Questions tagged [real-analysis]
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
1,376
questions with no upvoted or accepted answers
42
votes
0
answers
791
views
A kaleidoscopic coloring of the plane
Problem. Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap D)=\...
30
votes
0
answers
869
views
Three real polynomials
Theorem. Let $f,g$ be two real polynomials, and suppose that their Wronskian $W(f,g)=f'g-fg'$ has only real roots. Then on any interval $I\subset\mathbf{R}$ containing no roots of $W$ every non-...
28
votes
0
answers
936
views
Number of real roots of a polynomial
Let $P\in \mathbb{R}[x]$ be a polynomial such that $(P, P') = 1$. Suppose that we want to calculate the number of real roots of $P$ in the interval $[a, b]$ (to simplify, let us assume that $P(a), P(b)...
23
votes
0
answers
922
views
A question about small sets of reals
In ZFC, does there exist an uncountable set of reals $A$ such that for every closed measure zero set of reals $B$, we have that $ A + B = \{a+b : a \in A, b \in B\} \neq \mathbb{R}$?
This question is ...
21
votes
0
answers
633
views
A multiple integral
Let us consider the multiple integral
$$I_{n}=\int_{-\infty }^{\infty }ds_{1}\int_{-\infty}^{s_{1}}ds_{2}\cdots
\int_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos {(s_{1}^{2}-s_{2}^{2})}\;\cdots
\cos {(s_{2n-1}...
21
votes
0
answers
1k
views
Almost everywhere differentiability for a class of functions on $\mathbb{R}^2$
A while ago, I came across the following problem, which I was not able to resolve one way or the other.
Let $f,g\colon\mathbb{R}^2\to\mathbb{R}$ be continuous functions such that $f(t,x)$ and $g(t,...
20
votes
0
answers
622
views
Is $\sum_{n=1}^\infty \frac{n!}{n^n}$ rational?
Is $\displaystyle \sum_{n=1}^\infty \frac{n!}{n^n}$ rational?
This question has been posted in MSE for two years without an answer. A094082 seems to suggest that it is not rational. Is it still an ...
19
votes
0
answers
765
views
A Linear Order from AP Calculus
In teaching my calculus students about limits and function domination, we ran into the class of functions
$$\Theta=\{x^\alpha (\ln{x})^\beta\}_{(\alpha,\beta)\in\mathbb{R}^2}$$
Suppose we say that $...
18
votes
0
answers
1k
views
Does there exist a continuous open map from the closed annulus to the closed disk?
(Originally from MSE, but crossposted here upon suggestion from the comments)
In this MSE post, user Moishe Kohan provides an example of a non-continuous open and closed ("clopen") function $...
18
votes
1
answer
2k
views
Function of two sets intersection
Let $U$ be the set of all nonempty subsets of $[0,1]$ that are a union of finitely many closed intervals (where an "interval" that is a single point does not count as an interval). Does ...
15
votes
0
answers
463
views
Quantitative Skorokhod embedding
The Skorokhod embedding theorem says that any random variable $X$ with $\mathbb E X=0$ and $\mathbb E[X^2]<\infty $ can be written as $X=B_{\tau }$ where $B$ is a Brownian motion and $\tau $ is a ...
15
votes
0
answers
740
views
Prove $\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge {\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$
I would like to prove that
$$\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge
{\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$$
for any $\omega > 0$ and $...
15
votes
0
answers
404
views
Is there a continuous map $f:\mathbb R^\omega\to\mathbb R^\omega$ with dense countable preimage $f^{-1}(\mathbb Q^\omega)$?
Let $\mathbb Q^\omega_0:=\{(x_i)_{i\in\omega}\in\mathbb Q^\omega:\exists n\in\omega\;\forall m\ge n\;\;x_m=0\}$ and observe that $\mathbb Q^\omega_0$ is a countable dense set in $\mathbb R^\omega$ (...
14
votes
0
answers
708
views
Lower bounds on analytic functions connected to Fox H
The question is related to the one I asked before and never got an answer to. Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$ . I need to demonstrate that the ...
14
votes
0
answers
481
views
Lebesgue density 1/2 (or bounded away from 0 and 1)
From the work of Preiss, we know that in infinite-dimensional spaces, one has violations of the Lebesgue density theorem. In particular, he has constructed examples of probability spaces where a set ...
14
votes
0
answers
625
views
Classes of (non-continuous) functions with the fixed point property
Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$.
...
13
votes
0
answers
539
views
Is there a logical relationship between constructions of the reals and proof methods in real analysis?
In my elementary real analysis course three years ago, I remember noting that there seemed to be 3 main ways of proving the main theorems about continuity. There was Bolzano-Weierstrass, continuous ...
13
votes
0
answers
376
views
Converse to Riesz-Thorin Theorem
Let $T$ be an operator on simple functions on (say) $\mathbb{R}$.
The Riesz-Thorin interpolation theorem, in one form, says that the Riesz type diagram of $T$ is a convex subset of $[0,1]\times[0,1]$....
12
votes
0
answers
428
views
Uniform closure of subspaces of Baire class 1
Describe a uniformly closed linear subspace $A \subset C([0,1])$ such that the space $B_1(A)$ is not uniformly complete. Here $B_1(A)$ is the set of all bounded functions $f$ which are pointwise ...
11
votes
0
answers
520
views
Is every Baire metric space a complete metric space in disguise?
I am currently giving lectures in real analysis and a student asked an interesting question I couldn't answer, so I'm posting it here:
Let's say that a metric space $X$ is Baire if every countable ...
11
votes
0
answers
2k
views
A question on trig series
Assume $\{a_k\}_{k\ge1}$ is a real sequence such that $u(x) = \sum_{k\ge 1}a_k\sin(kx)$ is a smooth function, and for every $x \in [-\pi, \pi]$
$$\left(\sum_{k\ge 1}\frac{a_k}{k}\sin(kx)\right)\left(\...
11
votes
0
answers
319
views
Does any real function have a Lipschitzian restriction on $D$?
Does any real function have a Lipschitzian restriction on $D$, where $D$ is an infinite subset of $\Bbb R$ with an accumulation point?
11
votes
0
answers
136
views
Assymptotics of a Selberg type integral
Does any one know some references/ ideas on how to study the assymptotics as $N$ goes to $\infty$ of the following Selberg type integral
$$\int _{\mathbb R^N} e^{-|x|^2}\ \prod_{1\le i<j\le N} \...
10
votes
0
answers
263
views
On the infinity of $\{p\in \mathbb {N}:\exists n\in\mathbb{N}~p| \left \lfloor{r^n}\right \rfloor\}$
I've already asked this same question on MSE here, but didn't get much help, so I will try on this site as well.
For which $r\in\mathbb{R}$ is the set $\mathscr{P}_r=\{p \in \mathbb{P}:\ (\exists n\...
10
votes
0
answers
367
views
Concerning Luzin-(N)-property
Definition: a function $f:\mathbb{R}\to \mathbb{R}$ has Luzin-(N)-Property if $f$ maps any null set to a null set.
By https://www.encyclopediaofmath.org/index.php/Luzin-N-property, it is known that ...
10
votes
0
answers
164
views
Maximizing an integral w.r.t. a measure on the unit sphere
I would like to know if the answer to the following question is known.
Let $d \ge 3$. What is the value of
$$
\theta(d) := \max_{\mu} \int_{S^{d-1}} \int_{S^{d-1}} \cdots \int_{S^{d-1}} |x_1 \...
10
votes
0
answers
783
views
Witt's proof of Gelfand-Mazur / Ostrowski's Theorem
Previously asked on Math Stackexchange without answers.
Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the ...
10
votes
0
answers
313
views
Does antidifferentiability of continuous functions imply Dedekind completeness?
Let $R$ be an ordered field, and let $I$ be {$x \in R: a < x < b$} for some $a < b$ in $R$. Define notions of $R$-continuity and $R$-differentiability for functions $f : I \rightarrow R$ by ...
10
votes
0
answers
436
views
Evaluating Shintani cone zeta functions
Hi everyone
I am trying the evaluate sums of the form
$$ \sum_{n_1>0,n_2>0,\ldots,n_m>0} \frac{1}{\big((a_{1,1}n_1 +\ldots +a_{1,m}n_m)^k \ldots (a_{m,1}n_1+ \ldots +a_{m,m}n_m)^k\big)}$$
...
9
votes
0
answers
1k
views
How complicated can an elementary antiderivative get?
I asked this question on MSE here.
I recently learned that there are many very large numbers that have been defined, such as $\operatorname{TREE}(3)$ and many others that are too big to be written ...
9
votes
0
answers
473
views
Does the intersection of middle third and middle half Cantor sets contain an irrational number?
Let $C_\frac{1}{3}$ be the middle third Cantor set, that is, the set of real numbers in the interval $[0,1]$ which can be written in base $3$ using only digits $0$ and $2$.
Likewise let $C_\frac{1}{2}$...
9
votes
0
answers
161
views
Changing coordinate to smoothen a function
Let $U\subset \mathbb{R}^2$ be an open neighborhood of the origin $0$, and let $f:U\to \mathbb{R}$ be a continuous function which is smooth on $U\setminus\left\{0\right\}$. Let's say that $f$ is ...
9
votes
0
answers
301
views
A game of harmonic series(s)
Given a set $A\subseteq\mathbb{R}_{>0}$, consider the following (two-player, perfect-information, length-$\omega$) game $H_A$:
Players $1$ and $2$ alternately play strictly increasing natural ...
9
votes
0
answers
329
views
Can one prove Rademacher’s theorem via the rising sun lemma?
The classical Rademacher’s theorem states that Lipschitz continuous functions on $\mathbb R^n$ are differentiable almost everywhere.
In dimension one, a stronger result holds - it can be shown that ...
9
votes
0
answers
494
views
On Riesz criteria for Riemann hypothesis:
Marcel Riesz defined a function :
$R(x) = \sum_{n=1}^\infty \frac {(-1)^n x^n} {\zeta(2n)\Gamma(n)}$
The Riemann hypothesis holds if $R(x)= O( x^{1/4 + {\varepsilon}}$)
For any $\varepsilon$
We have ...
9
votes
0
answers
293
views
Constructing an infinite chain of subsets of 'hyper' algebraic numbers?
This question is cross posted from MSE.
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 ...
9
votes
0
answers
177
views
Infinite series identities in search of a proof
This comes in relation to the Fishburn numbers.
I stumbled on the following relation for which I ask a proof if true.
Let $Q_i(z):=1-(1-t)^{i-1}(1-zt)$. Then
$$\sum_{n=0}^{\infty}\frac{(n+1)zt}{...
9
votes
0
answers
567
views
A standard name for a function satisfying the intermediate value theorem?
Do you know any (standard) name for a function $f:\mathbb R\to\mathbb R$ having the following weak intermediate value property:
$(*)$ for any connected subset $C\subset \mathbb R$ and points $a,b\...
9
votes
0
answers
910
views
Strong convexity of the trace of the square root of a matrix function
Any clues about how to prove that the following function is strongly-concave in $x$? (We conjecture it is $2$-strongly concave but cannot prove it. We have already proved strict concavity through ...
8
votes
0
answers
223
views
The approximate mean value theorem / Rolle's theorem in pure constructive mathematics
In the replies of this very similar question, there is a fascinating answer that is beautiful in its simplicity. In particular, it seems to use perhaps the most minimal assumptions one can possibly ...
8
votes
1
answer
186
views
Sequential colimit of iterated quotients of Cauchy sequences
We work in constructive mathematics.
The sets and functions in the foundations form a Grothendieck topos, which means that all colimits exist, and in particular, that all sequential colimits exist. ...
8
votes
0
answers
280
views
Is there a real-analytic approach to evaluate a definite integral (with an elementary integrand) whose value involves Lambert $W$?
I have never seen a real-analytic approach to evaluate integrals of the form below
$$\int_a^b\text{elementary function}(x)\,dx=\text{constant non-trivially involving}\,W(\cdot)\tag1$$ The elementary ...
8
votes
0
answers
380
views
Non-affine smooth transformation of Gaussian is Gaussian
Suppose $Z\sim N(0,1)$ (standard Gaussian) and $f: \mathbb{R} \to \mathbb{R}$ is a differentiable function such that $f(Z)\sim N(0,1)$. My question is whether there exists any such $f$ other than $f(x)...
8
votes
0
answers
513
views
Concave and other bounded functions: Series representation and converging polynomials
Main Question
Suppose $f:[0,1]\to[0,1]$ is continuous, polynomially bounded, and belongs to a large class of functions (for example, the $k$-th derivative, $k\ge 0$, is continuous, Lipschitz ...
8
votes
0
answers
305
views
How to prove that $ \sum_{m=0}^{\infty} { \Gamma\{(1+2m)/\alpha\}\over \Gamma(1/2+m)} { (-t^2/4)^{m}\over m !} \ge (\alpha/2)^{3}\exp(-t^{2}/4) $
I would love to prove the following inequality
$$
{1\over \sqrt{\pi} } \sum_{m=0}^{\infty}
\Gamma\{(1+2m)/\alpha\}
{ (-t^2)^{m}\over (2m) !}=$$
$$
\sum_{m=0}^{\infty}
{ \Gamma\{(1+2m)/\alpha\}\over \...
8
votes
0
answers
251
views
Structural Stability on Compact $2$-Manifolds with Boundary
I'm studying the structural stability of vector fields and I'm interested in learning about this phenomenon on compact $2$-manifolds with boundary.
Let $M^2$ be a compact connected 2-manifold and $\...
8
votes
0
answers
110
views
Connected component optimization
For an open set $A\subset[0,1]^d$, denote the connected components of $A$ by $cc(A)$. Given a smooth symmetric function $f\colon[-1,1]^d\to\mathbb R$ with $f(0)>0$, I am interested in the ...
8
votes
0
answers
192
views
Concavity of product and ratio of sums
Apologies if this question is not appropriate for MathOverflow. I have asked at Math.StackExchange without success.
Consider the function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ defined as
$$
f(x)=\...
8
votes
0
answers
417
views
Heisenberg group: function without vertical derivative
Let $\mathbb H$ be Heisenberg group with vector fields
$$
X=\partial_x - \frac12y\partial_t,\quad Y=\partial_y + \frac12x\partial_t,\quad T=\partial_t
$$
and $U\subset\mathbb H$ is an open set.
I am ...
7
votes
0
answers
146
views
The space of analytic associative operations
This question is a follow-up to this old one of mine.
Let $\mathcal{A}$ be the set of functions $\star:\mathbb{R}^2\rightarrow\mathbb{R}$ which are associative and $C^\omega$ (real analytic entire) in ...