Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, integration through the fundamental theorem of calculus...
4
votes
0answers
130 views
Question regarding the Kolmogorov-Riesz theorem on relatively compact subsets of $L^p(\Omega)$.
Usually, the Kolmogorov-Riesz theorem is quoted for $L^p(\mathbb R^n)$, but I am looking for versions considering spaces over subsets in $\mathbb R^n$.
The following is from the book "Sobolev spaces" ...
4
votes
0answers
55 views
Constructing the support of a Borel measure
From Rudin, Real and Complex Analysis, Chapter 8, Problem 7, 1st Edition.
Suppose $E$ is a compact set in $\mathbb{R}^{k}$ without isolated points. Show that $E$ is the support of a continuous ...
4
votes
0answers
75 views
Inexact Newton method.
Let's a nonlinear function
$
f:[-1,+1]^N\subset\mathbb{R}^N\to\mathbb{R}^N,\; N\in\mathbb{N},
$
such that the the sequence generated by the method of Newton-Raphson
$$
...
4
votes
0answers
76 views
When $K$ is compact, if $S\subset C_b(K)$ is closed,bounded and equicontinuous, then $S$ is compact? (ZF)
Let $K$ be a compact metric space and $S\subset C(K,\mathbb{C})$.
Let $S$ be closed,bounded and equicontinuous. The usual proof for this is, using Arzela-Ascoli Theorem and Axiom of countable choice, ...
4
votes
0answers
51 views
Prove that there exists $t$ such that $0\le t\le T$ and $\int_0^Te^{-x}y'y''dx=\int_0^ty'y''dx$.
Let $y(x)$ be a solution to $y''+e^xy=0$. Prove that there exists $t$ such that $0\le t\le T$ and $\int_0^Te^{-x}y'y''dx=\int_0^ty'y''dx$.
4
votes
0answers
219 views
prove that every continuous function is integrable
Can someone tell me whether this is correct thank you!
We know that if a function f is continuous on $[a,b]$, a closed finite interval, then f is uniformly continuous on that interval. This means ...
4
votes
0answers
75 views
A question about functions in $L^p(E)$
I've been working on a problem. I found a couple of related questions on the site, but I was having a little bit of trouble clarifying everything. The goal is to show that given $f\notin L^p(E)$, ...
4
votes
0answers
112 views
Prove that liminf of functions is semicontinuous
I recall that, if $\psi:\Bbb{R}\longrightarrow \Bbb{R}$ is a function defined over $\Bbb R$ with euclidean topology, we have
$\liminf\limits_{y\to x} \psi(y) = \sup\limits_{U\in \mathscr{U}_x} ...
4
votes
0answers
76 views
How many points does one need for an epsilon-net
Does anyone know, how many points does one need to have an $\varepsilon$-net on a unit sphere sitting in the three-dimensional Euclidean space? Thanks!
4
votes
0answers
128 views
Show different limits under different mode of convergence equals almost everywhere
Suppose that a sequence of bounded and continuous functions $f_n$ converges uniformly to $f_1$ and $f_n$ converges to $f_2$ in $L^2$ sense, then how to show $f_1= f_2$ a.e.?
I tried the following: ...
4
votes
0answers
451 views
Characteristic functions based proof problem.
I am trying to show that if $T$ be a closed bounded interval and $E$ a measurable subset of $T$. Let $\epsilon >0$, then there is a step function $h$ on $T$ and a measurable subset $F$ of $T$ for ...
4
votes
0answers
201 views
Fourier dimension of sets of positive Lebesgue measure
Let $K$ be a compact set in $\mathbb{R}$ with positive Lebesgue measure. My question is whether there exists a probability measure $\mu$ supported on $K$ such that $\hat{\mu}(\xi)$, the ...
4
votes
0answers
244 views
Proving the measure of an increasing sequence of measurable sets is the limit of the measures
Show that if $A_1\subseteq A_2\subseteq A_3\cdots$ is an increasing sequence of measurable sets(so $A_j\subseteq A_{j+1}$ for every positive integer $j$),then we have
...
4
votes
0answers
115 views
German Analysis Texts
My question is somewhat related to this one but is somewhat more specific. Since a lot of good mathematics is written in German, I have decided to start developing my German reading abilities. So far, ...
4
votes
0answers
371 views
Question about limit points of a Subset of $\mathbb{R}$
The question :
Let D be a nonempty subset of the reals that is bounded above. Is the supremum of D a limit point of D?
My Reasoning: I think this is false for these two cases.
Case 1:If I look ...
4
votes
0answers
168 views
Simple formulation, nontrivial problem
There's a problem from calculus I remember:
$$\forall x\ \exists n.\ f^{(n)}(x) = 0 \iff \exists n\ \forall x.\ f^{(n)}(x) = 0\,.$$
Function $f \in C^\infty(\mathbb{R})$, and the notation $f^{(n)}$ ...
4
votes
0answers
232 views
What are the conditions sufficient and necessary on $g(t)$ for the Dirichlet integral to be equal to $\frac{\pi}{2} g(0+)$?
Dirichlet Integral of a function $g\colon \mathbb{R} \to \mathbb{R}$ is defined as $$ DI(\alpha) = \int_0^{\delta} g(t)
\frac{\sin(\alpha t)}{t} dt$$ assume $\alpha \in \mathbb{N}$
For the equality ...
4
votes
0answers
110 views
$C(K,R)$ continuous functions on compact metric spaces
How to do this problem?
Question: let $(K,d)$ be a compact metric space and $A$ is contained in $C(K,R)$, the set of all continuous functions from $K$ to $\mathbb R$, an algebra that separates the ...
4
votes
0answers
149 views
Help with removing singularities involving $ \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}$
This post can be thought of as the prototype proof and the motivation for the question posted here Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion
...
4
votes
0answers
157 views
Laplace-Beltrami Operator for Euclidean Space
Consider the space $\mathbb{R}^n$ and let $x_1,\ldots, x_n$ be the coordinates. Fix the orientation $dx_1\wedge dx_2\ldots\wedge dx_n$. Let $E^p$ denote the space of smooth $p$ forms and let $d$ ...
4
votes
0answers
148 views
Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion
Let $ 0 < r < 1$, fix $x > 1$ and consider the integral
$$ I_{r}(x) = \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}.$$
In the investigation of ...
4
votes
0answers
241 views
Interchanging the order of limits
Would you advise me on the references of Pringsheim Convergence about interchanging the order of limits? Where can I find the most general statement?
4
votes
0answers
337 views
First order variation and total variation of a function/stochastic process
The notions of first-order variation and total variation of a function or a stochastic process are equated in this book.
However, I found their definitions different from two other sources:
In ...
4
votes
0answers
116 views
Arclength integral
Suppose that $f: [a,b] \rightarrow \mathrm{R}^n$ is continuous with a derivative $f'$ whose norm is Riemann-integrable. To demonstrate the arclength integral formula, I'm trying to prove that, for ...
4
votes
0answers
407 views
Convergent sum with primes
If $f(n)$ is a strictly increasing elementary function from the reals to the reals, and $p(n)$ is the $n$'th prime number. Is there any $f(n)$ such that $\sum_{n=1}^\infty\frac{1}{f(p(n))}$ is ...
4
votes
0answers
216 views
existence of a sequence of continuous functions (pointwise convergence)
Prove that there no exist a sequence of continuous functions $
f_n :\left[ {0,1} \right] \to R
$ such that converges pointwise, to the function
$$f(x)=
\begin{cases}
0 & \text{if $x$ is ...
4
votes
0answers
672 views
Show that a differentiable everywhere function $f\colon \mathbb{R}\to\mathbb{R}$ has a derivative which is Borel measurable
So my homework problem is to show that if a function $f: \mathbb{R} \to \mathbb{R}$ is differentiable everywhere, then its derivative $f'$ is Borel measurable.
What I have for something to be ...
4
votes
0answers
369 views
characteristic function of the rationals
Let $\chi$ be the characteristic function of the rational numbers in $[0,1]$. Does there exist a sequence $\{f_n\}$ of continuous functions on $[0,1]$ that converges pointwise to $\chi$?
4
votes
0answers
130 views
Uniform convergence of piecewise linear interpolations
Let $$X^k (t) := X^0 (t+t_k) - X^0 (t_k)$$ where $X^0(t)$ is the piecewise linear interpolation of $X^0(t_k) \equiv X_k=\sum_{i=0}^{k-1} a_i b_i$ with interpolation intervals $a_k$. ...
4
votes
0answers
197 views
Equivalence of two sequences
I'm having some trouble showing that two things I really want to be the same are in fact the same. I want to show that these two sequences are, in fact, the same thing:
$$a_0=1,a_1=-1, ...
4
votes
0answers
236 views
Organizing types of functions by their calculus-related properties, in diagram form?
Does anyone know of a diagram that displays and organizes categories of functions according to their calculus-related properties (e.g. continuous, $C^\infty$, degrees of differentiability and ...
4
votes
0answers
188 views
Something connected with Arzelà-Ascoli theorem
Let $X$ be a Polish space. Assume that $(C_m)_{m\in\mathbb{N}}$ is an increasing sequence of compact subsets of $X$ and denote $C=\bigcup_{m}C_m$. Let $\{f_n:n\in\mathbb{N}\}$ be a family of ...
4
votes
0answers
333 views
How to construct a vector space and compute basis?
My professor demonstrated that in vector calculus that you can construct basis vectors for one, two, and three forms using the vectors $dx$, $dx$ and $dy$, as well as $dx \wedge dy$, $dy \wedge dz$, ...
4
votes
0answers
125 views
When is the $n$th term sufficient to guarantee convergence of a series
If $K$ is a field complete with respect to a non-archimedean absolute value, then the $n$th term test (checking whether the $n$th term of a series goes to zero) is sufficient to check convergence of a ...
3
votes
0answers
91 views
Prove that $\lim_{N\rightarrow\infty}(1/N)\sum_{n=1}^N f(nx)=\int_{0}^1f(t)dt$
Suppose $f$ is continuous and periodic on the reals with period 1. Prove that if $x\in[0,1]$ is an irrational number, then
$$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^N f(nx)=\int_{0}^1f(t)dt$$
...
3
votes
0answers
60 views
Computing the Fourier transform (in the sense of tempered distributions) from a sequence of samples
Let $f\in L_{loc}^2 (\mathbb{R})$ be a function of period $2\pi$. I'd like to get some help proving the following identity:
$\hat{f}=\Sigma_{n=-\infty}^{\infty} \hat{f}(n)\delta_n$
With some ...
3
votes
0answers
57 views
Prove that Riemann integrability implies boundedness.
Prove: If $f:[a,b] \to \mathbb{R}$ is Riemann integrable, then $f$ is bounded.
My class did this using proof by contradiction and epsilon-delta definition (which I have hard time understanding).
Is ...
3
votes
0answers
30 views
Existance of a simple function
Let $f : [a, b] \to \mathbb{R}$ be a measurable function. Suppose that $\varepsilon, M > 0$ are given. Show that there is some simple function $\varphi : [a, b] \to \mathbb{R}$ such that $|f(x) - ...
3
votes
0answers
79 views
For what values of x does it converge?
How can i study the character of the following series?
$$ \sum_{n=1}^\infty\,\,\frac{x^n(\sin1\cdot\sin2\cdot ...\cdot\sin n)^2}{(1+x\cos^21)\cdot(1+x\cos^22)\cdot...\cdot(1+x\cos^2n)},\qquad ...
3
votes
0answers
31 views
Double Integral exists or not
Let $ \Omega :[0,1] \times [0,1] $. Let $f:\Omega \rightarrow \mathbb{R}$ be $f(x,y) = 1 \ if\ x=0\ and \ y \in \mathbb{Q},\ \ \ 0\ otherwise $.
Which of the following exists, and find the integral ...
3
votes
0answers
89 views
Infinite self-convolution for a function
I have a mathematical problem that leads me to a particular necessity. I need to calculate the convolution of a function for itself for a certain amount of times.
So consider a generic function $f : ...
3
votes
0answers
76 views
Positive functions with zero integrals
I was a bit confused by this link mentioned in this question - in particular, in Remark 4.21:
Suppose that $f$ is a positive function on $[a,b]$. If $f$ is Henstock-Kurzweil integrable, then the ...
3
votes
0answers
42 views
Constructing an incomplete ordered field satisfying the nested interval property
Motivated by a problematic exercise in an analysis textbook, I decided to search for an example of an ordered field which satisfies the nested interval property yet fails to be complete.
I first ...
3
votes
0answers
57 views
State of the art of the Implicit Function Theorem
What is the most general form of the Implicit Function Theorem? Quite a general form of this theorem was given by Kumagai (1980): An implicit function theorem. So I am wondering what are the weakest ...
3
votes
0answers
53 views
Baby Rudin theorem 2.41
I am reading Baby Rudin chapter 2 and came across some question on Theorem 2.41.
When Rudin tries to prove that every infinite subset of $E$ has a limit points in $E$ implies $E$ is closed, he first ...
3
votes
0answers
43 views
Understanding the roots of homomorphism and homeomorphism
I understand the formal (i.e. mathematical) definitions of the terms "homomorphism" and "homeomorphism" as they relate to functions, but I am curious as to the origin of these terms. I don't know ...
3
votes
0answers
48 views
Different functional brachystochrone
Until today I thought that $$ \int_0^b \sqrt{\frac{1+y'(x)^2}{2gy(x)}} dx$$ would be the only functional to derive the brachystochrone, but in the textbook Variational Methods in Mathematical Physics ...
3
votes
0answers
51 views
Inverse Fourier transform and decay at infinity
Suppose $1<p<\infty$, $g:\mathbb{R} \to \mathbb{R}$. How can I see $\mathcal{F}^{-1}g \in L^p(\mathbb{R})$ as a condition on the decay of $g$ at infinity?
3
votes
0answers
72 views
What is the point of compactness and how does it look visually?
I just learned what the term compact means, as in compact sets. I know what the definition of compactness is, but I don't get what the real significance of it is. How should I try to think of it in a ...
3
votes
0answers
59 views
cantor set + cantor set =$[0,2]$
$C=$ cantor set to show $C+C =[0,2]$
ans
$x\in C,$ then $x= \sum_{n=1}^{\infty}\frac{a_n}{3^n}$ where $a_n=0,2$
so any element of $C+C $ is of the form $$\sum_{n=1}^{\infty}\frac{a_n}{3^n} ...
