Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.
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0answers
3 views
Antiderivative of an absolute value
$sgn(x)$ is the Sign-Function, $F$ is an antiderivative of $f$ and $S(x) := F(x) \cdot sgn(f(x))$
$$ \int \left|f(x)\right| \, dx = S(x) + \left(\sum\limits_{p=1}^{q}sgn(x-z_p) \lim_{x \to ...
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votes
0answers
6 views
Step functions are dense in the intergrable functions
I am trying to show that the set of step functions $X=\{\mbox{step functions} \ I\rightarrow\mathbb{C}\}$ are dense in the set of $Y=\{\mbox{intergrable functions}\ I\rightarrow\mathbb{C}\}$ with ...
0
votes
0answers
21 views
If a function $f:J\to\mathbb{R}$ satisfies the Zygmund condition, is it $C^1$?
A function $f\colon J\rightarrow \mathbb{R}$ on an open interval $J$ satisfies Zygmund condition if, for all
$x,y\in J$, $$f(x)+f(y)-2f\left(\frac{x+y}{2}\right)=o(|x-y|).$$ It is clear, if $f\in ...
1
vote
2answers
33 views
Can a function be uniformly continuous on an open interval?
I am learning analysis and all the uniformly continuous functions I have seen are over a closed interval. So, can a uniformly continuous function be defined on an open interval?
2
votes
2answers
39 views
Proof on showing $\frac{(b-a)}{2}(f(a) + f(b)) \leq \int_a^b f \leq (b-a) f(\frac{a+b}{2})$ for class $C^2$ function $f$
The task is as follows:
Given:
(a) function $f \in C^2$
(b) $f \geq 0$ and (c) $f'' \leq 0$ on $[a,b]$
Goal:
Show
$$\frac{(b-a)}{2}(f(a) + f(b)) \leq \int_a^b f \leq ...
0
votes
1answer
43 views
Mathematical Metric spaces
How can we show that any finite measure on a separable complete metric space is tight?
By tight, given $\epsilon > 0$, showing that there exists finitely many points $x_{1},\ldots,x_{n}$
3
votes
1answer
42 views
Question regarding $\lim_{x \to 0} \left(\exp(\sin (x)) + \exp \left(\frac{1}{\sin (x)}\right)\right)$
I wanted to find out whether the following limit exists, and find the value if it does.
$$\lim_{x \to 0} \left(\exp(\sin (x)) + \exp \left(\frac{1}{\sin (x)}\right)\right).$$
Attempt
After many ...
4
votes
1answer
46 views
Closed form of an integral
Is there a closed form of $$\int\limits_0^1\frac{\arctan(\sqrt{x^2+2})}{(1+x^2)\cdot(\sqrt{x^2+2})}dx \quad \text{?}$$
I just know that ...
1
vote
1answer
28 views
Is the following version of the fundamental lemma of the calculus of variations valid?
Let $U$ be an open subset of $\mathbb{R}^n$ with smooth boundary $\partial U$. Consider a function $f$ in $L^2(u)$. Suppose that for every $h$ in the Sobolev space$ H^2_0(U)$ it holds that
$$\int_U f ...
1
vote
1answer
24 views
Finding a strong enough solution to a specific PDE problem.
Let $U\subset \mathbb{R}^n$ with smooth boundary $\partial U$. And consider the expression
$$\Delta u = f.$$
$$\text{+"convenient boundary conditions"}$$
In my specific case $f\in H^2_0$. Under ...
-1
votes
0answers
13 views
Help with Toeplitz operators applications. [closed]
I am trying to find a physics problem which solution involves Toeplitz operators.
2
votes
4answers
57 views
The value of $\lim_{x\to +\infty} \dfrac{e^{x^2}}{10^{|x|}}$
I want to find the value of
$$\lim_{x\to +\infty} \dfrac{e^{x^2}}{10^{|x|}}.$$
Since $x \rightarrow + \infty$, I only consider the value of the function for $x \ge 0$, i.e.
$$\lim_{x\to +\infty} ...
0
votes
1answer
44 views
Show $C\geq \mathrm{max}\left \{ A,B \right \}$.
Let $\sum_{n=0}^{\infty}a_{n}x^{n}$ and $\sum_{n=0}^{\infty}b_{n}x^{n}$ be the power series with the convergent of radius respectively $A>0$ and $B>0$.
Define $c_{n}=\mathrm{min}\left \{ ...
0
votes
0answers
41 views
Uniform convergence of $\sum_{n=2}^{\infty}\frac{\mathrm{ln}(n)^{x}}{n}$.
Given that the series $\sum_{n=2}^{\infty}\dfrac{\mathrm{ln}(n)^{p}}{n}$ is convergent iff $p<-1$.
Show that $\sum_{n=2}^{\infty}\frac{\mathrm{ln}(n)^{x}}{n}$ is uniform convergent as $x \in ...
0
votes
1answer
40 views
Find for all value of constant $a>0$; the interval of convergence of the power series $\sum_{n=0}^{\infty}\frac{1}{1+a^{n}}x^{n}$
Find for all value of constant $a>0$; the interval of convergence of the power series
$\sum_{n=0}^{\infty}\frac{1}{1+a^{n}}x^{n}$.
What I have tried is; if we let $b_{n}=\frac{1}{1+a^{n}}x^{n}$ so ...
0
votes
2answers
25 views
How do I show that the degree $n$ Taylor polynomials of $f$ about two points are equal?
Question
Suppose that $f(x)$ is a polynomial of degree $d$, and that $n \ge d$. Let $x_0 \neq x_1$. Show that the degree $n$ Taylor polynomials of $f$ about $x_0$ and $x_1$ are equal.
Attempt
Let the ...
1
vote
2answers
39 views
corollary to the completeness axiom
The corollary states "Every nonempty subset $S$ of $\mathbb{R}$ that is bounded below has a greatest lower bound inf S.
The part I don't get in the proof is from where they came up with the set $-S$ ...
1
vote
0answers
55 views
minimization of function $F(a) = \int_0^1 (G(x) - P_a(x))^2\,dx$?
I have the following questions referring to this link to a previous question on this site : Approximate a function over the interval $[0, 1]$ by a polynomial of degree $n$ (or less).
a) Explain why ...
5
votes
1answer
79 views
Continuous function differentiable on $[0,1]\setminus\mathbb{Q}$, but nondifferentiable on all of $\mathbb{Q}\cap[0,1]$?
I'm trying to work out an example of a continuous function which is differentiable at all irrationals but nondifferentiable at all rationals in $[0,1]$.
Since $\mathbb{Q}$ is countable, list it as ...
6
votes
3answers
110 views
If $\lim_{x \to \infty}f(x)=\lim_{x \to -\infty}f(x)=0$ does it imply that $\lim_{x \to \infty}f'(x)$ = $\lim_{x \to -\infty}f'(x)=0$?
Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is everywhere differentiable and
$\lim_{x \to \infty}f(x)=\lim_{x \to -\infty}f(x)=0$,
there exists $c \in \mathbb{R}$ such that $f(c) \gt 0$.
Can we ...
1
vote
2answers
44 views
Prove the convergence of the sequence.
Prove the convergence of the following sequence:
$$x_1 = \sqrt{a}$$
$$x_{n+1} = \sqrt{a + x_n}$$
1
vote
0answers
28 views
What's the need of $^{S}_{T}$ in $f^{S}_{T}:S\rightarrow T$?
I'm reading Lang's Undergraduate Analysis:
In the chapter about mappings, he says that we should denote the set of arrival and the set of departure with the following notation:
...
0
votes
1answer
17 views
Is there extension of function from a curve on the whole space preserving smoothness?
Assume that $\alpha: (a,b) \rightarrow \mathbb R^3$ and $f: (a,b) \rightarrow \mathbb R$ are given smooth functions. Let $t_0 \in (a,b)$.
Do there exist a $\delta>0$ and a smooth function $V: ...
1
vote
2answers
140 views
A less known definition of the definite integral of a continuous function
The definite integral of a continuous function can be defined using the bounded monotone sequence property: see Osgood's Functions of Real Variables, p.110.
(link to full book) (screenshots: page ...
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0answers
22 views
question about Bernoulli number
we know that we can generate the Bernoulli number using this equation
$(1+B)^n=B^{[n]}$
but how we can prove it ?
please help
and thanks for all
2
votes
1answer
67 views
Prove that $(1+1/x)^x$ is concave for $x>0$
From the graph it looks like $(1+1/x)^x$ is concave for $x>0$. But in this post, I can only prove that it is concave for $x\ge 1$. It is of interest to see a proof for $x>0$.
5
votes
1answer
48 views
Chain rule proof
Let $a \in E \subset R^n, E \mbox{ open}, f: E \to R^m, f(E) \subset U
\subset R^m, U \mbox{ open}, g: U \to R^l, F:= g \circ f.$ If $f$ is
differentiable in $a$ and $g$ differentiable in ...
2
votes
1answer
26 views
Proof on showing function $f \in C^1$ on an open & convex set $U \subset \mathbb R^n$ is Lipschitz on compact subsets of $U$
The question is as follows:
Given:
(1) function $f: U \subset \mathbb R^n ==> \mathbb R$
(2) $U$ is open and convex set
(3) $f \in C^1$ in $U$
Goal: Show that $f$ is ...
4
votes
1answer
64 views
$\iint f(x,y)\,dxdy$ and $\iint f(x,y)\,dydx$ exist but $f$ not integrable on $[0,1]\times[0,1]$
I want to look for a function $f(x,y)$, whose support is inside $[0,1]\times[0,1]$, such that $\int_0^1\!\int_0^1\!f(x,y)\,dxdy$ and $\int_0^1\!\int_0^1\!f(x,y)\,dydx$ both exist, but $f(x,y)$ is not ...
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votes
0answers
16 views
Question on a third-order boundary value problems
This is the corollary $2.1$, from the article "Positive solutions of third order semipositone boundary value problems"
if $$u'''=\lambda \left(\sum_{i=1}^m c_i(t)u^{\mu_i}-d(t)\right)+e(t), t\in ...
0
votes
3answers
36 views
Finding sequence in a set $A$ that tends to $\sup A$
I have been reading the book at http://www.neunhaeuserer.de/short.pdf, and have noticed that in the proof of the intermediate value theorem (Theorem 5.8 in the book), it seems to be quietly assumed ...
4
votes
1answer
63 views
A question on limsup
Let $a_n>0$. Prove that $$\varlimsup_{n\to\infty}n\left(\frac{1+a_{n+1}}{a_n}-1\right)\geq 1.$$
I argue by contradiction. If it is not ture, then $$\exists\ N,\ \forall\ n\geq N, ...
2
votes
2answers
49 views
alternating series test for $\sum_{n=1}^{\infty}(-1)^n\frac{\sqrt{n}}{n+4}$
I must determine if this series converges (using specifically the alternating series test) $$\sum_{n=4}^{\infty}(-1)^n\frac{\sqrt{n}}{n+4}.$$
I know the necessary and sufficient conditions are:
The ...
2
votes
2answers
82 views
Lipschitz continuous
Let $\delta$ be an interval in $\mathbb{R}$. Recall that a function $f$ is called Lipschitz continuous on $\delta$ with Lipschitz constant $L$ if there holds
$|f(x) - f(y)| \leq L|x-y|$ for all $x,y$ ...
5
votes
2answers
48 views
“Nearly” Harmonic Series
It's well known that
$$
\sum_{n=1}^{\infty} \frac{1}{n^{1+\varepsilon}} < \infty, \ \forall \varepsilon >0.
$$
What happens if we replace $\varepsilon$ with $\varepsilon_n \downarrow 0$?
...
1
vote
0answers
38 views
Extending a rational entry matrix to an orthogonal matrix.
Suppose $M \leq N$ are positive integers and let $r_1,\ldots,r_M$ represent the rows of an $M\times N$ matrix $A$ in which: (i) the rows of $A$ are orthogonal and having the same norm, (ii) the ...
4
votes
3answers
102 views
Limit. $\lim_{x \to \infty}{\sin{\sqrt{x+1}}-\sin{\sqrt{x}}}.$
I want to compute $$\lim_{x \to \infty}{\sin{\sqrt{x+1}}-\sin{\sqrt{x}}}.$$
Is it OK how I want to do?
...
6
votes
1answer
80 views
Can someone clarify Example I.I.2 from Hardy's Course of Pure Mathematics?
"If $\lambda, m,$ and $n$ are positive rational numbers, and $m > n$, then $\lambda(m^2 − n^2), 2\lambda mn$, and $\lambda(m^2 + n^2)$ are positive rational numbers. Hence show how to determine any ...
3
votes
2answers
68 views
Limit: $\lim_{x \to 0}{[1+\ln{(1+x)}+\ln(1+2x)+\ldots+\ln(1+nx)]}^{1/x}.$
How can I find the following limit?
$$\lim_{x \to 0}{[1+\ln{(1+x)}+\ln(1+2x)+\ldots+\ln(1+nx)]}^{1/x}.$$
It's a limit of type $\displaystyle 1^{\infty}$ and if I note with $\displaystyle ...
2
votes
3answers
79 views
Show that $c$ is closed in $l^{\infty}$
Let $$c=\{ (a_i)_{i \in \mathbb{N}} \ ; \ \ a_i \in \mathbb{R}\ ,\ \forall i \in \mathbb{N} \ , \ \mbox{exist} \ \displaystyle \lim_{i \to \infty}(a_i)\}$$
$$l^{\infty}=\{ (a_i)_{i \in \mathbb{N}} \ ...
16
votes
6answers
246 views
Why is boundary information so significant? — Stokes's theorem
Why is it that there are so many instances in analysis, both real and complex, in which the values of a function on the interior of some domain are completely determined by the values which it takes ...
1
vote
2answers
26 views
Find $y$-Lipschitz constant
$$f(x,y)=x^3e^{-xy^2}, 0\leq x\leq a, y\in \mathbb R, a>0$$
I need to find $K>0$ such that $$|f(x,y_1)-f(x, y_2)|\leq K|y_1-y_2|$$ for all $0\leq x\leq a$ and $y_1,y_2\in \mathbb R$
I did this ...
0
votes
1answer
40 views
What is the Sobolev Lemma?
In the paper I am reading the authors state that $|\nabla u|_\infty$ can be replaced by $|u|_3$ using the Sobolev Lemma. I am trying to find this lemma but its turned out to be very difficult.
The ...
3
votes
0answers
55 views
Formula for Sum of Logarithms $\ln(n)^m$
As you know $\sum_{n=1}^k \ln(n) =\ln(k!)$ is there a formula for $\sum_{n=1}^k \ln(n)^m$?
0
votes
2answers
17 views
Some algebraic inequalities with the binomial theorem.
I am working on proving the following limits.
1), $\lim_{n \to \infty} \sqrt[n]{n} = 1$
2), If $p >0$ and $\alpha \in \Bbb R$, then $\lim_{n \to \infty} {n^{\alpha}\over{(1+p)^n}} =0$
...
2
votes
1answer
16 views
Proof of a theorem with upper/lower limits.
Theorem: If $s_n \le t_n$ for all $n$ greater than a fixed integer $N$, then $$\lim_{n \to \infty} \inf s_n \le \lim_{n \to \infty} \inf t_n$$
I would like to prove this and it would be nice if ...
3
votes
1answer
24 views
I'm having trouble with a definition of the upper and lower limits, and a theorem that follows it.
The following is the definition.
Let $\{s_n\}$ be a sequence of real numbers. Let $E$ be the set of numbers $x$ such that $s_{n_{k}}\rightarrow{x}$. This set $E$ contains all subsequential limits, ...
0
votes
0answers
33 views
Proving that $\bigotimes_{i=1}^n \cal{B}_{X_i} = \cal{B}_{X}$
Theorem: Given separable metric spaces $X_1,\ldots,X_n$ and $X=\prod_{i=1}^n X_i$, where $X$ has the product metric $d(f,g)=\sqrt{d_1 (f(1),g(1))^2 +\cdots + d_n (f(n),g(n))^2}$. Then ...
0
votes
2answers
42 views
Relationship between sobolev spaces
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. I would like to understand something about the relationships between these two spaces: $$H^1_0(\Omega) \cap H^2(\Omega)\quad \text{and}\quad ...
2
votes
0answers
32 views
Closed curves question
Can you give me some help on the following problem?
Given two closed curves $\alpha, \beta : \mapsto \mathbb{R}^3$ we define $\phi_{\alpha \beta}: I^2 \mapsto \mathbb{R}^3$ as $\phi_{\alpha \beta} ...
