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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14 views

About window function

In Charles.K.Chui's An introduction to wavelets, on Page54, the window function is a non-trivial function $w∈L^{2}(R)$ satisfying $tw(t)∈L^{2}(R)$. I want to ask how to understand the notion, and how ...
1
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1answer
23 views

Asymptotics of $(\cosh(x+c)-\cosh(c))^{-\frac{1}{2}}$

let $c>0$ be a constant and consider the function $$\frac{1}{\sqrt{\cosh(x+c)-\cosh(c)}}, x>0.$$ I'm wondering how the asymptotic expansion for $x\downarrow 0$ look like!? In case of $c=0$ the ...
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1answer
44 views

Continuous at exactly two points and differentiable at exactly one of them

Give an example of a function which is continuous at exactly two points and differentiable at exactly one of them. Justify your answer. This question is from a competitive examination. I have of ...
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0answers
25 views

convergence in $L^p$ implies convergence in measure

I am trying to show that if $f_n$ converges to $f$ in $L^p(X,\mu)$ then $f_n\to f$ in $L^p$ in measure, where $1\le p \le \infty$. Here is my attempt for $p>1$ - Let $\varepsilon>0$ and define ...
7
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1answer
83 views

Show that $ \sum_{n\in \mathbb {S}} \frac{1}{n} $ is convergent [duplicate]

Let $\mathbb {S} =\left \{ 1,2,3,...,9,11,12,...,19,21,...99,111,112,113... \right \} $ i.e, the positive integers set which contain zero digit is omitted. Now show that $ \sum_{n\in \mathbb {S}} ...
2
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1answer
42 views

Finding $p$ for which $f\in L^p(\mathbb{R}^2)$

Here's an old qual problem in analysis. Let $s=\Vert x \Vert$ and define $$f(x)=\frac{1}{s(1+s^{1/2}\log s)}$$, $x\in \mathbb{R}$. Find $1\le p\le \infty$ for which $f\in L^p(\mathbb{R}^2)$. Attempt ...
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0answers
16 views

Show that $\lim_{y\xrightarrow{nt}x}\int_{\Sigma}f(y-z)g(z)d\sigma(z)=\lim_{\epsilon\searrow 0}\int_{|x-z|>\epsilon}f(x-z)g(z)d\sigma(z)$,

Show that $\lim_{y\xrightarrow{nt}x}\int_{\Sigma}f(y-z)g(z)d\sigma(z)=\lim_{\epsilon\searrow 0}\int_{|x-z|>\epsilon}f(x-z)g(z)d\sigma(z)$, $\forall g \in L^2({\sigma})$ here $x\in \Sigma$ $\Sigma$ ...
3
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2answers
30 views

Is $\limsup_{z\to z_0}f(z)=\limsup_{k\to\infty}f(z_k)$?

Let $\Omega\in\Bbb C$ open, $f:\Omega\to\Bbb R$ a generic function. Let $(z_k)_k\subset\Omega$ s.t. $\lim_{k}z_k=:z_0\in\Omega$. The question is the following: is true that $$ \limsup_{z\to ...
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1answer
43 views

Application of the Banach fixed point theorem

Let $a > 0$. We consider the function: $f: (0, \infty) \to (0, \infty)$, defined by $f(x) = \frac{1}{2}(x + \frac{a}{x})$. Let $(x_n)_{n \in \mathbb{N}_0}$ be defined by: $x_0 \in (0, \infty)$, ...
2
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1answer
31 views

Solutions of autonomous ODEs are monotonic

Problem. Let $I,J$ be open intervals, $\,f:I\to \mathbb R$, continuous, $\,\varphi :J\to R$, continuously differentiable, with $\varphi[J]\subset I$, and $\varphi$ satisfying $$ ...
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1answer
20 views

Theorem 3.3-1, Lemma 3.3-2, and Theorem 3.3-4 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: How to write these as one?

I'm trying to prepare some ancilliary material on the following three results in sec. 3.3 in the book Introductory Functional Analysis With Applications by Erwine Kreyszig: (First, I'm giving ...
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1answer
12 views

inverse of a smooth function in R

Does a $C^1$ strictly increasing function from $\mathbb{R}$ to $\mathbb{R}$ admit an inverse which is also $C^1$ ?
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1answer
34 views

Maximum and minimum of a function from $\mathbb{R}^n$ to $\mathbb{R}$

Let $A \in \mathbb{R}^{n \times n}$ be a real $n \times n$-matrix. Consider the function $$q: \mathbb{R}^n \to \mathbb{R}, x \mapsto x^t A x$$ where $x^t$ is the transposed vector $x$. I now want ...
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1answer
14 views

Evaluating iterated limits and double limit of $\frac{p}{q^2}\sum_{n=1}^q \operatorname{sin}\frac{n}{p}$

Investigate the existence of the two iterated limits and the double limit of the double sequence $f$ defined by $$f(p,q)=\frac{p}{q^2}\sum_{n=1}^q \operatorname{sin}\frac{n}{p}$$ I think I need to ...
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1answer
28 views

If $f\in C^0(\overline{\Omega})$ is subharmonic and $-f$ is subharmonic, too, then $f\in C^2(\Omega)$ and $\Delta f=0$

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain and $f\in C^0(\overline{\Omega})$ be subharmonic, i.e. for each closed ball $\overline{B}\subseteq\Omega$ it holds: $$u\in C^2(B)\text{ is ...
0
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1answer
37 views

Prob. 2, Sec. 3.3 in Erwin Kreyszig's “Introductory Functional Analysis With Applications”: How to minimise the norm?

Let $z$ be a given complex number. Let $M \subset \mathbb{C}^n$ be given by $$M \colon= \left\{ (\xi_1, \ldots, \xi_n ) \in \mathbb{C}^n \mid \sum_{i=1}^n \xi_i = z \right\}.$$ Then $M$ is convex ...
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1answer
42 views

Covariant derivative (or connection) of and along a curve

Let $c:[a,b] \rightarrow M$ be a curve parametrized by arc-length. Then $c': [a,b] \rightarrow TM$ such that $c'(t) \in T_{c(t)}(M).$ So essentially, I want to understand how $\nabla_{c'}c'$ is ...
0
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1answer
37 views

If $f\in C^0(\mathbb{R}^n)$ is subharmonic and $\limsup_{|x|\to\infty}f(x)\le 0$, then $f$ must be non-positive in $\mathbb{R}^n$

Let $f\in C^0(\mathbb{R}^)$ be subharmonic, i.e. for each closed ball $\overline{B}\subseteq\mathbb{R}^n$ it holds: $$u\in C^2(B)\text{ is harmonic in }B\text{ and }f\le u\text{ on }\partial ...
2
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2answers
63 views

Prove a limit, going back to definition

I have to use only the definition of limits (ie I can't use algebra of limits) to prove the following: $$\lim_{x \to 2} x^2 = 4$$ I can't think of what to use as an arbitrary constant, or how to ...
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0answers
29 views

$\Delta u = 0 \Rightarrow \Delta (u\circ \phi)=0$?

Let $U,V \subset \mathbb{R}^n$ be open sets and $u \in C^2(U)$ satisfies $\Delta u =0.$ Let $\phi: V \rightarrow U$ be a surjective $C^2(V;U)$ function, then I am looking for sufficient conditions ...
0
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1answer
11 views

Express covariant transformation conveniently

Let $\omega = \sum_i \omega_i dx^i = \sum_{i} \nu_i dy^i$ be a 1-form in two different bases. Now, $(\omega_1,...,\omega_n)$ transform covariantly to $(\nu_1,...,\nu_n).$ My question is, can we ...
0
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1answer
36 views

Uniform continuity of $\sqrt{x^2+x}$

I have to say that I know the definition. I've tried to use is in practical way, but I still don't know how to do that and I don't truly understand that topic. Please show me step by step solution to ...
2
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1answer
19 views

Does $\log$ minimize this functional for its Abel equation?

Suppose that we have the functional equations ("Abel equation", it is called) for a function $F: [1, \infty) \rightarrow \mathbb{R}$ given by $$F(1) = 0$$ $$F(ex) = F(x) + 1$$ where $e$ is the ...
0
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1answer
23 views

Show that lower semicontinuous function is the supremum of an increasing sequence of continuous functions

Show that lower semicontinuous function $f:X\rightarrow [0,1]$ on metrizable X is the supremum of an increasing sequence of continuous functions. My attempt: I dont know how to approximate $f(x)$ to ...
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1answer
19 views

Strong convexity and strong smoothness duality

A function $f$ is said to be strongly convex with respect to a norm $\|\cdot\|$ at a point $y$ if $f(x) \geq f(y) + \nabla f(y)^T(x-y) + \frac{1}{2}\|x-y\|^2.$ It is said to be strongly smooth with ...
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0answers
25 views

Laplacian of composition

Let $U \subset \mathbb{R}^n$ be open and $u \in C^2(U)$ with $\Delta u(y)=0$ for all $y \in U.$ Let $\phi: V \rightarrow U$ be in $C^2(U)$, too with $V \subset \mathbb{R}^n$ open. Now, I want to ...
1
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1answer
28 views

Real Analysis: Determine $|J|$ where $J=J_1 \cup J_2$

Let $J_1= \{-2,-1,0,1,2\}$ and $J_2=\{-2,-\frac{2}{3}, \frac{2}{3}, 2\}$ be partitions of $[2,2]$. Let $J = J_1 \cup J_2$. Determine $|J| = (max_i (x_i - x_{i-1})$. Would the answer be $1$ as the ...
6
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1answer
50 views

the elements of Cantor's discontinuum

Let $(A_n)_{n \in \mathbb{N}}$ the sequence of subsets of $\mathbb{R}$, given by $A_0 := \bigcup_{k \in \mathbb{Z}}[2k, 2k + 1]$ und $A_n := \frac{1}{3}A_{n-1}$ for $n ≥ 1$. Also, we define $$ A := ...
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2answers
32 views

Periodic function

Considering the function $F(x)=x-E(x)$ such as $E(x)$ is the integer part of $x$ So here just with observation we can see that : $f(x+1)=(x+1)-E(x+1)=f(x)$. But here mathematics is not based just ...
2
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2answers
37 views

$\inf_{x\in[a,b]}f(x)=\inf_{x\in[a,b]\cap\mathbb{Q}}f(x)$ for a continuos function $f:[a,b]\to\mathbb{R}$

Let $f:[a,b]\to\mathbb{R}$ be continuos. I'm sure it's not hard, but I'm unsure what exactly we need to do to prove $$\inf_{x\in[a,b]}f(x)=\inf_{x\in[a,b]\cap\mathbb{Q}}f(x)$$
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0answers
19 views

Let $ f:R_{\ge 0} \rightarrow R_{\ge 0}$ for every $\lambda \in [0,1]$ and $x,y \ge 0$

Let $ f:R_{\ge 0} \rightarrow R_{\ge 0}$ for every $\lambda \in [0,1]$ and $x,y \ge 0$ such that $f(\lambda x +(1-\lambda) y) \ge \lambda f(x) +(1-\lambda)f(y)$.Know that $f(0)=0$ so that $f(x) > ...
1
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1answer
17 views

Proving an inequality involving the summation of Riemann zeta function.

If $0\lt a \le 1, s\gt 1,$ define $\zeta(s,a)=\sum (n+a)^{-s}$. Show that this series converges absolutely for $s \gt 1$ and prove that $$\sum_{h=1}^k \zeta(s,\frac{h}{k})=k^s\zeta (s)\ \ \ ...
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1answer
16 views

Proving convergence for series containing ln and factorial

I am trying to show whether or not the following series ${a_n}$ converges. Based on the hint, I have tried using Bertrand's test, but I am having a hard time simplifying the absolute value of the ...
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1answer
110 views

Translate this proof from German to English

I need your help to translate some exercises from German to English. I will attach like images. Thanks :) Satz 3. Es sei $(X,d)$ ein ultrametrischer Raum. $X$ ist genau dann transvollständig, wenn ...
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2answers
66 views

The necessity of the axiom of induction

$\underline{First\ question}$ Let $P(n)$ be a proposition about $n$. In standard mathematical induction, we require: (1)$P(0)$ holds. (2)If $P(n)$ holds, $P(n+1)$holds. Here we use "the axiom of ...
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1answer
40 views

What are all the indeterminate forms for which L'Hopitals rule is valid?

What are all the indeterminate forms for which L'Hopitals rule is valid? I know the basic ones are $\frac{\infty}{\infty}$ and $\frac{0}{0}$. Are there any other ones? Thanks,
2
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1answer
24 views

Minimizer of a quadratic form

Suppose I have a quadratic form of the form: $$q(x)=\frac{1}{2} x^T Q x$$ Now I want to find the minimum step length w.r.t the steepest descent. So I know the descent direction is $\nabla q(x)$. So ...
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0answers
23 views

convergence, contracting sequence

I need a little help on the following question: let $x(0)$ be an element of reals and define a sequence $\{x(n)\}$ by: $x(1) =f(x(0)), x(2)=f(x(1)),...., x(n+1)= f(x(n))...$ show that if $m>0, ...
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1answer
19 views

A continuously differentiable bijection implies its inverse is Lipschitz continuous

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuously differentiable bijection. Does this imply that $f^{-1}$ is Lipschitz continuous? (of course, not globally, take for instance $f(x)=x^3$) If ...
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87 views

Why are diffeomorphism-invariant PDE not elliptic?

In reading geometric analysis papers, I frequently encounter a statement of the form, "The PDE in question is diffeomorphism-invariant, and therefore cannot be elliptic." My vague understanding is ...
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41 views

On some counterexamples about continuity, intermediate value propriety, Riemann integrability, and antiderivatives

At school, I have been studying the relationship between continuity, monotonicity, and Riemann integrability. In doing so, I tried to make up some examples and counterexamples, but there are some ...
3
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2answers
55 views

Differential forms in Physics

In the context of physics, I just read about the the symplectic 2-form $\omega$ on a symplectic manifold $M$ of dimension $2n$. Unfortunately, I could not follow a few arguments. I.e. it was said ...
4
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3answers
72 views

Show that: $\int_{0}^{\infty} \frac{\sin{x^{q}}}{x^{q}} dx = \frac{\Gamma{\frac{1}{q}}}{q-1}\cos{\frac{\pi}{2q}} \mbox{, q > 1}$

How do you show that: $$ \int_{0}^{\infty} \frac{\sin{x^{q}}}{x^{q}} dx = \frac{\Gamma{\frac{1}{q}}}{q-1}\cos{\frac{\pi}{2q}} \mbox{, q > 1} $$ Without using Gamma function?
2
votes
2answers
27 views

problem of numeric sequence

Give an example of a sequence $(A_n)$ which is not convergent but the sequence $(B_n)$ defined by $\displaystyle B_n = \frac{A_1+A_2+\cdots+A_n}{n}$ is convergent.
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0answers
16 views

Difference of linear transformation of convex function

I'm trying to show that for constants $a,b > 0$, and a convex, continuously differentiable function $f$ with $f(0) = 0$ that $x_1 > x_2 > 0$ implies $f(-a-b x_1) - f(-b x_1) \geq f(-a-b x_2) ...
1
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1answer
63 views

Inequality with differential equations solutions

I would love some help working through this problem: Let $f_1,f_2,f : [0,\infty) \to \mathbb{R}$ be three bounded, continuous and absolutely Riemann integrable functions so that $|f_1(x)|, |f_2(x)| ...
2
votes
2answers
81 views

How to do this integral.

I need to do this: $$\int_0^\infty e^{ikt}\sin(pt)dt$$ I already have the solution, I'm just clueless on how to actually calculate it. I've tried several changes,integration by parts and extending ...
1
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0answers
17 views

Must the definition of the limit of a complex function be an inequality?

Looking at a few books, the definition of the limit of a complex function is of the form: DEFINITION. If $f :S\rightarrow \mathbb{C}$ is an arbitrary complex function and $z_0$ is a limit point of ...
0
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0answers
15 views

Show that if f is differentiable at $x_0$, then it is continuous at $x_0$. (Weierstrass-Caratheodory formulation)

this is an argument for a question which I am unsure whether it is sufficient or not. We are asked to try show the continuity at $x_0$ given that $f$ is differentiable at $x_0$. My argument goes as ...
0
votes
1answer
20 views

What is a Bi-Analytic function

I want to know what the definition of a Bi-analytic function is. I have tried looking it up online, but all I am able to find are research papers/books on the theory of bi-analytic functions. Can ...