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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0
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1answer
54 views

Is $S$ a regular submanifold of $\Bbb R^{3}$?

$$S=\{(x,y,z) \mid x^{2}+y^{2}=z^{2}\}$$ $g: \Bbb R^{3}\to \Bbb R$, $S=g^{-1}(0)$ Is $S$ a regular submanifold of $\Bbb R^{3}$? I'd be grateful for a clear and explicit explanation of why this is ...
3
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2answers
60 views

Inequality between norms in $\mathbb{R}^n$

I am trying to prove that given $p>1$ there exists a constant $C=C(p,n)$ such that $\big||x|^px-|y|^py\big|\leq C\big(|x|^p+|y|^p\big)|x-y|$ for all $x,y\in\mathbb{R}^n$. It seems useful to ...
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5answers
1k views

Use Cauchy product to find a power series represenitation of $1 \over {(1-x)^3}$

Use Cauchy product to find a power series represenitation of $$1 \over {(1-x)^3}$$ which is valid in the interval $(-1,1)$. Is it right to use the product of $1 \over {1-x}$ and $1 \over ...
2
votes
1answer
62 views

Separating points

Assume $A$ is a closed subalgebra of $C(S)$, the space of continuous complex functions on compact Hausdorff space $S$. Assume $A$ separates point on $S$ and and if $f \in A$ then $\bar{f}\in A$. ...
2
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1answer
109 views

Finding the max. of an integral

I have a question which asks: Let $g\in C[-1,1]$ and the usual inner product $\langle f,g\rangle = \int_{-1}^{1} f(x)g(x)dx$. Find the max value of $\int_{-1}^{1}g(x)x^3dx$ where $g$ is subject to ...
1
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1answer
73 views

Sign of a function

Suppose $\gamma$ is a real number with $|\gamma|\ll1$. The function $$ \theta(s)=s-\frac{\sin \left( \sqrt{1+\gamma} \, k \, \pi \, s \right)}{\sin \left( \sqrt{1+\gamma}\, k \, \pi \right)}, \qquad k ...
0
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1answer
469 views

Uniform sum distribution

I was wondering how to derive the probability density function for the sum of $n$ independent iid distributed random variables on the interval $[0,1]$. A formula for that is given on ...
2
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2answers
150 views

How does degree theory imply that this mapping $f$ is locally onto?

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a smooth vector field ($\mathcal{C}^1$ mapping). Let $0$ be a critical point of $f$, i.e. $H f(0) = 0$. Assume that the index of $f$ at $0$ is ...
2
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1answer
49 views

Showing a function is differentiable using definition of derivative

Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be defined by $f(x,y)=(x^2-xy, x+y^2)$. Use the definition of the derivative of a function to show that $f$ is differentiable at the point $p=(1,-1)$. My ...
0
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1answer
30 views

$f \in S \rightarrow \hat f \in S$

Let $S$ be the Schwartz space and $ \hat f$ be the Fourier transform of $f$. I hope to prove that $f \in S \rightarrow \hat f \in S$. I know some properties about Fourier transfrom but I do not know ...
2
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1answer
49 views

Bounded-ness of range

We are give a sequence of the form $x_n = 1/n$ in the complex metric space. This sequence of course has a limit at 0, the range is clearly infinite, however, it said that the sequence is bounded. ...
1
vote
1answer
387 views

Inverse fourier transform 3 dimensions

Hoi, I want to show that for $n=3$ that $$\mathcal{F}^{-1}\left(\frac{1}{1+|s|^2}\right) = \frac{1}{4\pi |x|}e^{-|x|} $$ As a hint I've been given: Its the unique solution to the equation ...
2
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0answers
111 views

Completeness proof.

I'm getting stuck showing a space is a Hilbert space. For $\Omega$ an open, connected and bounded set in $\Bbb R^2$ with regular boundary $\partial \Omega$, let $V=\{v \in H^1(\Omega)\ ;\ ...
1
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0answers
39 views

period of motion on the phase curve

I am interested in the following question. (It is a rephrased problem in Arnold's book "Mathematics methods of classical mechanics" (2nd ed. page 20)). Given are potential function $U(x)$ such that ...
3
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1answer
179 views

Cauchy's functional equation - a generalisation? (do additive maps have to be continuous?)

If a map $f : \mathbb{R} \to \mathbb{R}$ is additive, in the sense that $f(x + y) = f(x) + f(y)$, then it is simple to show that $f$ is $\mathbb{Q}$-linear, buy it does not need to be ...
3
votes
1answer
200 views

Intuition behind compact subspaces of a metric space

I've read up on compactness in a metric space and have found a few definitions (let $X$ be a metric space and $E \subset X$ in all the following): $E$ is compact in $X$ if for every open covering of ...
5
votes
3answers
965 views

Proving Newton's Binomial Theorem

So, I've done most of the problem to this point, but just cannot figure out the last piece. I may just be missing the math skills needed to complete the proof (differential equations). Problem (from ...
2
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2answers
136 views

Hahn-Banach Separation Theorem and Bishop's Theorem

I am looking at the proof of Bishop's Theorem on pages 122 and 123 of Rudin's Functional Analysis. The following quote is from the the last two sentences of the proof on pg. 123. "Every continuous ...
0
votes
1answer
79 views

An alternative definition for integral of a nonnegative measurable function in terms of infimum

How could I show "integral of a nonnegative measurable function f could be defined as the infimum of a set of integrals of simple functions g with f<=g for all g". We could assume f is bounded by ...
1
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2answers
50 views

On the limits, prove that and find its limit

Can any one tell me how to prove that: The sequence $x_{1}=\sqrt{2}$ , $x_{2}=\sqrt{2+\sqrt{2}}$, $\cdots$, $x_{n}=\underbrace{\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}_{n \text{ times}}$ converges and ...
2
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1answer
234 views

Differentiable but not Absolutely continuous

Please give an example (if it exists) for a function which is differentiable everywhere but not absolutely continuous.
2
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0answers
71 views

Construction of Monotone function which is differentiable on the given set

Given a set $A \subset \mathbb{R}$ of measure 0, is it possible to construct a monotone function whose set of non differentiable points is $A$ ?
1
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1answer
89 views

Multiplication of distributions by smooth functions

Let $u\in D'(\mathbb{R})$ and $f\in C^{\infty}$. I'm trying to figure which of the following statements is true: I. If $f\restriction_{supp(u)}=1$ then $f\cdot u=u$. II. If ...
0
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1answer
39 views

Removable discontinuity for $C^1$ functions on $\mathbb{R}^2$ with uniformly bounded partials.

I'm stuck on the following question: Say I have a function $f\in C^1(\mathbb{R}^2\setminus \{0\})$ with uniformly bounded partials, why must $f$ admit a continuous extension to $\mathbb{R}^2$? My ...
3
votes
2answers
145 views

Proving Bishop's Theorem using Krein-Milman Theorem

I am studying the proof of Bishop's theorem (generalization of Stone-Weierstrass) in Rudin's Functional Analysis 2nd edition. He make the following statement on the bottom of page 122, "Since $\mu ...
10
votes
2answers
347 views

Series $\sum\limits_{n=1}^\infty \frac{1}{\cosh(\pi n)}= \frac{1}{2} \left(\frac{\sqrt{\pi}}{\Gamma^2 \left( \frac{3}{4}\right)}-1\right)$

I was playing around with Mathematica and found that $$\sum_{n=1}^\infty\frac1{\cosh(\pi n)} = \frac12\left(\frac{\sqrt{\pi}}{\Gamma \left(\tfrac34\right)^2}-1\right)$$ Does anybody know how to ...
4
votes
1answer
724 views

Unit ball of a Separable Banach Spaces is metrizable

Please help me to understand why the unit ball of a separable banach space is metrizable, when it is given the induced topology from the weak topology. Specifically, my problem is I think I'd be able ...
2
votes
1answer
138 views

Equivalence of measures and $L^1$ functions

Suppose we have two probability measures $\mu$ and $\delta$ on $(X, \mathcal{B})$ such that $ \delta <<\mu << \delta $. How can I prove that $f \in L^1(X,\mathcal{B}, \mu)$ iff $f \in ...
2
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1answer
78 views

Prove that a set is a Linear subspace

While reading "Principles and Techniques of Applied Mathematics" from Bernard Friedman I stumbled on an exercise that I don't know how to properly solve... The question is as follows: Prove that ...
1
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4answers
215 views

Convergence of the infinite series $ \sum_{n = 1}^\infty \frac{1} {n^2 - x^2}$

How can I prove that for every $ x \notin \mathbb Z$ the series $$ \sum_{n = 1}^\infty \frac{1} {n^2 - x^2}$$ converges uniformly in a neighborhood of $ x $?
2
votes
3answers
79 views

Existence and value of $\lim_{n\to\infty} (\ln\frac{x}{n}+\sum_{k=1}^n \frac{1}{k+x})$ for $x>0$

Does the limit $$W(x)=\lim_{n\to\infty} \left(\ln\frac{x}{n}+\sum_{k=1}^n \frac{1}{k+x} \right)$$ exist for all $x>0$? If so, what is the limit $$\lim_{x\to\infty}W(x)?$$
4
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3answers
113 views

Showing that $\mathbb{Q}$ is not complete

Show that there is no least upper bound for $A=\{x: x^2<2\}$ in $\mathbb{Q}$. Suppose $\alpha \in \mathbb{Q}$ is the least upper bound of $A$. Then either $\alpha^2 < 2$ or $\alpha^2 > 2$. ...
1
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1answer
81 views

Paths on $\mathbb{Z}^d$

Let's say a path must be non-self-intersecting, and that we have the usual lattice structure. Then if $\sigma(n)$ is the number of paths of length $n$ then why do we have convergence of the sequence ...
1
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2answers
27 views

How I can calculate $g^{(1)}$?

Let $f$ be an analytic function defined over all complex plane. Now, consider the function $g:ℝ^{r+1}→ℝ$ defined by $$g(t₁,t₂,...,t_{r+1})=f^{(r+1)}(1-2∏_{j=1}^{r+1}t_{j})$$ where $f^{(r+1)}$ is the ...
1
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3answers
2k views

Definition of “not converging” and proving $(-1)^n$ does not converge to $1$.

Remember that a sequence $x_n, n = 1,2,3\cdots$ is said to converge to $x$ as $n → ∞$ if for all $ε > 0$ there exists an $N ∈ \mathbb{N}$ such that $|x_n − x| < ε$ for all $n ≥ N$. (a) Complete ...
0
votes
1answer
36 views

Terminology of functions in $L_2$

I am reading a text that states ... any function in $L_2(0,\pi)$ has a Fourier sine series that converges to it in $L_2(0,\pi)$ ... Unfortunately no definition of $L_2$ is given. What does ...
0
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4answers
121 views

Square and square root and negative numbers [duplicate]

Are they equal? -5 = $\sqrt{(-5)^2}$
2
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1answer
90 views

Isn't $f_n(x)=\frac{{n^2}\ln x}{x^n}$ with $x\geq1$ uniformly convergent by $T$-test?

One Dr. showed to me that the function $f_n(x)=\dfrac{{n^2}\ln x}{x^n}$, $x\geq1$ is not uniformly convergent by $T$-test, but I showed it to converge to $0$ anyway. $$\lim_{x\to \infty}T_n=\lim_{x\to ...
4
votes
2answers
175 views

If $x\mapsto \| x\|^2$ is uniformly continuous on $E$, the union of all open balls of radius $r$ contained in $E$ is bounded $\forall r > 0$

A subset $E$ contained in $\mathbb{R}^n$ is such that the function $x \mapsto \left\Vert x\right\Vert^2$ is uniformly continuous on $E$. For $r > 0$, let $E_r$ denote the union of all open balls ...
3
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2answers
362 views

A sequel for Elementary Analysis by Ross?

I've been learning real analysis from this book: Elementary Analysis, K.A. Ross I really liked the style of this book. It is quite old, and sometimes very difficult, but I guess I liked the way it ...
2
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2answers
1k views

Bounded partial derivatives imply continuity

As stated in my notes: Remark: Suppose $f: E \to \mathbb{R}$, $E \subseteq \mathbb{R}^n$, and $p \in E$. Also, suppose that $D_if$ exists in some neighborhood of $p$, say, $N(p, h)$ where ...
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2answers
54 views

What does it mean for $f$ to be continuous at $a$? [closed]

Let $a \neq 0$. Prove that $\displaystyle{f(x)=\frac{1}{x^2}}$ is continuous at $x=a$.
3
votes
1answer
281 views

A theorem about Lipschitz regularity and Fourier transform

How to prove that: A function $f$ is uniformly Lipschitz $\alpha$ over $\mathbb R$ if $$\int_{-\infty}^{+\infty}|\hat f(\omega)|(1+|\omega|^\alpha)d\omega<+\infty$$ A function $f$ is uniformly ...
21
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1answer
542 views

Can the chain rule be relaxed to allow one of the functions to not be defined on an open set?

I've written the question first, then the motivation behind it and lastly some background. Note that the question makes references to definitions and theorems written in the background bit at the end. ...
0
votes
1answer
99 views

Analysis of Algorithms Observation

a.Use induction to prove that $T(n) = \theta(n^k \lg n)$ b.The preceding fact shows that we could, in principle, extend Case $2$ of the Master Theorem to include more overhead functions than simply ...
1
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1answer
108 views

Why does the Gamma function interpolate $(n-1)!$?

Why does the Gamma function interpolate $(n-1)!$ and not $n!$ instead? What is the historical reason?
1
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1answer
116 views

Calculation of the Laplacian of a function in $\mathbb{R}^3$.

I have to calculate the Laplacian distributional sense) of the following function $$\frac{e^{i\sqrt{\lambda}|x|}-1}{4\pi|x|}$$ with $\lambda>0$, $x\in\mathbb{R}^3$. I've procedded in the following ...
1
vote
1answer
70 views

Show that this equation is true.

Consider the following function in $\mathbb{R}^n (n\geq 3)$: \begin{equation} H(y)=2b_n\int_{0}^{\infty}e^{\ as}D_n\Phi(y-\tilde{x}+bs)\text{ d} s,\quad (x, y\in\mathbb{R}_{+}^{n}, x\neq y), ...
1
vote
2answers
337 views

Convergence rate of a series

What is convergence rate of a series $ \sum_{k=1}^{n} k^\alpha \\ $ where $\alpha< -1$ ? For example, for $\alpha=-1$ it equals to $O(\log n)$.
1
vote
2answers
144 views

Convergence of $\sum\limits^\infty _{k=0} a_k \sin(kx)+b_k \cos(kx)$

Ok, for the infinite series: $$\sum^\infty _{k=0} a_k \sin(kx)+b_k \cos(kx)$$ How do I show that this converges on any finite interval if $\sum^\infty _{k=0} k(|a_k|+|b_k|)<\infty$? Also, do the ...