Questions tagged [real-analysis]

For questions about real analysis, a branch of mathematics dealing with 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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128
votes
0answers
2k views

What functions can be made continuous by “mixing up their domain”?

Definition. A function $f:\Bbb R\to\Bbb R$ will be called potentially continuous if there is a bijection $\phi:\Bbb R\to\Bbb R$ so that $f\circ \phi$ is continuous. So one could say a potentially ...
57
votes
0answers
711 views

Conjectured formula for the Fabius function

The Fabius function is the unique function ${\bf F}:\mathbb R\to[-1, 1]$ satisfying the following conditions: a functional–integral equation$\require{action} \require{enclose}{^{\texttip{\dagger}{a ...
52
votes
0answers
1k views

On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
38
votes
0answers
716 views

Convergence of $\sum_{n=1}^{\infty} \frac{\sin(n!)}{n}$

Is there a way to assess the convergence of the following series? $$\sum_{n=1}^{\infty} \frac{\sin(n!)}{n}$$ From numerical estimations it seems to be convergent but I don't know how to prove it.
31
votes
2answers
540 views

Can $ f\colon \mathbb{R}^k \to \mathbb{R}^n$ such that $ \forall y \in \operatorname{im}(f)$, $f^{-1}(y) = \{a_y,b_y\} $ be continuous?

This is the problem we want to solve: Can $f\colon \mathbb{R}^k \to \mathbb{R}^n$ such that $ \forall y \in \operatorname{im}(f)$, $ f^{-1}(y) = \{a_y,b_y\}, a_y \neq b_y $ be continuous? ...
28
votes
1answer
905 views

The Modulus of all the roots of a Polynomial are equal to $1$

Suppose the real number $\lambda \in (0,1)$, and let $n$ be a positive integer. Prove that all roots of the polynomial $$f\left ( x \right )=\sum_{k=0}^{n}\binom{n}{k}\lambda^{k\left ( n-k \right )}x^{...
27
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0answers
744 views

Constructing an infinite chain of subfields of 'hyper' algebraic numbers?

This has now been cross posted to MO. Let $F$ be a subset of $\mathbb{R}$ and let $S_F$ denote the set of values which satisfy some generalized polynomial whose exponents and coefficients are drawn ...
19
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0answers
517 views

Prove a strong inequality $\sum_{k=1}^n\frac{k}{a_1+a_2+\cdots+a_k}\le\left(2-\frac{7\ln 2}{8\ln n}\right)\sum_{k=1}^n\frac 1{a_k}$

For $a_i>0$ ($i=1,2,\dots,n$), $n\ge 3$, prove that $$\sum_{k=1}^n\frac{k}{a_1+a_2+\cdots+a_k}\le\left(2\color{red}{-\frac{7\ln 2}{8\ln n}}\right)\sum_{k=1}^n\frac 1{a_k}.$$ The case without $\...
19
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0answers
344 views

Witt's proof of Gelfand-Mazur / Ostrowski's theorem

Now asked on MathOverflow. Background: It seems that, after his groundbreaking work on quadratic forms and inventing Witt vectors, Ernst Witt developed the hobby of giving extremely short proofs to ...
18
votes
0answers
366 views

A subadditive bijection on the positive reals

Question. Does there exist a subadditive bijection $f$ of the positive reals $(0,\infty)$ such that $$ \liminf_{x\to 0^+}f(x)=0 \,\,\,\text{ and }\,\,\,\limsup_{x\to 0^+}f(x)=1\,? $$ Ps. I guess ...
17
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1answer
464 views

How to solve $\dot{x}=f(x)/||f(x)||$?

How to solve the following ODE? \begin{equation} \frac{d}{dt}x=\frac{f(x)}{\|f(x)\|}, \end{equation} where $x: \mathbb{R} \to \mathbb{R}^n$, i.e., $x(t)$ is the trajectory. The right-hand side $f: ...
16
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0answers
278 views

Norms and pointwise convergence

It is known that There is no norm $\|.\|$ on the space $E$ of continuous real-valued functions on an interval, say $[0,1]$ such that $f_n \to f$ for $\|.\|$ if and only if $f_n$ converges pointwise ...
16
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398 views

A question connected with the decomposition of a functional on $C(X)$ on Riesz and Banach functionals

Let $X$ be a metric space and let $C(X)$ be a family of all bounded and continuous functions from $X$ in $\mathbb{R}$. We call a positive linear functional $\varphi: C(X) \rightarrow \mathbb{R}$ the ...
15
votes
0answers
170 views

Prove that $\lim_{n \rightarrow \infty} \sum_{k=0}^{n} \frac{1}{k!} = e$

Define $e, e'$ by $$e: =\lim _{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^{n} \quad \text{and} \quad e' := \lim _{n \rightarrow \infty} \sum_{k=0}^{n} \frac{1}{k!}$$ Prove that $e' \in \...
15
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0answers
262 views

Extension of Vector Field in the $\mathcal{C}^r$ topology.

Let $M\subset \mathbb{R}^n$ be a compact smooth manifold embedded in $\mathbb{R}^n$, we define $$\mathfrak{X}(M) := \{X: M \to \mathbb{R}^n;\ X\mbox{ is smooth and }\ X(p) \in T_p M \subset \mathbb{R}^...
15
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0answers
152 views

$\frac{1}{p}+\frac{1}{q}=1$ vs $\sum_{n=0}^\infty \frac{1}{p^n}=q$

It just occurred to me that conjugate exponents, i.e. $p,q\in(1,+\infty)$ such that $$\frac{1}{p}+\frac{1}{q} =1$$ also satisfy the relations: $\sum_{n=0}^\infty \frac{1}{p^n}=q;$ $\sum_{n=0}^\infty \...
15
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0answers
291 views

Geometric representation of Euler-Maclaurin Summation Formula

When reading Tom Apostol's expository article (or the free link), I was expecting more diagrams to come that follow the figure below (or this from the Wolfram project). It was a disappointment not ...
15
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0answers
252 views

The sequence $\,a_n=\lfloor \mathrm{e}^n\rfloor$ contains infinitely many odd and infinitely many even terms

PROBLEM. Show that the sequence $\,a_n=\lfloor \mathrm{e}^n\rfloor$ contains infinitely many odd and infinitely many even terms. It suffices to show that the terms of the sequence $$\,b_n=\mathrm{e}^...
15
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0answers
272 views

A closed $1$-form on a convex open set is exact

Baby Rudin Exercise 10.24: Let $\omega = \sum a_i(\mathbf x) \, dx_i$ be a $1$-form of class $\mathscr{C}''$ in a convex open set $E \subset \mathbb{R}^n$. Assume $d \omega = 0$ and show that $\...
15
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0answers
680 views

Can fundamental theorem of algebra for real polynomials be proven without using complex numbers?

Update 24 Nov 2015: It is solved. Please refer to this arXiv paper. For polynomials with real coefficients, I am trying to prove the following version of fundamental theorem of algebra, which avoids ...
14
votes
0answers
300 views

If $V\subset L^\infty[0,1]$ with $\|f\|_\infty \leq c\|f\|_2$, then $V$ is finite dimensional

If $V$ is a linear subspace of $L^\infty[0,1]$ with $\|f\|_\infty \leq c\|f\|_2$ for all $f\in V$, then $V$ is finite dimensional. The proof is an explicit calculation: Since $L^\infty[0,1] \subset L^...
13
votes
0answers
538 views

Bounding a polynomial from below

Let $\sigma >0$ be fixed. For even $k \in \mathbb{N} \cup \{0\}$, we consider the polynomial \begin{equation} \varphi_k(x) = \sum_{j=0}^{k} (-1)^j {k \choose j} b_j \, x^{2j} \quad x \in (-1,1), \...
13
votes
0answers
138 views

How much can we rearrange a series?

There's a well-known result that if $\sum a_n$ is conditionally convergent, then for any real $c$ there exists a permutation $\pi:\mathbb{N} \to \mathbb{N}$ such that $\sum a_{\pi(n)} = c$. A ...
13
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0answers
104 views

A question about Existence of a Continuous function.

Let $f$ be a continuous function on the interval $[1,2]$. It follows from Stone-Weierstrass theorem that if $\displaystyle \int_1^2x^nf(x)dx=0$ for integers $n=0,1,2,....$, then $f$ must be ...
13
votes
0answers
724 views

Lagrange multipliers in Calculus of Variations

I am trying to learn about Calculus of Variations and I am beginning to see some constrained optimization problems in the domain of functionals, by using Lagrange multipliers. It seems that things ...
13
votes
1answer
248 views

Number of flips to get to a Set of Positive Lebesgue Measure

A consequence of Exercise 1.1.19 on page 13 of Stroock's "Probability Theory: An Analytic View" is that if a set $E\subset[0,1)$ has positive (Lebesgue) measure, then for almost every $x\in[0,1)$, a ...
12
votes
0answers
460 views

How to prove this lemma related to Rolle's theorem

For any function $f$ denote by $Z(f)$ and $Z_o(f)$ the cardinalities of $f^{-1}(0)\cap[0,1]$ and $f^{-1}(0)\cap(0,1)$, respectively. Let $H=\{f\in C^\infty(\mathbb{R}): \text{supp}(f) = [0,1]\}$ From ...
12
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0answers
449 views

How to construct examples of functions in the Spaces of type $\mathcal{S}$

There are $3$ $\mathcal{S}$-type Spaces, namely $\mathcal{S}_\alpha\:,\: \mathcal{S}^\beta\:,\:\mathcal{S}_\alpha^\beta$. They are defined by: $\mathcal{S}_\alpha: |x^k\varphi^{(q)}(x)|\le C_qA^kk^{...
12
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0answers
836 views

Heat Kernel Property

Let $\phi$ be the Heat Kernel in $\mathbb{R}^n$, i. e. $$\phi (x,t)={(4\pi t)}^{-n/2}\exp\left( - \frac{\mid x \mid ^2}{4t}\right)$$ and let $u$ satisfy Heat equation. Show that: $$\frac{d}{dt}\...
12
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0answers
1k 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" ...
11
votes
1answer
304 views

How to prove this integral inequality $ \int_0^{2\pi} p(x)[p(x)+p''(x)] dx \int_0^{2\pi}\frac{1}{p(x)+p''(x)} dx\geq 2\pi \int _0^{2\pi} p(x) dx $?

Let $p\in C^2(\mathbb{R})$ be a $2\pi$-periodic function such that $p(x)>0$ and $p(x)+p''(x)>0$ for all $x\in \mathbb{R}$. Then it holds $$ \int_0^{2\pi} p(x)[p(x)+p''(x)] dx \int_0^{2\pi}\frac{...
11
votes
0answers
375 views

Increasing derivatives of recursively defined polynomials

Consider recursively defined polynomials $f_0(x) = x$ and $f_{n+1}(x) = f_n(x) - f_n'(x) x (1-x)$. These polynomials have some special properties, for example $f_n(0) = 0$, $f_n(1) = 1$, and all $n+1$...
11
votes
0answers
207 views

Can (linear) differential equations of infinite order be recast into equations of first order?

In most analysis courses one sees that differential equations of order $n$ are basically a subset of higher dimensional differential equations of order $1$, for example the equation: $$f^{(n)}(t)=F\...
11
votes
0answers
367 views

A very useful lemma for Henstock-Stieltjes integration

I'd like to see a proof (or hints and outlines) for the following lemma, which is very useful to prove some interesting properties, including an Integration by Parts theorem for Henstock-Stieltjes ...
11
votes
0answers
126 views

Evaluate the following integral involving $\sin \pi x$

Let $F: \Bbb{R} \to \Bbb{R}$ be defined by $$F(s)=\begin{cases}1, & \text{if }s\ge \dfrac12 \\[0.2cm]0, & \text{if }s< \dfrac12 \end{cases}$$ I need to evaluate $$\int^{1}_{0} F(\sin \pi ...
11
votes
0answers
621 views

norm of a singular integral operator

My question is from Harmonic Analysis, about the study of singular kernels (in the Calderon Zygmund sense.) Suppose that a kernel $K$ is a singular kernel, extending to a bounded operator on $L^p$, ...
11
votes
0answers
586 views

Egorov's theorem for this Lebesgue integral

I want to prove Egorov's theorem using this Lebesgue integral defined by the upper integral $$\int^*f:=\left\{\int h ; h \ge f \text{ and h upper-continuous }\right\}$$ $$\int_*f:=\left\{\int h ; h \...
11
votes
0answers
350 views

Combinatorial Proofs of Real Analysis Identity

In this question, a proof using real analysis is given of the following identity: $$ \sum_{n=1}^{\infty} \frac{(n-1)!}{n \prod_{i=1}^{n} (a+i)} = \sum_{k=1}^{\infty} \frac{1}{(a+k)^2}$$ Is there a ...
11
votes
1answer
2k views

Equality condition in Minkowski's inequality for $L^{\infty}$

I am trying to find out when equality holds in Minkowski's inequality for $L^{\infty}$ (i.e. a necessary and sufficient condition for equality). I did a search and there was a discussion for the case ...
11
votes
1answer
283 views

A Question on certain Hilbert space of continuous functions, and a characteristic of convergence in it

Define $T^k(\Omega)$, $\Omega$ an open subset of $\mathbb{R}^m$ (with a smooth boundary), as a space of function equivalance classes, with the norm defined as $$ \|f\|_{T^k(\Omega)}^2 = \|f\|_{L^2(...
11
votes
1answer
308 views

Approximating the area below average of a concave function

Given a non-decreasing concave function $f:[0,1]\rightarrow \mathbb{R}^+$. Define \begin{align*} F(n)=\displaystyle\sum_{i=1}^n \min\left\{\frac{1}{n},\frac{f\left(\frac{i}{n+1}\right)}{n+1}\right\} ...
10
votes
0answers
44 views

A nice Combinatorial Identity

I am trying to show that $\forall N\in\mathbb{N}$, $$\sum\limits_{n=0}^{N}\sum\limits_{k=0}^{N}\frac{\left(-1\right)^{n+k}}{n+k+1}{N\choose n}{N\choose k}{N+n\choose n}{N+k\choose k}=\frac{1}{2N+1}$$ ...
10
votes
1answer
188 views

Trying to compute limit of singular integrals : $L= \lim_{s\to 1}(1-s)\int_{\Omega}\frac{(u(x)-u(y))}{|x-y|^{d+2s}} d y.$

Let $\Omega\subset \Bbb R^d$ be a bounded $C^1$ domain. Let $u:\Bbb R^d\to \Bbb R$ be a function in $C^2_b(\Bbb R^d)$. I would like to compute the following limit: for $x\in \partial \Omega$ $$L= \...
10
votes
0answers
116 views

On an asymptotic improvement of AMM problem 11145 (April 2005)

Motivation Motivated by this question, I tried improve the inequality $$\sum_{k=1}^{n}\dfrac{k}{a_{1}+a_{2}+\cdots+a_{k}}\le2\sum_{k=1}^{n}\dfrac{1}{a_{k}}$$ asymptotically. In other words, with ...
10
votes
0answers
100 views

proof verification: $f(x) = 1/x$ is not uniformly continuous on the open interval (0,1).

I've written a proof that $f\left(x\right)=\frac{1}{x}$ is not uniformly continuous on the interval $(0,1)$ and would like to know if it is correct. Here's what I've got. In order to show a ...
10
votes
0answers
106 views

Symbolic approximation through integration by parts

This is a slightly soft question. Suppose I have an integral $f(x) =\int_a^x g(t) dt $ which cannot be expressed in terms of elementary functions. One might still be able to integrate by parts to get ...
10
votes
0answers
81 views

Understanding the Legendre transform

In physics, I've seen the Legendre transform motivated by "changing the variable $x$ of a function $x \mapsto f(x)$ to the variable $u = \frac{df}{dx}$." I don't quite see what that means and why the ...
10
votes
0answers
149 views

How to get the idea of the formula for the mean value property for the heat equation

From the mean-value property of the Laplace's equation, we have the following mean-value property: $$ u(x)=\frac{1}{a(n)r^n}\int_{B(x,r)}u\,dy. $$ But for the mean-value property of the Heat equation, ...
10
votes
0answers
371 views

Fréchet L-Spaces

NOTE: The question has now been posted on MathOverflow: Fréchet L-Spaces According to the paper The emergence of open sets, closed sets, and limit points in analysis and topology famous ...
10
votes
0answers
843 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.