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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3
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1answer
109 views

Derivative and lipschitz

If I have a real-valued continuous function defined on a compact subset of real line, such that its derivative(wherever it exists) is bounded. Is such a function necessarily Lipschitz? Additionally, ...
1
vote
0answers
32 views

understanding topological argument in rado-kneser theorem

Rado-kneser choquet theorem states that Poisson integral of a homeomorphism of unit circle is a homeomorphism. It's proof goes like proving it local homeomorphism by proving non vanishing of jacobian ...
1
vote
2answers
180 views

Lebesgue integral of a non-negative function.

I have been looking at Kolmogorov's book "Introductory Real Analysis" and have become stuck at the problem 4a) on page 301. In this problem we are given $f$ a nonnegative and integrable function on ...
4
votes
0answers
80 views

Prove that $\lim\limits_{t \rightarrow 0} \int\limits_{0}^{\infty} e^{-tx} f(x) dx = 1$ for $t>0$.

Suppose $f \in \mathcal{R}$ on $[0,A]$ for all $A < \infty$, and $f(x) \rightarrow 1$ as $x \rightarrow + \infty$. Prove that $\lim\limits_{t \rightarrow 0} \int\limits_{0}^{\infty} e^{-tx} f(x) ...
1
vote
0answers
42 views

Question about Morse index

in general the Morse index of a critical point $p$ is the suprimum of the dimensions of sub spaces where $f''(p)$ is negative definite but whene $f''(p)=I-T$ ($f''(p)$ is a compact perturbation of ...
0
votes
1answer
29 views

If $a>0,b>0$ show that exists $x,y\in \mathbb{I}$ near to $a,b$ such that $x^y\in \mathbb{Q}$

If $a>0,b>0$ show that exists $x_0,y_0\in \mathbb{I}$ (irrationals) near to $a,b$ respectively such that $x_0^{y_0} \in \mathbb{Q}$. I was trying this way: Defining the function ...
0
votes
1answer
57 views

what are closed sets in $L^{1}(\mathbb R)$?

Consider, $L^{1}(\mathbb R)$= The space of Lebesgue integrable functions on $\mathbb R$; for $f\in L^{1}(\mathbb R),$ we define its norm, by $\|f\|_{L^{1}}=\int_{\mathbb R}|f(x)| dx$; It is well-known ...
1
vote
1answer
347 views

Problem about $G_{\delta}$-set and $F_{\sigma}$-set

Prove if $E$ is any measurable subset of $\mathbb{R}$, then there are a $G_{\delta}$-set $G$ and a $F_{\sigma}$-set $H$ such that $H \subseteq E \subseteq G$, and such that $m(G$\ $H)=0$. In order to ...
0
votes
1answer
75 views

Can we take the infimum over a variable set?

Suppose that we have a family of functions $\lbrace f_{\alpha}(x)\rbrace_{\alpha}$ define on an open set of $\mathbb{R}^{m}$, and $\alpha$ runs over a set $\Gamma$. Assume that the family is ...
1
vote
0answers
33 views

Reference needed for the following sobolev inequalties

I'm reading a paper and the authors applied the following sobolev type estimates $$ ||(Dv)^{2}||_{H^{3k-2}(\Omega)}\leq C||v||_{H^{3k-1+\alpha}(\Omega)}^{2} $$ for $\alpha>\frac{1}{4}$, where $v$ ...
5
votes
4answers
109 views

Does there exist an $x$ such that $3^x = x^2$?

I tried solving for $x$ by using $x \log(3) = \log(x^2) $$\log(3) = \frac{\log(x^2)}{x}$$ I'm stuck on this part. how do I isolate $x$ by itself? Any help would be appreciated.
1
vote
2answers
121 views

Locality of Inverse Function Theorem

For the Inverse function theorem, the theorem proved the existence of a inverse relation on a local scale, that is if $Df(x)$ is invertible, $f$ is $C^1$ function and $f$: open set E $\subset R^2$ -> ...
1
vote
1answer
59 views

How to find the Fourier transform of $\mathbf 1_{[0,2\pi]}(x)\sin(x)$?

How does one find the Fourier transform of $f(x):=\mathbf 1_{[0,2\pi]}(x)\sin(x)$? I have tried to use the definition from my text: \begin{align*} \hat f(\xi) & = \frac{1}{\sqrt{2\pi}} ...
0
votes
1answer
39 views

Proof of $\left |\frac{\sin(n+1/2)t}{\sin{t/2}}-\frac{\sin{nt}}{\tan{t/2}}\right| \leq 1$

I need help to proof $$\left |\frac{\sin(n+1/2)t}{\sin{t/2}}-\frac{\sin{nt}}{\tan{t/2}}\right| \leq 1$$
1
vote
1answer
35 views

Calculating similarity of gaps between integers in a set

First off I should state that I'm not a mathematician, I'm a programmer (Python, Javascript). But I thought this was more of a mathematical question than a programming one, so I'm asking it here. I ...
0
votes
1answer
46 views

Show that the norms $ | p | _1 $ and $ | p | _2$ are not equivalent

Let P be a vector space of polynomials with real coefficients. Show that the norms $ | p | _1 $ and $ | p | _2$ are not equivalent, where $|p|_1$=max$ \{|p(t)|$; $0\leq t \leq 1 \}$ and $|p|_2$ = max ...
2
votes
1answer
151 views

Convergence in Probability of a Sequence of Exponential Random Variables

If $X$ is an exponential random variable with $\lambda = 3$ and $Y_n = \frac{X^n}{n}$, I am trying to prove whether or not $Y_n$ converges in probability. My original approach was the following: ...
2
votes
0answers
63 views

Notation for near optimal solution

Usually, $x^*$ is used to denote the optimal solution to a maximization problem. I need a notation to describe a solution that is not optimal but "good enough." In my case this solution is the first ...
0
votes
3answers
85 views

Can two function $f$ and $g$ have same values through out a given interval and different values outside that interval?

Is it possible that for two functions $f$ and $g$ and some interval $(a,b)$ we have $f(x)=g(x)$ for all $x\in(a,b)$ and $f(x)\neq g(x)$ for $x$ outside the interval $(a,b)$? $f$ and $g$ are ...
2
votes
1answer
33 views

Contradiction between $a_0$ and $a_k$ for Fourier Series

I need to calculate the Fourier Series for the function $f(x) = |x| \; f:[-\pi,\pi] \to \mathbb{R}$ When calculating $a_k = {1 \over \pi} \int_{-\pi}^{\pi} f(x) \cos{(kx)} dx \; (k \in \mathbb{N_0})$ ...
0
votes
1answer
43 views

Question on Morse inequalities

I want to understand why: if i have then $(4.1)$ is formal : it means that please help me Thank you EDIT1: $(4.1)$ tel us that $\displaystyle\sum_{q=0}^{\infty} (M_q-\beta_q)t^q=(1+t)Q(t)$ ...
1
vote
1answer
61 views

proving differentiability of functions in $\mathbb{R}^2$

Define $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ by $$f(x,y):= \begin{cases} \frac{xy(x^2-y^2)}{x^2+y^2} &\text{when} \, (x,y) \neq (0,0) \\ 0 &\text{when} \, (x,y)=(0,0)\end{cases}$$ Prove ...
2
votes
1answer
488 views

Proof concerning equivalent definition of supremum using limits

I believe I have a proof to the following theorem: Let $S$ be a subset of $\mathbb{R}$ that is non-empty and bounded above. $s \in \mathbb{R}$ is the supremum iff $s$ is an upper bound of $S$ and for ...
1
vote
1answer
104 views

Suprema Proof: $|\sup A-\sup B| \leq \sup|A - B|$

I am trying to prove that $|\sup A(x)-\sup B(x)| \leq \sup|A(x) - B(x)| \quad \forall x$ in some arbitrary set $S$. It is clear why this is true: the difference between the maximum values of each ...
0
votes
1answer
95 views

Example of continuous function on a closed unit ball with no minimum point at its sphere

Find a continuous function f from a closed unit ball $\subset R^2$ -> $R^2$ that is continuously differentiable on the unit ball (B(0,1)), but 0 is not in the range of f. and there is no point $X_0$ ...
2
votes
1answer
61 views

Recover Fundamental solution of wave equation on $\mathbb{R}^n$ by on the sphere

It's well known that $\frac{\sin{t\sqrt{-\Delta}}}{\sqrt{-\Delta}}\delta$, the fundamental solution of wave equation on the $\mathbb{R}^n$ can be expressed as the form \begin{equation} \lim_{t\to ...
25
votes
1answer
574 views

An awful identity

We take place on $\mathbb C(x_1,...,x_r,x'_1,...,x'_p,u_0,...,u_r,u'_0,...,u'_p)$ with $r,p\in \mathbb N$ Show that : $$\displaystyle{\sum_{i=1}^r \left( \frac{\prod_{j=0}^r (u_j-x_i) ...
2
votes
1answer
24 views

Suprema and Infima of nonpositive functions

I am trying to get some estimates using the time-dependent infimum and supremum of a function $g(t,x)$. I have the following question. Suppose $g(t,x)\leq0$ for all $x\in\mathbb{R}$ and $t\geq0$. ...
3
votes
1answer
82 views

Integral inequality problem

Let $f:[0,1]\to\mathbb R$ be a differentiable function with $f(0)=0$ and $f'(x)\in(0,1)$ for every $x\in(0,1).$ Show that $$\left(\int_0^1f(x)dx\right)^2>\int_0^1(f(x))^3dx$$ I am not even sure ...
11
votes
1answer
580 views

A variation of the isoperimetric problem in the plane

The isoperimetric problem in the plane: « The classical isoperimetric problem dates back to antiquity. The problem can be stated as follows: Among all closed curves in the plane of fixed ...
1
vote
1answer
134 views

integration of product of even and odd function

I have a problem like this: Let $f:[-a,a]\to\mathbb R$ be a continuous function where $a>0$. If $f$ satisfies that $$\int_{-a}^a f(x)g(x)dx=0$$ for every integrable even function ...
2
votes
2answers
50 views

Explore the convergence of a series

I have to explore the convergence of a series. At this picture I used radical Cauchy indication. But I don't now what to do with a denominator to find a limit. Help me please ! Thank You so much :) ...
0
votes
2answers
32 views

Find a sum of a series

Help me find a sum of this series I tried to excrete as (2/7)^n * 3^(n+2) and use De Lamber indication. It gives me a result 6/7. I checked it in Wolfram Math but the result was 54. Where did I go ...
2
votes
1answer
231 views

compactness in topology of pointwise convergence

I started reading about the topology of pointwise convergence. So far I do not feel quite comfortable with this theory. Maybe one can help me out in a more concrete example case. Let's consider ...
2
votes
0answers
22 views

Convergence of Iteration with Sum

Consider a continuous function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$, and the iteration $$ x_{k+1} = \frac{1}{k} \sum_{i=1}^{k} f( x_i ) $$ for some given initial condition $x_1 \in ...
3
votes
1answer
67 views

difference between weak* convergence and convergence

I am trying to prove the following: If $X$ is a finite-dimensional space, then for sequences $\left\{x_n\right\}\subseteq X$ and $\left\{f_n^*\right\}\subseteq X^*$, if there exists an $x\in X^*$ ...
1
vote
1answer
91 views

How to prove it?

Let $y_0\geqslant 2$, $y_n=y_{n-1}^2-2$, $n\in\mathbb{N}_+$, set $\displaystyle S_n=\sum_{k=0}^{n}\frac{1}{y_0\cdots y_k}$, how to prove $$\lim_{n\to\infty}S_n=\frac{y_0-\sqrt{y_0^2-4}}{2}.$$ Do you ...
2
votes
1answer
128 views

Why is $-\Delta+1$ an isomorphism between Sobolev space $W^{2,p}$ and $L^p\,$?

Consider the linear elliptic operator $L: W^{2,p}(\mathbb{R}^N)\to L^p(\mathbb{R}^N)$ defined by $Lu=-\Delta u + u$. Can we prove that $L$ is an isomorphism for all $p\geq 1$? It can be proved that it ...
2
votes
1answer
93 views

Product of Hölder and Sobolev functions

Here $C^{\kappa , \lambda} ( \overline{\Omega} ) = \left\{ h|_{\overline{\Omega}} :h \in C^{\kappa , \lambda} ( \mathbb{R}^{n} ) \text{ and } h \text{ has compact support} \right\}$ denotes $\kappa$ ...
2
votes
0answers
49 views

Proving this is Lebesgue integrable using radial functions

Show that $f:\Bbb R^n\to\Bbb R$, given by: $$ f(x) = \begin{cases}\sin\left(\frac{1}{\|x\|}\right)\|x\|^{-n-\arctan(\|x\|-1)} & x\not=0 \\ 0 & x=0 \\ \end{cases}$$ is Lebesgue ...
2
votes
0answers
55 views

Essential hypothesis of Fourier Inversion formula

Let $f\in L^{1}(\mathbb R)$ and we define its Fourier transform as follows: $\hat{f}(\xi)=\int_{\mathbb R} f(x)e^{-2\pi i \xi\cdot x} dx, (\xi \in \mathbb R);$ and we define $f^{\vee}(x):=\hat{f}(-x)= ...
1
vote
1answer
67 views

Show uniform convergence of a series of complex function on every compact subset

Let $f:B(0,1)\rightarrow \mathbb{C}$ be an analytic function. Suppose $\sum^\infty_{n=0}f^{(n)}(0)$ converges absolutely. Show that there exists an entire function $g(z)$ such that $g(z)=f(z)$ for ...
0
votes
1answer
97 views

Prove $\nabla f(\mathbf x) = \mathbf 0.$

Suppose that the function $f:\Bbb R^n \to \Bbb R $ has first-order partial derivatives and that the point $\mathbf x$ in $\Bbb R^n$ is a local minimizer for $f:\Bbb R^n \to \Bbb R $, meaning that ...
0
votes
1answer
37 views

showing that a sequence converges in the dual space of a normed vector space

Suppose that $S=\left\{s_\alpha: \alpha \in A\right\}$ is a set of points in a normed vector space $X$ such that $\overline{span}(S)=X$. If $\left\{f_n\right\}$ is a bounded sequence in $X^*$ and ...
0
votes
1answer
45 views

Is it possible to integrate this asymptotic expansion?

Suppose that some smooth function $f \in C^\infty\bigl(\mathbb R^n \times (0,+\infty)\bigr)$ possesses an asymptotic development $$ f(x,t) \sim t^{-\alpha} e^{ith(x)} \sum\limits_{k=0}^{+\infty} ...
0
votes
2answers
74 views

Understanding the Definition of $\int_\gamma f\ \overline{dz}$

Definitionally, we have that $$ \int_\gamma f\ \overline{dz} = \overline{\int_\gamma \overline{f}\ dz} $$ Now let $\int_\gamma f\ dz = w = x +yi$. Question 1: Is it not the case that $\int_\gamma ...
0
votes
1answer
21 views

Defining the Complex Line Integral w.r.t. $x$ and $y$

Ahlfors defines line integrals with respect $x$ as follows: $$ \int_\gamma f\ dx = {1 \over 2} \left( \int_\gamma f\ dz + \int_\gamma f\ \overline{dz} \right) $$ From this I take it as obvious that ...
0
votes
1answer
133 views

Find the rate of convergence?

Given is the iteration $x_{k+1}=\frac{1}{11}(1-\cos(x_{k}))$ with $x_{0}\in (-\frac{\pi }{2},\frac{\pi }{2})$ without $0$. Check if the sequence converges to $x^{*}=0$ and find its convergence rate. ...
1
vote
1answer
42 views

Fix point of a continuous function under some conditions [closed]

Prove that under each of the following conditions the continuous function $f:[a,b]\to\Bbb{R}$ has a fix point: $f([a,b])\subset [a,b]$ $f([a,b])\supset [a,b]$ When $f$ is bijective and ingective.
0
votes
2answers
38 views

Explore the convergence of a series with ln

How to explore the convergence of this series: $$ \sum_{n=1}^{\infty}\dfrac{1}{\ln^n(n+1)} $$ What would be better to use: De Lamber indication or feature comparison. And if comparison is a good ...