac{1}{n}(\sqrt{n+1}-\sqrt{n-1})$

Test for convergence $$\sum_{n=1}^\infty \frac{1}{n}(\sqrt{n+1}-\sqrt{n-1})$$ So far I attempted to use the ratio test, but I'm stuck on what to do after. ...
0
votes
1answer
16 views

Is an open connected subset of Euclidean space a countable sum of open precompact connected subsets?

Let $U$ be an open subset in $\mathbb R^n$. Then there exists a sequence $(U_n)_{n=1}^\infty$ of open precompact subsets of $\mathbb R^n$ such that $U_n \subset cl U_{n+1} \subset U$ and ...
-1
votes
0answers
37 views

A question about Littlewood Paley Inequality

Classical Littlewood-Paley theory affirms that, for $1 < p < \infty$, we have $$ \Vert Sf \Vert_p \sim \Vert f \Vert_p, $$ where $$ Sf := (P_k f)_{k \in \mathbb{Z}}, \quad \vert Sf \vert := ...
0
votes
1answer
31 views

Need a formula to calculate rating, based on 3 factors?

I have multiple users.... millions actually, each user has two values: "Talk About Theft" and "Stole Something". "Talk About Theft" - 1 to Infinity "Success Rate" - 0 to Infinity I need to have ...
2
votes
0answers
35 views

Why is the derivative of the translates of a measure measurable?

Let G be a topological group and X a measure space. Let $G \times X \rightarrow X$ be a measurable group action, $\mu$ a $\sigma$-finite measure on $X$, and $g\mu$ (for any $g \in G$) the measure ...
2
votes
0answers
29 views

$E \subseteq [0, 1]$, $m(E) > 0$. Show that there are $\alpha$ and $\beta$ such that $\alpha, \alpha + \beta, \alpha + 2\beta \in E$.

Let $E$ be a Lebesgue measurable subset of $[0, 1]$ with positive measure. Show that there are $\alpha$ and $\beta$ such that $\alpha, \alpha + \beta, \alpha + 2\beta \in E$. The only idea I have ...
0
votes
1answer
29 views

Volume of a solid in R3

How can I find the volume of this field? : $$ G=\{\left. (x,y,z) \, \right| \, x^2+y^2+z^2 \le 16 \wedge 1 \le x+y+z \le 2\}. $$ Can anybody help me? Thanks.
2
votes
0answers
32 views

If $\phi\in \mathcal{S}(\mathbb R) $ then $\phi_{t}(x)=\frac{1}{t} \phi(x/t)\in\mathcal{S}(\mathbb R)$?

Let $\phi:\mathbb R \to \mathbb C$ be a function; and define $$\phi_{t}(x):=\frac{1}{t}\phi (x/t), (t>0).$$ We note that, if $\phi\in L^{1}(\mathbb R),$ then $\phi_{t}\in L^{1}(\mathbb R);$ in ...
0
votes
0answers
27 views

Just a curious question. What type of mathematics do we need to deal with a vector valued- function that depends on multiple vectors?

We have a scalar valued function that depends on multiple variables (takes in a vector), what about a vector valued function that depends on multiple vectors??
0
votes
0answers
36 views

Question about the Continuity of the derivative of a function

I have that $$\begin{cases}-(p(t) u'(t))'=f(t,u(t))\\ u(0)=u(+\infty)=0\end{cases}$$ where $f:[0,+\infty)\times \mathbb{R}\rightarrow \mathbb{R}$ is continuous and $\displaystyle\frac1p\in ...
2
votes
1answer
74 views

Show that $\arctan(n)$ is irrational for all $n \in \mathbb{N}$

Question : Show that $\arctan(n)$ is irrational for all $n \in \mathbb{N}$. Hint: My solution doesn't use continued fraction. I am interested in other possible proofs for this question.
2
votes
1answer
43 views

Series for reciprocal of polylogarithm?

The series representation of a polylogarithm of order $s$ is given by $$\text{Li}_s(z) = \sum_{k=1}^{\infty}\frac{z^k}{k^s}$$ Are there any simplified expressions for $\dfrac{1}{\text{Li}_s(z)}$? ...
0
votes
0answers
64 views

Is Fourier transform density preserving?

I know my question is not well-defined since Fourier domain and codomain are not the same, but one knows that they are actually homomorphic. Now what I mean by density preserving is as follows: ...
1
vote
1answer
19 views

Primitive and continuity

I have that $$-(p(t) u'(t))'=f(t,u)$$ where $f:[0,+\infty)\times\mathbb{R}\rightarrow \mathbb{R}$ is continuous i want to prove that $p(t) u'(t)$ is continuous! for this i do the primitive of the ...
3
votes
1answer
43 views

$\int_{\mathbb R^{2}} |\int_{\mathbb R} (f(t-y)- f(t)) g(t-x) e^{-2\pi i w\cdot t} dt|dx dw \to 0 $ as $y\to 0$?

Fact: It is well-known that translation is continuous in the $L^{1}$ norm, that is, if $f\in L^{1}(\mathbb R)$ then $\lim_{y\to 0} \|f_{y}-f\|_{L^{1}(\mathbb R)}=0;$ (where, $f_{y}(x)= f(x-y)$, ...
0
votes
0answers
10 views

Produce a list of the most-similar units, given various correlations/relationships

I have a database full of units (U1 - U50, U51...) where every unit has the same standard attributes (A1 - A10) and where a % of each attribute defines the amount of that attribute for that particular ...
0
votes
1answer
47 views

Green's representation on a compact domain

This is from page 17-18 of Trudinger and Gilbarg Let $\Omega$ be a domain for which the divergence theorem holds. Let $\Gamma(x-y)$ be the normalised fundamental solution of the Laplace's equation, ...
3
votes
1answer
26 views

Example of a function f that is Generalized Riemann Integrable, but its square is NOT Generalized Riemann Integrable.

I am reading a section about Generalized Riemann Integral (Kurzweil-Henstock), and there was a problem on that section to provide an example of a function $f$ on $[0,1]$ that is Generalized Riemann ...
0
votes
1answer
43 views

Suggestions for a real analysis reference.

Can anyone suggest some real analysis book which has a geometric presentation of the concepts with pictorial representation.
0
votes
1answer
30 views

Proof of an open set or closed set

I'm struggling on a proof that I can't proof correctly. Let $A=\mathbb{Z}$, $B=\{n-\frac{1}{2n} | n \in \mathbb{N}*\}$ I could prove easily that A is a closed set and B as well : $\overline A =$ ...
0
votes
2answers
61 views

Is it true that $f \in L_1([a,b])$ is the uniform limit of polynomials?

Is it true that $f \in L_1([a,b])$ is the uniform limit of polynomials? And why? I know it is the uniform limit on a set take out some finite measurable set but not sure if I can say more. Thanks.
1
vote
1answer
42 views

How to evaluate this integral?

How to evaluate $$ \int_{a}^{b} [x] \ dx \ \ + \int_{a}^{b} [-x] \ dx \ ? $$ I know that $[-x] = -[x]$ if $x$ is an integer, whereas $[-x] = -[x] - 1$ if $x$ is not an integer. So is it about ...
4
votes
1answer
48 views

References for mathematical theory of summability of divergent series

Once in a while, I can't help it to ask very broad questions. I have read (a portion of) Hardy's Divergent Series. Back then, I think besides in mathematics, divergent series and the need to assign ...
2
votes
2answers
34 views

Proove of equality of integrals

I'm currently sitting on the following problem: Let f be in the set of the integrable functions(:=$L^¹(\mathbb{R}^n))$, A $\in \mathbb{R}^{n\times n}$ invertible. Therefore define g:=$\mathbb{R}^n ...
5
votes
3answers
85 views

prove that $a^b\ge{b}^a$ where $a\le{b}$.

prove that $a^b\ge{b}^a$ for all $a,b\ge3$. given that $a\le{b}$. I was trying to solve the question by graph. Can anyone help me please?
1
vote
2answers
95 views

Finding a root of a function via Rolle's theorem

Consider the function $f(t)=a(1-t)\cos(at)-\sin (at)$, where $a\in\mathbb R$. To show that it has a root in the unit interval I am urged to integrate $f$ and apply Rolle's Theorem. Attempt: $$\int ...
2
votes
1answer
28 views

Does there exists $f\in \mathcal{S} (\mathbb R)$ so $\hat{f}=1$ on a comapct set $C$ and $\hat{f}=0$ outside $C\subset W$ (open set)?

Let $C$ is a compact subset of $\mathbb R,$ $V\subset \mathbb R,$ and $0<m(V)<\infty,$ where $m$ is a Lebsgue measure on $\mathbb R.$ My Question is: Can we expect to find $k\in ...
0
votes
1answer
84 views

If $f(z):=\sum_{n=0}^\infty a_nz^{-n}$ is compact convergent, then $f$ is holomorphic

Let $\left(a_n\right)_{n\in\mathbb{N}}\subset\mathbb{C}$ such that $$f(z):=\sum_{n=0}^\infty a_nz^{-n}$$ is compact convergent on $B_r(0)\setminus\left\{0\right\}$. I want to show: $f$ is ...
1
vote
2answers
13 views

Basic question about interior sets property

Reading a textbook of mine, I've encountered with a simple property and I couldn't prove it is true, I would like if someone could show me why the statement holds so I'll textually copy it. Statement ...
0
votes
0answers
13 views

Deriving lower bound for eigenvalues of laplace operator

Let $u$ be a function such that $$ \Delta u + \lambda u = 0 \quad \mbox{ on } \quad D, \quad u = 0 \quad \mbox{ on } \partial D $$ and let $w$ be a function such that $\Delta w + \beta w < 0$ ...
2
votes
0answers
29 views

Show that $L[v^2] := \Delta(v^2) + \frac{2}{w} \sum_{i=1}^n w_{x_i} (v^2)_{x_i} \ge 0$

Let $u$ be a function such that $$ \Delta u + \lambda u = 0 $$ for some $\lambda \in \mathbb R$, also let $w$ be a function such that $$ \Delta w + \beta w < 0. $$ for some $\beta \in \mathbb R$. ...