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)

0
votes
1answer
123 views

Prove that the equation has no solution

Prove that there exist infinitely many positive real numbers r such that the equation 2$^x$ +3$^y$ + 5$^z$ = r has no solution (x, y, z) $\subseteq$ Q × Q × Q. First, I prove that set S = {2$^x$ +3$^...
5
votes
2answers
203 views

Showing that the norm of the canonical projection $X\to X/M$ is $1$

How do I show that given $M$ a closed subspace of a normed space $X$, and let $\pi$ be the canonical projection of X onto $X/M$. Prove that $\|\pi\| = 1$. I figure I could use Riesz' lemma and set $\|...
2
votes
3answers
109 views

Regarding limits

If $f$ is positive and differentiable in $(0,\infty)$, then I want to find the following limit. $\lim\limits_{n\to \infty}\left(\dfrac{f\left(x+\dfrac{1}{n}\right)}{f(x)}\right)^n$. I have done as ...
0
votes
1answer
46 views

Left and right continuity

I was wondering if someone can help me write down (or perhaps just check my answer) the proof for the following theorem formally. I feel that what I wrote down is too easy to be correct ... Suppose ...
1
vote
2answers
105 views

Logical Quantifiers

I am wondering if there is a reference or book that clearly translates all English forms of logical quantifiers to mathematical quantifiers. For example, when we say for any element $ x \in S$, is ...
0
votes
1answer
33 views

Why $\displaystyle\rho_{\alpha, \beta}(f)=\sup_{x\in\mathbb R^n}|x^\beta \partial^\alpha f(x)|$ is not a norm on $\mathcal{S}(\mathbb R^n)$?

The Schwartz space $\mathcal{S}(\mathbb R^n)$ is the set of all function $f:\mathbb R^n\longrightarrow \mathbb C$ such that $\displaystyle \sup_{x\in \mathbb R^n}|x^\beta \partial^\alpha f(x)|<\...
2
votes
2answers
63 views

$\lim \sup$ of a sequence

Let $\{A_n\}$ be a sequence and $\frac{1}{R} = \lim \sup A_n$. Let $\alpha < R$. My question: Why is there $n_0\in \Bbb N $ such that $$A_n < \frac{1}{\alpha}\text{ for any } n\geq n_0$$ Thanks ...
1
vote
3answers
133 views

Use L'Hopital's Rule to Prove

Let $$f: \mathbb R\rightarrow \mathbb R$$ be differentiable, let a in $\mathbb R$. Suppose that $f''(a)$ exists. Prove that $$\lim_{h\rightarrow0}\frac{f(a+h)-2f(a)+f(a-h)}{h^2}=f''(a) $$ Suppose ...
1
vote
1answer
26 views

If T is a top-linear map what does $(T^\ast T)^{1/2}$ mean?

If $T: H_1 \rightarrow H_2$ is a continuous linear map between two Hilbert spaces, $H_1$ and $H_2$, what does the notation $(T^\ast T)^{1/2}$ mean? The book I'm reading defines $|T|$ to mean $(T^\ast ...
1
vote
2answers
45 views

Showing $ u =0 $ for a particular laplace equation.

Given open $\Omega\subset \mathbb{R}^n $ with smooth enough boundary, if for $x_0 \in \Omega$ there exists $ u \in C^2(\Omega\setminus\{x_0\}) $( we may as well take $u$ as smooth as we want) that ...
1
vote
1answer
245 views

If an entire function grows slower than a polynomial, then it is a polynomial!

I was investigating the following corollary to Liouville's Theorem in Complex Analysis: if $f(z)$ is entire and $\lim_{z\rightarrow \infty}z^{-n}f(z)=0$, then $f(z)$ is a polynomial in $z$ of degree ...
46
votes
9answers
2k views

Is $dx\,dy$ really a multiplication of $dx$ and $dy$?

On the answers of the question Is $\frac{dy}{dx}$ not a ratio? it was told that $\frac{dy}{dx}$ cannot be seen as a quotient, even though it looks like a fraction. My question is: does $dxdy$ in the ...
0
votes
1answer
63 views

Equivalence between dual estimates

Let $H$ a Hilbert space and $X$ a measure space; let $U(t):H\to L^2(X)$ an operator defined for all real $t$. I'm considering the following estimates: $$\Vert U(t)f\Vert_{L_t^qL_x^r}\leq\Vert f\Vert_H$...
2
votes
0answers
32 views

Can we expect, $\left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$ in the Banach algebra $A(\mathbb R)$?

Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$ and consider Fourier algebra $$A(\mathbb R):= \{f\in L^{1}(...
1
vote
0answers
29 views

Question in the Continuity of a function

I have this function: $$(J''(u)v,w)=(v,w)-(KN_{f'}(Ku)Kv,w)$$ for all $u,v,x\in L^2[0,1]$ such that $f\in C^1([0,1]\times\mathbb{R},\mathbb{R})$ and $Ku(t)=\int_0^1 G(t,s) u(s)ds$ $K$ is symetric, ...
0
votes
1answer
32 views

Examples of function the following three conditions

Let function f:(0,∞)→(1,∞) satisfying the following conditions: (i) f is nondecreasing; (ii) for each sequence $({x_n})⊂(0,∞),lim_{n→∞}f(x_{n})=1$ if and only if $ lim_{n→∞}x_{n}=0⁺$; (iii) there ...
1
vote
2answers
34 views

Finite vs infinite distinction in Rudin's Analysis

I'm starting to self-study Rudin's Principles of Mathematical Analysis. I'm up to the second chapter, theorem 2.24. For any collection $\{G_i\}$ of open sets, $\bigcup_i^nG_i$ is open. For ...
0
votes
1answer
37 views

asymptotic analysis

For each of the following sentences involving functions f and/or g, find a counterexample to show that it is false: What is meant by counterexamples?
1
vote
1answer
127 views

Stacked with this Problem of Calculus

I have been struggling for quite some time with the following problem and I would really appreciate some help: Consider $f(d)=\frac{(1-d)\left(1-d^{(\frac{t}{2}-1)}\right)}{(1-d)\left(1-d^{(\frac{t}{2}...
4
votes
0answers
195 views

Integral $=\int_0^\infty x^{\alpha -1}Li_n (-\sigma x) Li_m(-\omega x^r)dx$.

I am trying to calculate an integral that can be expressed in terms of infinite hypergeometric series by using transforms and Residue method, the integral is $$ I_{n,m}(\alpha,\sigma,\omega,r)=\int_0^...
1
vote
1answer
27 views

Applying theorems on differential functions

I'm thinking that I need to do a proof by counter example? Is it possible to use Rolle's theorem: f is cts on [a,b] and differentiable on (a,b) and f(a)=f(b), so there exists a c in (a,b) such that f'...
1
vote
1answer
98 views

Serie of functions : interchange of limit of series

Let $\{f_n\}_{n=1}^\infty$ be a sequence of real-valued functions on $\mathbb{R}$. Show that if $f_n$ is continuous for all $n \in \mathbb{N}$ and the series $\sum_{n=1}^\infty f_n$ converges ...
0
votes
2answers
67 views

Complex plane and $\mathbb{R}^2$. [duplicate]

What differences -if any- are there between the complex numbers $\mathbb{C}$ and $\mathbb{R}^2$? I am taking multi variable analysis now and I was wondering what possible changes there might be from ...
0
votes
1answer
28 views

What is $|x|$ for $x\in \mathbb T^n$?

The $n$-dimensional torus is $\mathbb T^n=\mathbb R^n/\mathbb Z^n$. Let $|x|$ be the Euclidian norm. What is $|x|$ for $x\in \mathbb T^n$?
5
votes
1answer
183 views

An analysis problem about convergence

Suppose that $f$ is a continuous function from $[a,b]$ to $[a,b]$. Let $x_0\in [a,b]$, and define by induction that $x_{n+1}=f(x_n)$. Show that $$\lim_{n \rightarrow \infty} (x_{n+1}-x_n)=0$$ implies $...
2
votes
0answers
42 views

Applications of Vito Volterra's theorem

We know from Volterra's theorem that: There cannot exist two pointwise discontinuous functions on an interval $(a,b)$ for which the continuity points of one, are the discontinuity points of the other,...
0
votes
2answers
48 views

$\lim_{ x\to 0^+}f(x)=?$

Suppose $f(x)$ is bounded on $[0,1]$, and for all $0\le x\le\frac{1}{a}$ satisfis $f(ax)=bf(x)$. ($a,b>1$) $\lim_{ x\to 0^+}f(x)=?$
1
vote
1answer
41 views

Solution to a general scaling problem $G(\lambda z)=\frac{G(z)}{\gamma z^n}$

When playing with the scaling problem $$G(4z)=\frac{G(z)}{2z}$$ (see also this question) I discovered, that the general problem $$G(\lambda z)=\frac{G(z)}{\gamma z}$$ with two constants $\lambda,\...
0
votes
1answer
240 views

Proving the convergence and divergence of the p-series

We know from calculus that $\sum_{n=1}^{\infty} \frac{1}{n^p}$ diverges if $p \in [0,1)$ and converges if $p > 1$. I want to use analysis to prove these two statements. For the case where $p > 1$...
5
votes
1answer
315 views

Convolution of a probability measure with a smooth function

If $f\in L^1(\mathbb{R}^n)$ and $g\in L^p(\mathbb{R}^n)$ then by Young's convolution inequality we have the estimate: $$ \|f*g\|_{L^p}\leq \|f\|_{L^1}\|g\|_{L^p}.$$ Question: Let $\mu$ be a ...
0
votes
2answers
30 views

Evaluating Data

I have a set of data that I need to evaluate. The data came from 15 people we asked to evaluate several project proposals using 6 different criteria. For each weighted criteria, they selected 1 of 4 ...
0
votes
1answer
69 views

find an odd differentiable function $f: \mathbb{R} \to \mathbb{R}$ s.t. $f'(x) = e^{-x^2}$

find an odd differentiable function $f: \mathbb{R} \to \mathbb{R}$ s.t. $f'(x) = e^{-x^2}$ This is for my analysis course and we are studying power series; my attempt: $e^x = \displaystyle \sum_{k=0}...
2
votes
3answers
334 views

Continuity of Fixed Point

For all $a \in \mathbb{R}$, let $f_a: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous and contractive, that is, there exists $\epsilon \in (0,1)$ such that $\left\| f_a(x)-f_a(y) \right\| \leq (1-...
2
votes
0answers
89 views

If $f(z) = \sum a_n z^n$, what is $\sum n^3 a_n z^n$?

Problem: If $f(z) = \sum a_n z^n$, what is $\sum n^3 a_n z^n$? Attempt: First we note that $$ f'(z) = \sum_{n=1}^\infty n a_n z^{(n-1)} $$ so that $$ z f'(z) = \sum_{n=1}^\infty n a_n z^n $$ ...
0
votes
1answer
41 views

Showing $\lim\frac{\left| a_{n+1} z^{n+1} \right|}{\left| a_n z^n \right|} = |z| \lim \frac{ a_{n+1} }{ a_n }$

Let $\sum a_n z^n$ be a complex power series. I've seen it asserted without explanation in a text that $$ \lim\frac{\left| a_{n+1} z^{n+1} \right|}{\left| a_n z^n \right|} = |z| \lim \frac{ a_{n+1} ...
0
votes
1answer
79 views

Showing the radius of convergence of $\sum a_n b_n z^n$ is at least $R_1 R_2$

Problem: If $\sum a_n z^n$ and $\sum b_n z^n$ have radii of convergence $R_1$ and $R_2$, show that the radius of convergence of $\sum a_n b_n z^n$ is at least $R_1 R_2$. Is the following proof ...
2
votes
2answers
229 views

Relationship Between Ratio Test and Power Series Radius of Convergence

Let $ \{a_k\} $ be a sequence of positive real numbers. Why does it hold that $$ \liminf \frac{a_{k+1}}{a_{k}} \leq \liminf (a_k)^{\frac{1}{k}}\leq\limsup (a_k)^{\frac{1}{k}} \leq \limsup \frac{a_{...
5
votes
0answers
156 views

Gradient on Sobolev spaces

Suppose I have a function $\Phi \in L_p(\Omega)$ and $\nabla\Phi \in W^{-\alpha}_p(\Omega)$, $1<p<\infty$ and where $\Omega\subset\mathbb{R}^n$ is a bounded domain. Can I conclude that $\Phi\in ...
1
vote
1answer
561 views

When can we use Fubini's Theorem?

I am using Munkres' Analysis on Manifolds textbook. Munkres defines Fubini's Theorem on rectangles and on simple regions (at least till the point that I have read). Now, according to the book, we ...
5
votes
3answers
255 views

Prove that the logarithmic mean is less than the power mean.

Prove that the logarithmic mean is less than the power mean. $$L(a,b)=\frac{a-b}{\ln(a)-\ln(b)} < M_p(a,b) = \left(\frac{a^p+b^p}{2}\right)^{\frac{1}{p}}$$ such that $$p\geq \frac{1}{3}$$ That is ...
0
votes
1answer
118 views

Convergence of a series if limit goes to 0

I am stuck on the following question. We are asked to prove the following: Assume $\lim \limits_{n \to \infty}$ $a_n$ = L 1) prove that if L>1, then $\sum_{n=1}^{\infty}$ ${1\over n^{a_n}}$ ...
0
votes
2answers
71 views

Integrals on unlimited sets

How do you evaluate this expression $$ \left| \int_{1}^{\infty} 1 \; dx - \int_{1}^{\infty} 1 \; dx \right| \quad ? $$ Using improper integral definition, this should be an indeterminate $\infty - \...
3
votes
1answer
50 views

Integral with absolut-value function

How do I seperate the following integral? The integral of $|x^2-y|$ with $|y| \leq 1$ and $|x| \leq 1$. I know that the absolute value is positive for $x^2 \geq 1$ and negative for $x^2$ but I am ...
0
votes
1answer
41 views

Showing that $\underset{n \rightarrow \infty}{\lim} |a_n| / |a_{n+1}| = R$ implies that the radius of convergence of $\sum a_n z^n$ is also $R$

Hypothesis: Suppose that $\underset{n \rightarrow \infty}{\lim} |a_n| / |a_{n+1}| = R$. Goal: Show that $\sum a_n z^n$ has radius of convergence $R$. Attempt: The radius of convergence of $\sum ...
1
vote
2answers
502 views

Finding Radii of Convergence for $\sum a_n z^{2n}$ and $\sum a_n^2 z^n$

Setting: Let $\sum a_n z^n$ have radius of convergence $R$. We have that $$ R = \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n \right|^{1/n}} $$ via Hadamard's formula for the radius ...
1
vote
3answers
466 views

Using Hadamard's Formula to show that the radius of convergence of $\sum z^{n!}$ is $1$

Background: Recall that Hadarmard's formula for the radius of convergence of a complex power series $\sum a_n z^n$ is as follows: $$ R = \frac{1}{\underset{n \rightarrow \infty}{limsup} \left| a_n \...
2
votes
1answer
164 views

Finding the radius of convergence for $\sum n^p z^n$ (Proof Verification)

Goal: Find the radius of convergence for the following complex power series: $$ \sum n^p z^n $$ Attempt: We have by Hadamard's formula for the radius of convergence that the complex power series $\...
0
votes
1answer
49 views

Why this set $M$ isn't compact in $X$?

I want to prove this set $M=\{U\in X, \|U\|≤1\}$ is compact in $X=C[a,b]$ I must show with sequence & use piecewise linear continuous function such that $U_n(x)\to0$ as $n\to$ where this ...
1
vote
1answer
111 views

Characterisation of the limit superior and limit inferior

in my notes (University 1st year Analysis) is the following proposition : ! with the proof ! I don't understand what it means for the set to be finite/infinite and I am therefore a little hazy ...
0
votes
1answer
42 views

when one can expect $\left \| |x|^{2}x- |y|^{2}y \right \| \leq \frac{1}{2} \left \| (x-y) \right \|$?

Suppose $(X, \left\|\cdot \right \|)$ is Banach algebra with the property that $\left\| (|x|^{2}x) \right\| \leq \left\|x\right \|^{3};$ for every $x\in X.$ Let $y_{0}\neq 0 \in X$ and fix it; and ...