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)

1
vote
1answer
20 views

$\{f\in L^{1} : \hat{f} \in L^{p} \}$ closed under convolution?

Let $f, g\in L^{1}(\mathbb R),$ we may define the convolution of $f$ and $g$ as follows: $f\ast g(x)= \int_{\mathbb R} f(x-y)g(y) dy, (x\in \mathbb R).$ We note that $L^{1}(\mathbb R) \ast ...
5
votes
1answer
59 views

Fourier transform of the Heaviside function

As you can see from the title I want to calculate the Fourier transform of the Heaviside function $u(t)$. Proven the the Heaviside function is a tempered distribution I must evaluate: $$ \langle ...
2
votes
0answers
73 views

Sign fluctuation in the harmonic series (sum)

Lets begin with a few simple rules that we know. -$\sum_{k=1}^\infty \frac{1}{n}=\infty$ -$\sum_{k=1}^\infty \frac{\color{red}{(}-1\color{red}{)}^{n-1}}{n}= \ln2 $ Going off of this knowledge, we ...
2
votes
2answers
152 views

Prove that if $B = \{x-y : x,y \in A\}$, where $A$ is a Borel measurable subset of $R$ with positive measure

Suppose that $m$ is Lebesgue measure, and $A$ is a Borel measurable subset of $R$ with $m(A) > 0$. Prove that if $B = \{x - y : x,y \in A\}$, then $B$ contains a non-empty open interval centered at ...
0
votes
1answer
33 views

Gauss divergence theorem, $div(F) = 0$?

I'm trying to solve the following problem using the Gauss divergence theorem. I have to calculate the Flux of $$ f(x,y,z) = (\sin(yz),y+\sqrt{x^2 + z^2}, 1-z) $$ through the surface $$ \Omega = ...
0
votes
2answers
63 views

If $d_1(x,y)$ and $d_2(x,y)$ are metrics, prove that $d'(x,y)= \sqrt{d_1^2(x,y)+d_2^2(x,y)}$ is a metric.

$$d'(x,y)= \sqrt{d_1^2(x,y)+d_2^2(x,y)}$$ The first three properties are trivially proven. The triangle inequality, not so much. I tried using the triangle inequalities that apply to $d_1$ and $d_2$, ...
0
votes
1answer
30 views

Define a metric using scalar product and prove that it is indeed a metric

So this is how I went about this: $\langle\,\cdot\,,\,\cdot\,\rangle: X \times X \to \mathbb{R}$ such that (by definition I list the properties of scalar product) and I can east prove the first three ...
0
votes
4answers
67 views

Secondary solving method of polynomial

$$x+1+\frac{1}{x}=0$$ This is a fairly trivial and possibly bland equation to solve. But for the sake of the question I will display them here: $$x\left(x+1+\frac{1}{x}\right)=x(0)$$ $$x^2+x+1=0$$ ...
3
votes
0answers
59 views

Understanding tensors

Locally in a chart, a tensor field looks like $$T= T^{i_1,...i_n}_{j_1,...,j_m} dx^{j_1} \otimes...\otimes dx^{j_m} \otimes \partial_{i_1} \otimes ... \otimes \partial_{i_n},$$ where ...
-5
votes
1answer
59 views

Why is this statement true? [closed]

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be the function $f(x,y)=(y-x^2)(y-2x^2).$ Why is this statement true: $t\mapsto f(t\xi)$ has in $t=0$ a local minimum for every $\xi\in\mathbb{R}^n$
0
votes
1answer
30 views

Heat equation with fourier transformation

I want to understand a solution from an exercise where we should find a solution of the heat equation: $$\frac{\partial u(x,t)}{\partial t}=\sum_{j=1}^{n}\frac{\partial^2 u(x,t)}{\partial x_j^2} $$ ...
0
votes
0answers
13 views

Primitive-ability of a function

Prove that the function $f:{R}\to{R}$ is primitive-able(does this term exist in English?) and find one of its primitives. $$f(x)=\left\{\begin{array}{cc} 1-x & x<1 \\ x^2-2x+1 & x \geq 1 ...
-4
votes
1answer
42 views

Is every continuous 1-1 function onto? [closed]

Is it true that every continuous 1-1 function on the interval [0,1] to [0,1] onto?
0
votes
0answers
18 views

Convergence of Laplace transform and its inverse

There is a sequence of functions $F^{\epsilon}(\lambda)$ which converges to 0 as $\epsilon \rightarrow 0$. Assume that each $F^{\epsilon}(\lambda)$ has a inverse Laplace transform f(s) such that ...
1
vote
1answer
53 views

$f$ is continuous on $E$ if and only if its graph is compact.

This question may be asked before under different formulation, the original problem is Chapter 4, Exercise 7 of Rudin's text: The Principles of Mathematical Analysis: Problem: If $f$ is defined on ...
0
votes
2answers
41 views

Proving Lower bounds on an Approximately Linear Function

We are looking for a lower bound on the function, $\frac{1.31}{e^{\frac{1.31}{x+1}} - 1}$ for $x \geq 2$. This function seems to behave linearly. We believe that the following statement holds: ...
1
vote
1answer
38 views

Closed form defines locally a function?

In my particular example, I have a volume form on a one-dimensional manifold, so i.e. this volume form $dx$ is closed. Now, I was wondering, does the integral function $f(y):=\int_{x_0}^{y} dx$ define ...
3
votes
1answer
45 views

Covariant and partial derivative commute?

I know that we have for a function $\Gamma: (-\varepsilon,\varepsilon)^2 \rightarrow M$ we have (at least I think I know that this is true) $$\nabla_{\frac{\partial \Gamma}{\partial s}} ...
1
vote
0answers
40 views

Jacobi fields and variations

I want to show that every Jacobi field is a variation of geodesics, i.e. let $Y: I \rightarrow TM$ be a Jacobi field along a geodesic $\gamma$, then I want to show that $Y$ can be written as a ...
4
votes
2answers
58 views

Every neighborhood $N_r(x)$ in $\mathbb{R}^n$ is connected

I am working on an exercise in baby Rudin (Ex 2.20 in particular) and as part of that I am trying to show that any neighborhood in a metric space is connected. I've seen several differing definitions ...
2
votes
2answers
25 views

If I have let's say a finite number of points in a metric space, and lets say that space has a couple of different metrics.

If I have let's say a finite number of points in a metric space, and lets say that space has a couple of different metrics defined. Is the diameter of a subset unique with respect to the two most ...
1
vote
1answer
28 views

Why do we study multi-valued(set valued) mappings?

I am working on a problem that has to do with multi-valued mappings. Precisely, Iterative methods for Fixed points for multivalued mappings. However, I have no clear motivation for studying such ...
1
vote
0answers
28 views

If $\partial E$ has Jordan outer measure zero, then $E$ is measurable.

I am going through Tao's measure theory book, and have to prove If $\partial E$ has Jordan outer measure zero, then $E$ is measurable. where $\partial E$ denotes the boundary of the set $E$. I ...
1
vote
0answers
31 views

Convergence of a sequence of integration

I am considering one problem and I am stuck in this step. The problem is that What conditions on function $f(u,\epsilon)$ are required to satisfy $$ \int_0^\epsilon f(u,\epsilon)\,du \rightarrow 0 ...
0
votes
0answers
14 views

About a property of the upper triangular projection of a matrix

I need a hand checking that a property about the upper triangle projection of an infinite matrix holds. $\bullet$ Let A be an infinite matrix $A=(a_{ij})_{i\geq 1\;j\geq 1}$. We define its upper ...
2
votes
1answer
17 views

$1$-form of a antiholomorphic function, Cauchy-Goursat Theorem

Let be $f:U\to \Bbb C$ antiholomorphic function. Show that the 1-form $f(z)d\overline{z}$ is closed. We have that $\overline{f}$ is a holomorphic function, so by Cauchy-Goursat Theorem the ...
4
votes
1answer
52 views

Schwarz Lemma of Complex Analysis

Let be $f:B(0,1)\to B(0,1)$ holomorphic function such that $$f(0)=f'(0)=\cdots=f^{(n-1)}(0)=0$$ but $f^{(n)}(0)\neq 0.$ Show that $|f(z)|\le |z|^n,$ for every $z\in B(0,1)$ and $|f^{(n)}(0)|\le ...
0
votes
1answer
53 views

What is the difference between single and double layer potential

I want to know the difference between single layer and double layer potentials. Is there a link between the choice of single/double layer potential and the boundary condition of a PDE or an ...
0
votes
0answers
15 views

Inverse Laplace operator $\Delta^{-1}$ in $H^1_0, \ H^{−1}$.

Let open domain $\Omega \subset \mathbb R^n$, $u : \Omega \rightarrow \mathbb R$, $u \in H^1_0 (\Omega)$ and $f \in H^{−1}(\Omega)$. I'm looking for some known expressions of the inverse Laplace ...
2
votes
3answers
81 views

Show that $\frac{n}{n^2-3}$ converges

Hi I need help with this epsilon delta proof. The subtraction in the denominator as well as being left with $n$ in multiple places is causing problems.
3
votes
0answers
29 views

Operator Norm and Submultiplicativity against the Spectral Norm

Consider $\mathcal{A}:\mathbb{R}^{n\times m}\to \mathbb{R}^{p\times q}$ to be a linear operator. I know that by considering the trace norm and using the submultiplicativity of the operator norm we ...
1
vote
0answers
38 views

local inverse functions

consider $f(x,y)=(x\sin y,x\cos y),\; (x,y)\in (0,\infty)\times (0,3\pi)=U$. f is locally invertible at every point in U, because $\det(Df(x,y))\not= 0$ for all $(x,y)\in U$. I want to know : What are ...
3
votes
1answer
44 views

Jacobi field strange condition.

I am currently reading a textbook (Kuehnel) saying that if $V,W \in T_pM$ are such that $\langle V,W \rangle =0$ and $\|V\|=\|W\|=1,$ then $Y(t):=D \exp(tV)(tW)$ is a Jacobi field. The thing is, I ...
2
votes
5answers
129 views

Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$”

If $C$ and $A$ are the circumference and area of a circle, then $$ \frac{C^2}{A}=4 \pi\; . $$ I'm looking for a reference to an elementary but rigorous proof of it. I'm particularly interested in ...
3
votes
1answer
52 views

Assumptions on functions so that integral is zero

Let $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be two arbitrary functions. Assume $g\in L^2(\mathbb{R})$. I'm looking to find out the minimal set of assumptions on $f$ and $g$ such ...
0
votes
1answer
46 views

convolution and integral limits

Let $\xi$ be an increasing function , and $f$ be a continuous function on the interval $[0,1]$. Take $\phi$ a smooth function such that $\int_0^1 \phi(s)\, ds= 1 $ and consider an approximation of ...
0
votes
1answer
27 views

Local Lipschitz continuity

In some proof I have seen the author use that if $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and bounded, then it is locally Lipschitz continuous. I have never seen that before and I don't find ...
1
vote
0answers
146 views

Show a function defined by summation is increasing, another is decreasing

Problem: For real numbers $x\ge1$ and $k>0$, let $f:R\rightarrow R$ and $g:R\rightarrow R$ be defined as follows. $f(x) = -\frac{1}{x}+\sum_{n=1}^{\infty}\frac{1}{(nk+x)^2}$ , ...
-1
votes
1answer
71 views

Zeros of an analytic function [duplicate]

How to prove zeros of a real analytic function (non-zero function) is always countable?
2
votes
1answer
67 views

Rigorously proving $\displaystyle\int^{\infty}_0 \frac{\sin{x}}{x} dx= \frac{\pi}{2}$ [duplicate]

I want to prove the famous formula: $\displaystyle\int^{\infty}_0 \frac{\sin{x}}{x} dx = \frac{\pi}{2}.$ There are many ways to do it, for example, by some Fourier analysis. But how about a simple ...
0
votes
1answer
27 views

Show that the Lebesgue Stieltjes measure corresponding to $\alpha(x) = \mu((0,x])$ is $\mu$.

This is exercise 4.1 from Bass: Let $\mu$ be a measure on the Borel $\sigma$-algebra fo $R$ such that $\mu(K) < \infty$ whenever $K$ is compact, define $\alpha(x) = \mu((0,x])$ if $x \ge 0$ and ...
1
vote
1answer
45 views

Is $C^{\infty}$ dense in $W^{k,p}$?

The $C_c^{\infty}$ are certainly not sense in an arbitrary $W^{k,p}$ space. Despite, I started wondering whether at least $C^{\infty}$ is dense? Now, this can certainly not be true for general ...
0
votes
0answers
9 views

Calculate factor for FWHM in a sech(x)-function

I have a $sech\left(\frac{\pi}{2}a\cdot x\right)$-function, and I want to calculate $a$ such that the FWHM of the function meets a specific width $\Delta x$. So I started with ...
1
vote
1answer
28 views

ODE, Picard approximation of a second order equation: How do I make sure that this is correct.

I have the following problem: $$\ddot{x} + \dot{x}^2-2x=0$$ and I.V are: $x(0)=1 \qquad$ $\dot{x}(0) = 0$. and I need to find two first "Picard" approximations. I first arranged it in the form ...
2
votes
1answer
77 views

Is the space of continuous functions with bounded variation separable?

Let $$BVC([0,1]) = \{f:\mathbb [0,1] \to \mathbb R,\, f\in BV([0,1])\cap C([0,1]),\, f(0)=0\},$$ and $$\|f\|_{BVC}=\mathrm{Var}(f).$$ Is $BVC([0,1])$ separable?
0
votes
1answer
26 views

Nest of intervals explanation

I am currently reading Konrad Knopp book about infinite series, I just don't get the part where he mentions that the nest of intervals would determine or define as he said a rational number s if it ...
3
votes
2answers
48 views

When is a continuous function piecewise monotone?

Given a continuous function $f:[a,b]\mapsto \mathbb{R}$, are there known additional conditions that ensure $f$ is piecewise monotone? Like this question, my motivation is to decompose the interval ...
0
votes
1answer
19 views

For every $n \ge 2 $ , existence of uncountably many mutually disjoint closed balls , complement of whose union is path connected

For every $n \ge 2 $ , does there exist an uncountable family $\{D_\lambda: \lambda \in \Gamma\}$ of mutually disjoint closed balls in $\mathbb R^n$ , such that $\mathbb R^n \setminus \cup_{\lambda ...
3
votes
2answers
41 views

For which infinite dimensional real normed linear spaces $X$ , can we say that every infinite dimensional subspace of it is closed in $X$ ?

For which infinite dimensional real normed linear spaces $X$ , can we say that every infinite dimensional subspace of it is closed in $X$ ? Or , does every infinite dimensional normed linear space has ...
2
votes
0answers
34 views

The question about the support of Fourier transform of $|f|^p$

Suppose $f$ is a smooth function with $\mathbb{supp}{(\mathcal{F}{f})} \subset B(0,1)$. In addition, assume $f$ is non-negative. We can observe that the function $|f|^2$ has a nice property : ...