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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4
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6answers
154 views

How to prove $(1-\frac1{36})^{25}\lt\frac12$?

How to prove the inequality? $(1-\frac1{36})^{25}\lt\frac12$ I'm in trouble. Thank you very much for your help
0
votes
2answers
41 views

Firstly what is an $O(h^3)$ formula? Also I am not quite sure how to answer the question?

The forward-difference formula can be expressed as $$f'(x_0)=\frac{1}{h}(f(x_0 +h)- f(x_0))-\frac{h}{2}f''(x_0) - \frac{h^2}{6}f'''(x_0) + O(h^3).$$ Use Richardson's extrapolation to derive an ...
2
votes
0answers
27 views

Uniformly bounded sequence of $L^{2}$ functions and a limit

Let $f_{n}: \mathbb{R}^{d} \rightarrow \mathbb{R}$ such that $\sup_{n}\|f_{n}\|_{L^{2}} < \infty$. Furthermore suppose $f_{n} \rightarrow f$ pointwise almost everywhere for some $f$. The problem I ...
1
vote
2answers
30 views

Property of the variation of a function

I need help with the following: given $ f:[a,b]\rightarrow \mathbb{R}$, show that $$V_f (a;b)=V_f(a;c)+V_f (c;b)$$ with $a< c <b$. We know that $$V_f(a;b) \geq \sum_i |f(x_i)-f(x_{i-1})|$$ for ...
2
votes
2answers
44 views

Find an equation for a moving rod

The two endpoints of a 1-metre long rod have an initial position at $(0,0),(0,1).$ The rod slides continuously to the position $(1,0),(0,0)$ sweeping out a region in the positive quadrant. Determine ...
2
votes
1answer
35 views

Help with Rudin rank theorem proof!

I am struggling through Rudin's proof of the rank theorem (9.32) in the baby Rudin book. There is a part in the proof where he claims that for a finite-dimensional linear operator A, if the set V is ...
0
votes
1answer
87 views

Proof of $\displaystyle\lim_{x‎\to‎ 0} \frac{e - (1+x)^\frac{1}{x}}{x}=\frac{e}{2}$

I want to prove $$\lim_{x‎\to‎ 0} \frac{e - (1+x)^\frac{1}{x}}{x}=\frac{e}{2}$$ without useing L'Hôpital's rule.
9
votes
1answer
134 views

An integral with $e^{1+e^x}$ I had trouble working through

I had an analysis test earlier this morning and came across this integral, which I couldn't figure out. Parts of it are easy, but after integrating $y$ you're left integrating $xe^{1+e^x}$ which had ...
0
votes
0answers
33 views

Rudin rank theorem proof

I am studying rudin's proof of the rank theorem, (theorem 9,32 in baby rudin.) We have an invertible function H(x) defined on an open set. He claims we can "shrink" the open set on which the function ...
2
votes
0answers
48 views

Interesting examples of switching limit and integral

We learn many theorems regarding the relationship of limit and integral (Dominated/ Monotone Convergence, Fatou, Semicontinuity of norms, etc...). As I'm working on my research, I find that I often ...
0
votes
1answer
22 views

Test for uniform convergence on $\sum_{k=0}^\infty(\frac{1}{kx+2}-\frac{1}{(k+1)x+2})$

Test for uniform convergence on $0 \le x \le 1$: $$ \sum_{k = 0}^{\infty} \left[{1 \over kx + 2} - {1 \over \left(k + 1\right)x + 2}\right] $$ I think I'm supposed to use the weierstrass m-test. The ...
0
votes
1answer
17 views

Continuity along different spaces

1) Say I have a function that is continuous along $\mathbb{R}.$ Would that function be then continuous along $\mathbb{Q}$ ? How about the other way around? 2) If I have two functions that are not ...
1
vote
1answer
37 views

Deck transformations

We have a theorem that says that if a group $G$ acts on a path-connected space $Y$ properly discontinuously, then $\pi: Y \rightarrow Y/G$ is a covering map. Especially, $G$ is isomorphic to the group ...
1
vote
3answers
92 views

Is the function $f(x) = 1/x$ continuous?

A function f is mapped from the non-zero reals to the reals . We assume the natural topology to be induced on the domain. Then is the function f(x) = 1/x continuous ? EDIT Suppose I use this ...
1
vote
1answer
45 views

To control first derivative with the function itself.

Let $f$ be a compactly supported positive $C^2$ function. I want to show that there exists $C$, such that for all $x\in \mathbb R$, we have $f'(x)^2< C f(x) $ by showing that for every point ...
0
votes
1answer
15 views

Determining the semigroup and relating it to the generator via the functional calculus

In the theory of Hille Yoshida, we have a (definitely real) banach space $L$ and on it a contractive, strongly continuous semigroup of operators. ($T_tT_s=T_{t+s}$ and $T_0=I$.) We then define the ...
0
votes
0answers
22 views

ODE Initial value problem formualtion

If I have the following ODE initial value problem, $$\begin{align} y'(t) &= f(t), \quad t>0, \\ y(0) &= y_0. \end{align}$$ Then I was taught that a solution to the problem is given by ...
2
votes
1answer
34 views

Showing $f_{n} \rightarrow f$ in $L^{1}$ given an integral condition

Let $f_{n}: [0, 1] \rightarrow [0, \infty)$ be a Borel measurable function such that $$\int_{0}^{1}f_{n}(x)\log(2 + f_{n}(x))\, dx < \infty.$$ If $f_{n} \rightarrow f$ Lebesgue almost everywhere. ...
1
vote
0answers
11 views

Request for information about certain linear transformations of functions on subsets

Suppose I have an infinite set $U$ and let $M$ be the linear subspace of all real-valued functions $\nu$ on $2^U$ such that $\nu(\emptyset) = 0$. Here the sum of two such functions (and the product of ...
0
votes
0answers
25 views

Proving $\cos$ is Lipschitz continuous with $L=\frac{\sqrt3}2$ on $[-\frac12,1]$, using $\frac{\sqrt3}2=\cos\frac\pi6=\sin\frac\pi3$

I'm working my way through some analysis exercises to gain a better understanding and I stumbled upon an exercise where I could really use a hint. The task is to show that the inequality $|\cos ...
2
votes
2answers
35 views

Which $f \in L^\infty$ are the Fourier transform of a bounded complex measure?

A measure on $\mathbb R$ is a set function $\mu,$ defined for all Borel sets of $\mathbb R,$ which is countably additive(that is, $\mu(E)=\sum \mu(E_{i})$ if $E$ is the union of the countable family ...
0
votes
0answers
34 views

Proving the existence of limit of an integral

Let $g:\mathbb{R}^d\to\mathbb{R}$ a smooth function and $B:\mathbb{R}^d\to\mathbb{R}^d$ a Lipschitz continuous vector field. I have to study the limit of the following integral ...
4
votes
2answers
81 views

Spectral theory - continuous spectrum

imagine that I have some differential operator $D$ that is defined on an interval $[a,b]$. Now, assume that we take the boundary conditions in such a way that this operator is self-adjoint. Then, I ...
0
votes
0answers
21 views

Integral inequality with gamma function

I have some trouble with paper I'm reading. The goal is this: let $s=\frac{1}{2}+\frac{1}{\log n}+it$. $M$ is a function such that $M(s)=O(\log^{3}(N(|t|+2)))$. Define $$U(s)=\frac{1}{2\pi ...
1
vote
1answer
24 views

holomorphic functions with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
0
votes
0answers
9 views

Upper semicontinuous function and equivalent statements

Problem Let $f:\mathbb R^n \to \overline{\mathbb R}$, then the following statements are equivalent: (1) $f$ is upper semicontinuous; (2) for every $t \in \overline{\mathbb R}$, $\{x \in \mathbb ...
2
votes
0answers
47 views

A hard Conformal Mapping problem

I am trying to construct a conformal map from $R = \{z \in \mathbb{C} : -1 < Re(z) < 1$ and $Im{(z)} > 0\} \cap \{z \in \mathbb{C} : |z| > 1\}$ to the unit disk $\mathbb{D}$. I am really ...
1
vote
1answer
56 views

Existence of solution of ordinary differential equation

I am reading a proof of the existence of solutions for ordinary differential equations and I have some basic doubt. I'll copy the statement, the part of the proof I don't understand and my question: ...
2
votes
2answers
55 views

When do evaluation and the integral sign “commute”?

This is a difficult question to put into words so it's much easier to write the math. Let $a$ and $b$ be given constants and $g(y) \equiv \int_a^b f(x,y) dx$. When is $g(c) = \int_a^b f(x,c) dx$? I ...
9
votes
0answers
141 views

How prove that $\lim\limits_{x\to+\infty}f(x)=\lim\limits_{x\to+\infty}f'(x)=0$ if $\lim\limits_{x\to+\infty}([f'(x)]^2+f^3(x))=0$?

Question: Let $f$ be differentiable on $[0,+\infty)$, such as$$\lim_{x\to+\infty}\left([f'(x)]^2+f^3(x)\right)=0$$show that $$\lim_{x\to\infty}f(x)=\lim_{x\to\infty}f'(x)=0$$ I think this ...
1
vote
2answers
40 views

Multivariable version of the extreme value theorem

The Wikipedia entry on the extreme value theorem says that if $f$ is a real-valued continuous function on a closed and bounded interval $[a,b]$, then $f$ must attain a maximum value, i.e. there exists ...
0
votes
1answer
23 views

Can't see how this equality that involves the gamma function holds

$$\frac{1}{\sqrt{2\pi}}\sigma^m \gamma\left(\frac{m+1}{2}\right) 2^{\frac{m+1}{2}} = 2^{\frac{m}{2}}\sigma^m \left(\frac{m-1}{2}\right) \left(\frac{m-3}{2}\right)\cdots \left(\frac{3}{2}\right) ...
0
votes
0answers
25 views

Di Perna-Lions theory

I'm reading the paper of Di Perna and Lions "Ordinary differential equations, transport theory and Sobolev spaces". I'm not understanding the proof of corollary II.2; in particular I don't understand ...
0
votes
0answers
17 views

Finiteness of the lower integral implies finiteness a.e. of the function

I want to prove that if a function $f$ is $\mu$-measurable, $f\geq 0 $ $\mu$-a.e., then the integral of $f$ exists, that is its upper and lower integrals coincide. I've found the proof in Modern and ...
6
votes
1answer
62 views

Show that there is sequence of homeomorphism polynomials on [0,1] that converge uniformly to homeomorphism

Let $f:[0,1]\rightarrow [0,1]$ be a homeomorphism. Show that , there exists a sequence of polynomials $$(P_n(x))_n$$ such that $P_n(x)$ converge uniformly to $f$ on $[0,1]$ and every $P_n(x)$ is a ...
0
votes
0answers
19 views

A Good Picture of Analytic?

What is a good mental image of analyticity, for real and complex valued functions? What things are there to look out for to know if something isn't analytic? Anything from the basic axioms of being ...
1
vote
1answer
38 views

Could not find equation of function from data

I have a problem to find equation of function of my data: 0.00 0.007 0.20 0.041 0.40 0.165 0.60 0.449 0.80 0.816 1.00 0.982 1.20 0.741 1.40 0.212 1.60 -0.362 1.80 ...
2
votes
0answers
40 views

when we can say that $\limsup_{n\to\infty}f(a_n)=f(\limsup_{n\to\infty}a_n)$?

when we can say that $\limsup_{n\to\infty}f(a_n)=f(\limsup_{n\to\infty}a_n)$? (What conditions must have the function $f$?)
1
vote
0answers
15 views

the continuity of total variation function of a continuous function of bounded variation [duplicate]

Let f be a continuous function of bounded total variation (refer to http://en.wikipedia.org/wiki/Total_variation for the definition) on $[0,1]$, i.e., $\text{Var}_{[0,1]}f<\infty$. Then the total ...
4
votes
1answer
172 views

Can we express the following in a closed form? [duplicate]

I want to evaluate the integral: $$I=\int_{0}^{\pi/2}\ln \left ( \frac{(1+\sin x)^{1+\cos x}}{1+\cos x} \right )\,dx$$ Well, the sub $u=\pi/2-x$ does not give me any result. In fact it makes the ...
1
vote
0answers
28 views

Roots of this trigonometric polynomial

Let $f:[0,2\pi) \rightarrow \mathbb{R}$ with $f(x):=\sum_{n=0}^{k}a_n \left(1+\cos(x)\right)^n$ for arbitrary $a_n$ with $a_k \neq 0$. My question is: What is the maximum number of zeros that this ...
3
votes
1answer
21 views

What is the value of $a$ so that this condition holds?

Let $f(x) \colon= x-x^2$, $g(x) \colon= ax$. Determine the value of $a$ so that the region above the graph of $g$ and below the graph of $f$ has area equal to $9/2$. Here $f(x) - g(x) = (1-a)x - x^2 ...
1
vote
2answers
56 views

How are these two integrals related?

How to express the integral $$\int_{-2}^{2} (x-3) \sqrt{4-x^2} \ dx $$ in terms of the integral $$ \int_{-1}^{1} \sqrt{1-x^2} \ dx?$$ I know that the latter integral is equal to $\pi / 2$. We can't ...
2
votes
1answer
58 views

Sequence of orthogonal vectors in a Hilbert space

Let $\{x_n\}_{n\in\mathbb{N}}$ be a sequance of pairwise orthogonal vectors in a Hilbert space $H$. Show that the following are equavalent: (a) $\sum_{n=0}^\infty x_n$ converges in the norm topology ...
4
votes
1answer
96 views

Question about path method for multivariable limits

I have to prove that the limit $$\lim\limits_{(x,y) \to (0,0)}\dfrac{x^2}{x+y}$$ does not converge. This is fairly 'easy' to do, but while I was doing it I came across some doubts. I took the limit ...
3
votes
2answers
77 views

Does a nondecreasing, differentiable function have continuous derivative?

Are the following statements true? How to prove or disprove? (1). Let $f$ be a nondecreasing, differentiable function on $[0,1]$. Then $f$ is absolutely continuous? To be stronger, (2). Let $f$ ...
1
vote
1answer
54 views

Does the following sequence converge?

Suppose $a_i>0$ for all $i$, $\frac{\sum_{i=1}^n a_i}{n}\to \infty$ and p>1. Let $$y_n = \frac{(\sum_{i=1}^n a_i)^p}{n^{p-1}\sum_{i=1}^n(a_i^p)}.$$ Is $y_n$ monotonic? How can you prove or disprove ...
1
vote
2answers
49 views

Difference between the two definitions about the equality of two functions

From a long time I have found there are two definitions about the equality of two functions (or identity of two functions). I quoted the two definitions in the following: Zorich's definition ...
1
vote
1answer
80 views

Is there a differentiable function f which the differential function f' is bounded but has no maximum on a closed interval.

Is there a differentiable function $f$ in which the differential function $f'$ is bounded but has no maximum on one closed interval? Thanks
-1
votes
1answer
65 views

Topology question about open spaces of a topological space homeomorphic to the full set. [closed]

Let $\mathcal{U}$ be an open subset of $\mathbb{R}^m$ such that there is homeomorphic $f$ from $\mathcal{U}$ to $\mathbb{R}^m$ and also $f$ is an uniformly continous function. Show that ...