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, and integration through the fundamental theorem of calculus.

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249 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 ...
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257 views

Behaviour at infinity of a function in terms of first and second derivatives

In a paper (dealing with spectra of certain Schrodinger operators) I found the following assumption for a function $f\in C^\infty(\mathbb R^n;\mathbb R)$: there exists a constant $C>0$ and a ...
6
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562 views

Distinct eigenvalues for integral operator?

Is there some sufficient condition on the kernel $K$ of a (say, finite rank, to simplify) integral operator $$ \mathcal K:f(x)\in L^2(\mathbb R)\mapsto \int K(x,y)f(y)dy $$ so that it has all its ...
6
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223 views

Characterize the continuous functions with finite right-hand derivative for at least one point of $[0, 1]$

Let $(E,d_\infty)$ the metric space of continuous functions defined on $ [0,1] $, with $$ d_\infty(f,g)=\sup\{ |f(x) - g(x)| : x\in [0,1] \}. $$ For all $ n\in \mathbb{N} $ let $$ F_n = \{ f: \exists ...
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75 views

Sobolev type embedding

Consider a compact manifold $M$ and a point $q \in M$. Let us say that that the following inequality holds: $$ \Vert \varphi u\Vert_{L^p} \leq C\Vert \varphi u\Vert_{H^1},$$ where $\varphi \in ...
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60 views

Disprove, fix, prove: If {$a_n$} and {$b_n$} are increasing, then {$a_n b_n$} is increasing

Prove the statement wrong, fix it, then prove the new statement: If {$a_n$} and {$b_n$} are increasing, then {$a_n b_n$} is increasing I think I'm headed in the right direction with this but I'm ...
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47 views

$L^2$ convergence of this sequence

I am given the following sequence of functions $(f_m)_{m \in \mathbb{N}}$. They are defined by $$ f_m(x):=\left( \frac{e^{-ix}-1}{-ix} \right)^m \left( \sum_{l \in \mathbb{Z}} \frac{\left|e^{-ix}-1 ...
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43 views

Importance of compactness in Rudin problem.

Okay the problem goes like this: Suppose X, Y, Z are metric spaces, and Y is compact. Let $f$ map X into Y, let g be a continuous one-to-one mapping of Y into Z, and put $h(x)=g(f(x))$ for $x\in ...
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35 views

Proof that a number uniquely determines its multiplicative inverse

I'm currently working through Pugh's Analysis (for fun.) Currently, I'm working on the following problem: A multiplicative inverse of a nonzero cut $x=A|B$ is a cut $y=C|D$ such that $x*y=1^*$. If ...
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40 views

Find $f:C\to\mathbb{R}^2$ continuous and bijective but not open, $C\subset\mathbb{R}$ is closed and connected

Are there a closed connected subspace $C$ of $\mathbb{R}^2$ and a continuous, bijective function $f:C\to\mathbb{R}^2$ that is not open? If we remove the condition for $C$ to be connected, we have ...
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85 views

The completion of the Borel $\sigma$-algebra the same as the completion of the Lebesgue outer measure?

My study group and I were discussing this question today. We can construct the Lebesgue measure using Caratheodory's extension theorem in the usual way: Given the function $F(x) = x$, we can ...
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293 views

Finding the inverse of a function.

Let $f:\mathbb{R}\to \mathbb{R}_+$ with $f\geq\epsilon>0$ be smooth and define $G:\mathbb{R}\to\mathbb{R}$ thus $$G(x):=\int_0^x\frac{1}{f(u)}\mathrm{d}u$$ Then it is clear that $G$ is ...
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76 views

Prove $(\overline{A \cap B}) \subseteq \overline{A} \cap \overline{B}$.

Let $\overline{A}$ be the closure of $A$. My attempt: Since $A \subseteq \overline{A}$ and $B \subseteq \overline{B}$, we have $$A \cap B \subseteq \overline{A} \cap \overline{B}.$$ Since ...
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93 views

Proving the function $ f $ is continuous on $ [0,1] $

I'm trying to prove that the following function $ f $ is continuous on $ [0,1] $. The function $ f:[0,1]\rightarrow [0,1] $ is defined as follows. Let $ x\in [0,1] $. Then $ x= \sum\limits_{n = ...
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83 views

Convergence of the solution of Volterra integral equation with convergent kernel.

Consider the following Volterra integral equation $$ g(t) = \int_0^t K_n(t,s)w_n(s) ds $$ where $g(t)$ and $K_n(t,s)$ are known(continuous) and $K_n(t,s)\geq K_{n+1}(t,s)$ for all $t,s$. Moreover, ...
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70 views

Conditions for Taylor formula

I know that, if $F:X\to Y$, where $X,Y$ are Banach spaces, is a map whose $n$-th Fréchet derivative $x\mapsto F^{(n)}(x)$ is continuous as a function of $x$ in a neighbourhood of $x_0\in X$, then the ...
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55 views

Estimating the (double) Riesz transform.

I'm trying to verify the following estimate, which appears in a paper I'm reading. It seems I'm missing something easy, I just can't figure this out. $\textbf{Background}:$ For a function $f \in ...
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87 views

Generating Function for Associated Stirling Numbers $b(n,k)$

I am trying to identify or find the ordinary generating function (not the exponential generating function) for the Associated Stirling numbers of the Second kind, denoted $$b(1;n,k)=b(n,k)$$ These ...
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267 views

Explain this step in lecture notes

The bounty offered is for the person that explains me how the author gets from equation 3.19 to equation 3.20 in these lecture see here. Normally I would agree that copying the relevant equation would ...
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138 views

Show that it is an algebra.

This excercise is a little struggling for me. The part I need help with is showing that $D$ is closed under complements. Let $C$ denote the collection of all intervals on $\mathbb{R}$, including ...
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277 views

Taylor Formula: Lagrange's remainder vs Cauchy's remainder (and other less known forms)

While solving problems and exercises, so far I've only used Lagrange's form of the remainder. Indeed, it must be said that many textbooks don't even mention other forms of the remainder for Taylor's ...
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204 views

Functions with compact support

I have a question about a convergence of functions with compact support. SETTING Let $d\geq 3$ and $U \subset \mathbb{R^{d}}$ be open and $dx$= Lebesgue measure on $U$. Let $b_{i},c,d_{i} \in ...
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75 views

An inequality between integrals of series of characteristic functions of cubes

Let $1\leq p<\infty$. Prove that there exists $C>0$ such that $$ \left(\int\left|\sum_{i=1}^\infty a_i\chi_{2Q_i}\right|^p \, dx\right)^{1/p} \leq C\left(\int\left|\sum_{i=1}^\infty ...
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250 views

Analysis or (abstract) algebra first?

Which one would you recommend? I only know calculus and linear algebra when it comes to university-level mathematics. Is one required to understand the other?
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97 views

Generating function of the squared Riemann zeta function

It's a well known fact that $$\sum_{k=2}^{\infty} \zeta(k) x^k=-x \psi(1-x)-x\gamma \space (|x|<1) $$ but I didn't meet yet a version for squared Riemann zeta function $$\sum_{k=2}^{\infty} ...
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236 views

Integral $I=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx$

Hi I am trying to integrate and obtain a closed form result for $$ I:=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx. $$ Here is what I tried (but I do not think this is ...
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71 views

Exponential integral equation

I want to find the function $q(x)$ such that the following integral equation is satisfied for all $s\ge 0$: $$ \frac{p}{as + 1}\int_0^\infty \exp \left( -\frac{xs}{as + 1} \right)q(x)dx = ...
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122 views

$f$ continuous nowhere implies $f$ not Riemann integrable

I am trying to prove the statement in the title, without using the theorem which says that the set of discontinuities of a Riemann integrable function must have measure $0$. I am aware that similar ...
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142 views

Showing some complicated integral expression is bounded

In my research, I come across some expression I need to bound. I wish to show that the following integral is bounded in $t$ and $x$, i.e. the following supremum is finite: $$\sup_{t,x\in ...
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126 views

Uniqueness solutions of $dx/dt = f^2(x) + e^{-t}$.

Someone can help me in the following problem? Is a question of Zhang. Let $f(x)$ be continuous for $x \in \mathbb{R}$, show that $dx/dt = f^2(x) + e^{-t}$ has the property of uniqueness of ...
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152 views

A Curious Identity

I met the following equations when I was trying to solve a complex line integral (W.Rudin, RCA, p.228 ex.13). My question is how to prove them: We have to show that for $n>2$ even $$ ...
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108 views

How can I show it?

Let $f$ be an absolute continuous function in $(0, 1)$ and satisfies $$ |f(x+h)+f(x-h)-2f(x)|\leq \text{const}\frac{|h|}{(\log\frac{1}{|h|})^{\gamma}} , $$ where $\gamma \in (0, 1)$ and $|h|$- is ...
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202 views

Decrease of $L_1$ norm of piecewise constant functions after some “averaging”

In the course of a project, I ended up looking at what happens to the $L_1$ of real-valued piecewise-constant functions after some particular king of smoothing — but am currently stuck at a particular ...
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82 views

$f \in L^1(- \infty , + \infty)$ , Find the limit $ \lim_{n \to \infty} \int^{{+ \infty}}_{- \infty} \frac{f(x)e^{nx}}{1+e^{nx}} dx $

$f \in L^1(- \infty , + \infty)$ , Find the limit $ \lim_{n \to \infty} \int^{{+ \infty}}_{- \infty} \frac{f(x)e^{nx}}{1+e^{nx}} dx $ My Attempt $$ \lim_{n \to \infty} \int^{{+ \infty}}_{- \infty} ...
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147 views

My (simpler) proof of the divergence of a harmonic series.

Let $H=1+\frac{1}{2}+\frac{1}{3}+\cdots$ . Proving that it diverges this is what I did. I supposed that the series converges to $H$, : ...
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130 views

Can one use $e^n$ instead of $2^n$ in Cauchy condensation test?

Cauchy condensation test is useful for testing the convergence of infinite series. The test is stated here as follows: for a positive non-increasing sequence $f(n)$, the sum $\sum_{n=1}^\infty f(n)$ ...
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89 views

What kinds of structures support integration?

I am doing topology which neatly generalizes analysis, which led me to wonder naturally about generalizations of calculus. Specifically I'm interested in knowing what is required of a mathematical ...
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199 views

Dyadic approach to Marcinkiewicz interpolation for Lorentz Spaces

In Exercise 21 , in a note for professor Terrence Tao on his own blog http://terrytao.wordpress.com/2009/03/30/245c-notes-1-interpolation-of-lp-spaces/ Exercise 21 Suppose we are in the ...
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287 views

If a Radon measure is a tempered distribution, does it integrate all Schwartz functions?

The question might at first sight sound like the answer is trivially "yes", so let me clarify the question a bit. Consider given a Radon measure $\mu$ on $\mathbb{R}^n$. Let ...
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72 views

A problem about Cantor set and found when learning dynamical systems.

Consider the family of functions F(x)=$x^3 -\alpha$x, for $\alpha \gt 0$ Prove that if $\alpha$ is sufficiently large, then the set of points |$F^n(x)$| which do not tend to infinity is a Cantor ...
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177 views

Is there an orthonormal basis for $L_2[0,1]$ consisting of convex functions?

Is there an orthonormal basis $\{\phi_{\alpha}\}$ for the space $L_2[0,1]$ of square-integrable functions from $[0,1]$ to $\mathbb{R}$ such that every $\phi_{\alpha}$ is convex? Edit: A helpful ...
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120 views

Existence of smooth extension of a function defined on a closed interval

Suppose $ f: [0,1] \rightarrow \mathbb{R}$ is a function such that derivatives of all orders exist ( at the end points of the interval the appropriate one-sided derivatives exist) and are continuous $ ...
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120 views

How to find a homeomorphism $\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ having certain properties

Let $n\ge 2$ and let $C$ be a cantor space in $\mathbb{R}^{n}$. That is, $C$ is homeomorphic to the cantor ternary set. Let $x$ and $y$ be two points in $\mathbb{R}^{n}-C$, and let $L_{xy}$ be the ...
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209 views

Why can I integrate something like “infinitesimal part” to calculate the length, area, volume, etc.?

Let me try to elaborate the question by an example: I want to calculate the length of a straight line like $y=x$ between $[0, 1]$. By conventional way I would do a definite integration of integrand ...
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271 views

Does every smooth surjective function have a smooth right inverse?

If you feel this question might be too broad, let me know and I’ll try to get more specific. If $r \colon I → J$ is a smooth surjective function between perfect subspaces $I$ and $J$ of $ℝ$, can ...
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175 views

Algorithm to calculate multiple integral.

One of the major difficulties of student in advanced calculus (including myself when student) is to obtain the extremes of repeated integrals to calculate the volume integral in $R^n$ i.e. transform ...
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186 views

Proving that a set is closed in $L^2(\mathbb{R})$

this is my first question here so i hope i don't do anything wrong. Excuse any spelling or grammar mistakes, english isn't my mother tongue. I'm reading this paper for my bachelor thesis and have ...
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76 views

$x\mapsto f(x^{1/p})$ is smooth if and only if $f^{(n)}(0)=0$ whenever $p\nmid n$

This is a slight generalization of something I got stuck on when trying to do Problem 2-5 from Introduction to Smooth Manifolds by John M. Lee (which uses the case $p=3$). Let $p\ge 1$ be an ...
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0answers
637 views

“Simple” proof of Lebesgue's Differentiation Theorem for dimension 1?

Lebesgue Differentiation Theorem for $\mathbb{R}$: Let $f:[a,b]\to \mathbb{R}$ be intergable and $F(x)=\int_a^xf$. Then $F$ is differentiable almost everywhere in $[a,b]$ and $F'=f$ a.e. Is there a ...
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1k views

Closure, Interior, and Boundary of Jordan Measurable Sets.

This question has a number of parts. Let $E\subset\mathbb{R}^{d}$ be a bounded subset. (1) Show that $m^{\star,(J)}(E)=m^{\star,(J)}(\bar{E})$ (closure) (2) Show that ...