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).

learn more… | top users | synonyms (1)

2
votes
1answer
79 views

Proof of $\int_0^\infty t^{a-1}e^{it}\,dt=\Gamma(a)e^{ia\pi/2}$?

Can anyone show a proof of $$\int_0^\infty t^{a-1}e^{it}\,dt=\Gamma(a)e^{ia\pi/2}$$ where $0<a<1$, and $$\Gamma(a)=\int_0^\infty t^{a-1}e^{-t}\,dt.$$ Thank you.
7
votes
1answer
178 views

$f:[a,b]\to(a,b)$ be continuous how prove $f(c)+f(c+d)+\cdots+f(c+nd)=(n+1)(c+\frac{nd}{2})$

let $f:[a,b]\to(a,b)$ be continuous how prove $\forall n\in\mathbb N$ $\exists d\gt0$ ,$\exists c\in(a,b) $ such that $$f(c)+f(c+d)+\cdots+f(c+nd)=(n+1)\left(c+\frac{nd}{2}\right)$$thanks in advance ...
7
votes
1answer
376 views

Wave Equation, Energy methods.

I am reading the book of Evans, Partial differential Equations ... wave equation section 2.4; subsection 2.4.3: Energy methods. Arriving at the theorem: Theorem 5 (Uniqueness for wave equation). ...
3
votes
1answer
120 views

Versions of L'Hôpital's rule

I am familiar with the following version of L'Hôpital's rule: Let $f,g:I\to\Bbb R$ be differentiable on the interval $I$, further assume that $g'(x)\ne0$. Let $a\in I$ and assume the limit $\lim_{x\to ...
2
votes
2answers
119 views

nested compact set question

Suppose $A \subset \mathbb R^n$ is not compact. Show that there exists a sequence $F_1 \supset F_2\supset F_3\supset\cdots$ of closed sets such that $F_k \cap A\ne\emptyset$ for all $k$ and ...
1
vote
1answer
107 views

Why is the following function not càdlàg?

I have constructed the following function but I can't see why it is not càdlàg on $[0,1]$: $$f(x)=\begin{cases} 1, & ...
1
vote
1answer
573 views

What are the range and the norm of this bounded linear operator?

Given the normed space $\ell^\infty$ of all bounded sequences of (real or complex) numbers with the norm given by $$||x||:= \sup_{j\in Z^+} |\xi_j|,$$ for each $x:=(\xi_j)_{j=1}^\infty$ in ...
3
votes
1answer
117 views

A problem about $C^1$-convergence! (Elliptic theory)

Let a function $u:\overline\Omega\subset\mathbb{R}^n\rightarrow\mathbb{R}$ that satisfies $$\Delta u+f(u)=0 \ \ \ \mbox{in} \ \ \Omega,$$ and consider ...
3
votes
1answer
43 views

$ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right).$

Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: $ \log_{\frac ...
4
votes
0answers
207 views

Disintegration of Measures

I was thinking about this exercise and I can't see how to end it. I'm sorry about the long post and thank you for the attention. Before asking the question, I need some background. Let $(\Omega, ...
0
votes
2answers
135 views

How to compute the norm of this particular bounded linear functional?

On the Hilbert space $l^2$, let $f$ be the functional defined by $$f(x):= \sum_{j=1}^\infty \alpha_j \xi_j$$ for each $x:=(\xi_j)_{j=1}^\infty$ in $l^2$, where $a:= (\alpha_j)_{j=1}^\infty$ is a fixed ...
5
votes
3answers
103 views

Stuck on this integral involving exp and the floor function

Here is the integral $$\int_0^\infty \lfloor x \rfloor e^{-x}dx$$ Here is what I have so far: $$I = \sum_{n=0}^\infty \int_n^{n+1} n e^{-x}dx$$ $$ = \sum_{n=0}^\infty -ne^{-n-1} + ne^{-n}$$ $$ = ...
1
vote
1answer
194 views

How to find the range and inverse of this linear operator?

Given $T \colon C[0,1] \to C[0,1]$ defined by $$Tx(t):= \int_0^t x(r) dr$$ for each $t\in [0,1]$, where $C[0,1]$ is the normed space of continuous real-valued (or complex-valued) functions defined on ...
7
votes
0answers
167 views

Prime numbers, analysis of polylogarithms

Can any interesting results concering prime numbers be obtained using the analytic properties of the polylogarithm, similar to how analytic methods are used on the zeta function to obtain results ...
3
votes
0answers
74 views

Is the graph of every real function a null set? [duplicate]

This question popped to my mind during an analysis lecture: Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a (general) function. Is there an $N\subset \mathbb{R}^2$ with $\lambda^2(N)=0$, such that ...
0
votes
0answers
68 views

Uniform best approximation in Chebyshev/Haar systems and the necessity of compactness of the function domain.

A great deal of Chebyshev/Haar systems are given for intervals $]-\infty,\infty[$, $[0,\infty[$ and other noncompact subsets of $\mathbb{R}$. Nonetheless, the theory of uniform best approximations in ...
3
votes
1answer
59 views

The area of the set in which a polynomial is “small”

Prove that there exist a constant $C$ such that for every monic polynomial $P$, the area of the set $A=\{x : |P(x)|<1\}$ is at most $C$. Remarks: This puzzle holds for both the real and the ...
1
vote
1answer
32 views

Help with limit $\lim_{h\to 0}\frac{1}{h}\int_{x}^{x+h}P(t, y)\ dt$..

Let $D\subseteq \mathbb R^2$ be an open set and $P:D\rightarrow \mathbb R$ continuous. For $y$ fixed how to evaluate, $$\displaystyle\lim_{h\to 0}\frac{1}{h}\int_{x}^{x+h}P(t, y)\ dt?$$ I know the ...
1
vote
1answer
521 views

About an extension of Riesz' Lemma for normed spaces

The Riesz' Lemma is as follows: Let $Y$ and $Z$ be subspaces of a normed space $X$ of any dimension (finite or infinite) such that $Y$ is closed (in $X$) and is also a proper subset of $Z$. Then for ...
0
votes
1answer
233 views

Chebyshev rational approximations to $\cos x$

How can we construct all the Chebyshev rational approximations of degree $3$ for $f(x) = \cos(x)$. So, I note that we first get the Taylor series of $\cos(x) = 1 - \frac{x^2}{2} + \frac{x^4}{24} ...
0
votes
1answer
235 views

How to define limsup of a function

Let $f: [0, \infty) \rightarrow \mathbb R$. What is the definition of $$ \limsup_{x \rightarrow a} f(x)$$ for $a \in [0, \infty)$?
3
votes
1answer
311 views

Unions of disjoint open sets.

Let $X$ be a compact metric space (hence separable) and $\mu$ a Borel probability measure. Given an open set $A$ and $r,\epsilon>0$ $\ $does there exist a finite set of disjoint open balls ...
1
vote
0answers
353 views

Derivation of poisson kernel for disk of radius $R$ from unit disk

Is there a way to derive poisson kernel for disk of radius $R$ from unit disk?
5
votes
2answers
393 views

characteristic curves for second-order equations

Reading about characteristic curves for second-order equations, in particular semi-linear equations of second order with two independent variables: ...
5
votes
0answers
93 views

Can we do some scaling argument in the presence of inhomogeneous norms?

Notation: $B^n_R$ stands for the ball of radius $R$ in $\mathbb{R}^n$. $\hat{f}$ stands for the Fourier transform of $f$. Question. The following inequality holds true for all $f\in ...
4
votes
2answers
214 views

If the sum converges to zero, does that mean that each sequence converges to zero?

If we are given that the sequence sum of two sequences of positive real numbers converges to zero, does that mean that each sequence converges to zero? (by the squeeze theorem)
0
votes
1answer
47 views

Characterization homeomorphism between real intervals.

Suppose $I$ and $J$ are intervals in $\mathbb{R}$ and $f: I \rightarrow J$ is a continuous bijection. Can we state that $f$ is a homeomorphism?
1
vote
3answers
162 views

Linear instability implies nonlinear instability

I am trying to understand the following proof that linear instability implies nonlinear instability. Suppose we have the ODE, $ \frac{du}{dt}=A(u)$ for which $0$ is a solution. Suppose $L $ is the ...
0
votes
1answer
46 views

Question about moving planes method.

In the paper Inequalities for second-order elliptic equations with applications to unbounded domains I, by Berestycki, Caffarelli and Nirenberg (page $486$), they considered a set ...
1
vote
1answer
478 views

Is $\sum \sin^2(k)/k$ Convergent? [duplicate]

A student recently used the series $\displaystyle\sum_{k=1}^\infty\frac{\sin^2k}{k}$ as an example of a divergent series whose terms tend to $0$. However, I'm having trouble convincing myself that ...
1
vote
2answers
234 views

A sequence converging weakly in $\ell^p$, for $p >1$ and failing to converge weakly for $p=1$

For $1 \le p < \infty$ and each index $n$, let $e_n \in \ell^p$ have $n$-th component 1 and all other componenets $0$. I want to show that $p>1 \Rightarrow \{e_n\} \to 0$ weakly in $\ell^p$ and ...
1
vote
1answer
265 views

Automorphisms on Punctured Disc

I have to find the automorphism group of the punctured unit disc $D = \{|z| <1\}\setminus \{0\}$. I understand that if $f$ is an automorphism on $D$, then it will have either a (i) removable ...
0
votes
1answer
55 views

Failure of convergence to 0

Consider the interval $I = [0,1]$ and the sequence of functions: $$f_n(x) = (-1)^k \ \text{for} \displaystyle \frac{k}{2^n} \le x < \frac{k+1}{2^n} \ \text{where} \ 0 \le k < 2^n - 1$$ I want ...
1
vote
1answer
109 views

Total Ordering and relation

Consider the relation $<$ on $\mathbb{Q}$ defined by: $(m, n) < (j, k) \iff jn-mk \in \mathbb{N}.$ Where $m, j \in \mathbb{N}$ and $n, k\in\mathbb{Z}$ I want to show that $<$ is a total ...
3
votes
1answer
175 views

Weak convexity and continuity

For any open interval $(a, b)\subset {\mathbb R}\,$, define a weakly convex function $f:(a, b) \rightarrow {\mathbb R}$ as one for which $$f(q\;x_0 + (1 - q)\;x_1) \leq q\;f(x_0) + (1-q)\;f(x_1)$$ ...
1
vote
0answers
65 views

what are the borders of the convergence disks of series?

Let $\mathbb{T}=\{z\in \mathbb{C}\mid |z|=1\}$. For which $S\subseteq \mathbb{T}$, is there a sequence $(a_n)\subseteq \mathbb{C}$ such that the series: $$\sum_{k=1}^\infty{a_kz^k}$$ is convergent on ...
1
vote
2answers
969 views

The Least Upper Bound and The Greatest Lower Bound

I am taking math class, and I am not sure about LUB and GLB. I need someone to give this dummy a short explanation about them.... On interval (0,10), 0 is a lower bound and 10 is upper bound, but ...
0
votes
1answer
79 views

Conditions of parameter $\lambda$ ensuring integral is 0

Let $1 \le p \le \infty$. I am seeking to find the values of the parameter $\lambda$ such that: $$\displaystyle \lim_{\epsilon \to 0^+} \frac{1}{\epsilon^\lambda} \int_{0}^{\epsilon} f = 0 \ \ \forall ...
7
votes
3answers
367 views

Why does metric space which has the countable chain condition implies separable?

I've looked around but all I could find is that if X is separable then X has ccc. Can anybody give me some help?
2
votes
2answers
3k views

Monotonically increasing vs Non-decreasing [duplicate]

Is monotonically increasing is same as non-decreasing? Thank you for answer beforehand.
4
votes
2answers
54 views

Limit sequence sets

In my measure theory book I came across the following definition: Let $(A_n)_{n\ge1}$ be a sequence of subsets of some set $X$. Define: $\limsup_{n\to\infty} A_n:=\bigcap_{n\ge1}\bigcup_{k\ge n}A_k$ ...
0
votes
2answers
117 views

How do I prove $f:[0,1]^3\rightarrow\mathbb R$ has a minimum and maximum?

How do I prove continuous function $f:[0,1]^3\rightarrow\mathbb R$ has a minimum and maximum? Using compact, sequentially compact, continuous, or closed theorems.
0
votes
2answers
172 views

Supremum Axiom of $S = \{a+b\sqrt{2}: a,b \in Q\}$

Let: $$S = \{a+b\sqrt{2}: a,b \in Q\}.$$ It can be shown that $S$ ordered field. Dose the Supremum Axiom hold in $S$? Why? (with proof please)
1
vote
3answers
147 views

Question regarding definition of limit point and uniqueness

I found the following definition in a paper A sequence in a space $X$ is an ordered family $(x_j)_{j\in \mathbb{N}}$ where $x_j \in X$ but not necessarily $x_i \ne x_j$ for $i \ne j$, that is, ...
0
votes
1answer
49 views

Non-negative functions bounded from below.

Is there a result (or source) that says if I have a function $f$ defined on some set $S$ and that $f(y)>0~\forall y\in S$, then there exists a constant $K>0$ such that $f(y)\geq K~\forall y\in ...
1
vote
3answers
247 views

How many points of discontinuity?

I need to prove that any monotonic function whose domain is an interval $[a;b]$ can have only finite or countable number of discontinuity points... I don't seem to have any insightful ideas. It even ...
5
votes
1answer
145 views

d'alembert's formula

I'm studying the Cauchy problem for the wave equation $n=2$; $$\begin{cases}u_{tt}=\alpha^{2} u_{xx}, x \in\mathbb{R}, t>0\\[8pt] u(x,0)=f(x), x\in\mathbb{R}\\[8pt] u_{t}(x,0)=g(x), x\in\mathbb{R} ...
0
votes
2answers
254 views

Taking supremum of inequality

If I have something like $$\int_0^T F(t,f) \leq \int_0^T G(t,f)$$ where f is a function, can I take supremums: $$\sup_{|f| = 1}\int_0^T F(t,f) \leq \sup_{|f| = 1}\int_0^T G(t,f)?$$ The integrands are ...
0
votes
1answer
32 views

Finding a minimal non-empty closed $G$-invariant set of a compact metric space

Let $G$ the abelian group generated by commuting homeomorphisms $f_1,\dots,f_q:M\rightarrow M$, where $M$ is a compact metric space. Show that there is $X\subset M$ minimal with respect to the ...
5
votes
1answer
80 views

Bounding $\liminf_{n} n |f^n(x)-x|$

I solved an exercise in which the first part asks to prove that for any measure preserving measurable transformation $f:[0,1]\rightarrow [0,1]$ we have $$\liminf_{n} n |f^n(x)-x| \leq 1, \ ...