Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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Are the two properties of a function equivalent?

$f(x)$ is a function defined on $\Bbb R^n$. $A$: $\forall x,y$ $$ |f(y)-f(x)-\nabla f(x)^T(y-x)| \le \frac{\beta}{2}\|y-x\|_2^2 $$ $B$: $\forall x,y$ $$ \| \nabla f(y)-\nabla f(x)\|_2 \le \beta ...
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30 views

(Strictly) concave/convex function: Increasing / decreasing slope triangle?

I have the following (possibly quick) question. In a paper I am working with, the following conclusions are drawn which I have a hard time to understand. Since they are given without proof, I assume ...
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1answer
46 views

Is $AA^*$ and $A^*A$ self-adjoint?

if I have a densely defined closed linear operator $A$ and $A^* = -A$(same domain also closed). Is this sufficient that $AA^*$ and $A^*A$ are proper self-adjoint operators, assuming that we can also ...
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1answer
30 views

Prove the uniform convergence of a Fourier series

Suppose that $f$ is a $2\pi$-periodic function that satisfies the estimate $$|f(x)-f(y)|\leq M|x-y|^\alpha$$ for an $0<\alpha<1,$ and let ...
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1answer
66 views

Method of Characteristics for a non-linear PDE

I've been trying to work through some of the more difficult questions we've been given in class in regards to the method of characteristics for solving PDEs, but I've come a bit unstuck. I've been ...
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2answers
49 views

Proving some statements only by the definition of Real numbers.

Let $f \colon \ [0.1] \to \mathbb R$ is monotonically increasing function and $f(0)>0$ and $f(x)\neq x $ for all $x\in [0,1]$. $$A=\{x\in [0,1] : f(x)>x \}$$ We know: every non-empty subset ...
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33 views

If $f_{n}(x_{0}) \rightarrow 0$ for some $x_{0}$ and $f_{n}$ converges uniformly, does $f_{n} \rightarrow 0$ uniformly?

Suppose $f_{n}$ is a sequence of smooth functions on $\mathbb{R}$. Let $K \subset \mathbb{R}$ be compact. Suppose $f_{n}(x_{0}) \rightarrow 0$ as $n \rightarrow \infty$ for some $x_{0} \in K$ and ...
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404 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
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2answers
44 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 ...
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0answers
30 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 ...
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2answers
39 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 ...
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2answers
51 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 ...
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1answer
42 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 ...
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1answer
97 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.
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1answer
140 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 ...
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53 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 ...
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1answer
30 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 ...
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1answer
18 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 ...
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1answer
38 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 ...
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3answers
94 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 ...
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1answer
58 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 ...
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1answer
24 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 ...
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26 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 ...
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1answer
35 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. ...
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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 ...
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0answers
26 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 ...
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2answers
46 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 ...
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35 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 ...
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2answers
97 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 ...
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23 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 ...
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1answer
26 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 $f(D)$?
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1answer
35 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 ...
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56 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 ...
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1answer
61 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: ...
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2answers
56 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 ...
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1answer
218 views
+50

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 ...
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2answers
43 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 ...
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1answer
38 views

Understanding an equality that involves the gamma function

$$\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) ...
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26 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 ...
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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 ...
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1answer
70 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 ...
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20 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 ...
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1answer
44 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 ...
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0answers
42 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 $f$?)
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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
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1answer
174 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 ...
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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
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1answer
22 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
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2answers
58 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
60 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 ...