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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91 views

Asymptotic property of integral involving Bessel function.

Consider the following integral $$ I(s)=\int_{0}^{\infty}{J_{\frac{n-2}{2}}}(sr)r^{A+1}(e^{-r^{2\alpha}}-1)dr, $$ where $J_{\frac{n-2}{2}}$ is the Bessel function of order ${\frac{n-2}{2}}$, $s, A, ...
6
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669 views

Proof that a sequence of continuous functions $(f_n)$ cannot converge pointwise to $1_\mathbb{Q}$ on $[0,1]$

As a homework question, we got asked the following: Construct a function $f:[0,1] \rightarrow \mathbb{R}$ which is not the pointwise limit of any sequence of continuous functions Thinking about ...
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103 views

Sum-Product Generating Functions

Let $A_n$ be a family of sequences $\{a_i\}_{i=1}^n$ of length $n$. I'll refer to sequence elements of $A_n$ as $a$. Then define $$G(z):=\sum_{a\in A_n}\prod_{i=1}^n(z+a_i).$$ Here's one possible ...
6
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154 views

Is there a subsequence of $a_n = n \sin(n)$ which tends to $0$?

I know there is such a subsequence for $b_n = \sin(n)$. What about $a_n = n\sin(n)$?
6
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90 views

Inverse of a differentiable function equal to its derivative then f is analytic

I've found a nice problem concerning analytic functions. Here it is: Let $f: (0, \infty) \rightarrow \mathbb{R}$ be a function differentiable on $(0, \infty)$ and such that $f^{-1} = f'$. Prove that ...
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229 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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250 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 ...
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548 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 ...
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222 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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54 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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77 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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111 views

Show that f is periodic if $f(x+a)+f(x+b)=\frac{f(2x)}{2}$?

Suppose $a$ and $b$ are distinct real numbers and $f$ is a continuous real function such that $\frac{f(x)}{x^2}$ goes to 0 when $x$ goes to infinity or minus infinity. Suppose that$ ...
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69 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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66 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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79 views

Find the closed form of the digamma related series

The question I asked here Computing $\sum_{n=1}^{\infty} \left(\psi^{(0)}\left(\frac{1+n}{2}\right)-\psi^{(0)}\left(\frac{n}{2}\right)-\frac{1}{n}\right)$ made me think to ask for your support for ...
5
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259 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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124 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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153 views

Is there a book only about epsilon delta proofs?

I want to know if there is such book, with beautiful epsilon delta proofs of all kind.
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189 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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69 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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161 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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92 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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110 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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137 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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116 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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144 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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107 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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200 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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77 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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134 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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122 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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88 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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187 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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132 views

Operator norm of a convolution

Consider the operator on $L^2(\Bbb R)$, $f\rightarrow f*g$, where $g\geq 0$ is some $L^1$ function. Show the operator is a bounded linear operator with operator norm equal to $||g||_1$. Showing ...
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235 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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71 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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154 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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296 views

How prove this $f^{(n)}(x)=0$ has at least $n-1$ distinct roots

Let $f\in C^{(n)}\left(]-1,1[\right)$ and $\displaystyle\sup\limits_{-1<x<1}|f(x)|\le 1$. Let $m_{k}(I)=\inf\limits_{x\in I}|f^{(k)}(x)|$, where $I$ is an interval contained in $]-1,1[$. ...
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108 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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114 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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168 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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249 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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81 views

$M$ is compact, non-empty, perfect, and $M \cong M \times M$. Must $M$ be homeomorphic to the Cantor set, the Hilbert cube, or some combination?

Assume that $M$ is compact, non-empty, perfect, and homeomorphic to its Cartesian square, $M \cong M \times M$. Must $M$ be homeomorphic to the Cantor set, the Hilbert cube, or some combination of ...
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167 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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180 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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522 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 ...
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109 views

Proofing an inequality with a slowly varying function

I am working right now with "Independent and Stationary Sequences of Random Variables" from Ibragimov 1971. I am trying to understand the proof of the following Lemma (18.2.4): $h: \mathbb N ...
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595 views

one to one mapping between the floor function and the Riemann prime counting function

We have the following 'transform' of a real valued, piecewise continuous function $f(x)$ : $$T[f(x)]=1+\sum_{n=1}^{\infty}\int_{\mathbb{R}^{n}_{+}}f\left(\frac{x}{\Lambda _{n}} \right ...
5
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245 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 ...