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2 votes
Accepted

Baire class $1$ functions and Baire's characterization theorem

I had a similar problem some time ago and I had the impression that a general review on Baire classes is in fact missing, and would be useful. Probably you already saw it, but in this chapter there is ...
1 vote

Representation of the Dirac delta function

Here is a (quick) derivation that $(\ast\ast)$ is the Dirac delta. I notice $$\frac{\epsilon\theta(x)}{x^{1-\epsilon}}= \frac{d}{dx}(x^\epsilon)\theta(x)\,.$$ so that for $\epsilon>0$ $$\int_{\...
3 votes
Accepted

An inequality for harmonic functions

Consider first the $d=2$ case. Then, $u$ is a real part of an analytic function. We can write $$u(z)=\frac12\sum_{n=0}^{\infty}(a_nz^n+\overline{a}_n\overline{z}^n)$$ and $$\partial_\nu u(z)=\frac12\...
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4 votes

Representation of the Dirac delta function

Your transformation (*) is related to the Mellin transform (with $\epsilon=s$). In particular, your result is obtained as the limit $s\to0^+$ of Ramanujan's master theorem $$ \frac{\sin(\pi s)}{\pi}\...
11 votes

Representation of the Dirac delta function

As usual in such examples, there is no need to integrate against a test function. One can simply use the fact that if a sequence (or net) of distributions converges in the distributional sense, then ...
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6 votes

Representation of the Dirac delta function

$\newcommand\ep\epsilon\newcommand\de\delta\newcommand\R{\mathbb R}$Consider any family of nonnegative measurable kernels $K_\ep\colon(0,\infty)\to\mathbb R$ for $\ep\in(0,\infty)$ such that $\int_0^a ...
12 votes

What is the origin/history of the following very short definition of the Lebesgue integral?

This approach was used in the German Analysis (Calculus) textbook MR0222221 Hans Grauert and Ingo Lieb, Differential- und Integralrechnung. Band I: Funktionen einer reellen Veränderlichen, ...
17 votes
Accepted

What is the origin/history of the following very short definition of the Lebesgue integral?

This definition is due to Jan Mikusiński, see Mikusiński, Jan, The Bochner integral. Basel, Stuttgart: Birkhauser, 1978. Mikusiński has co-authored another book on integration with Hartman in 1961, ...
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0 votes

In choiceless constructivism: If $f'=0$ then is $f$ constant?

Here's a stab at the question that attempts to make use of the locale of real numbers. (If someone has published something similar, please point me towards it). The reasoning behind this is that if ...
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1 vote

Derivative of the absolute value

Here is a result that proves part 3. It is copied from the paper: P. Hajłasz, J. Maly, Approximation in Sobolev spaces of nonlinear expressions involving the gradient Ark. Mat. 40 (2002), 245--274. ...
1 vote
Accepted

Derivative of the absolute value

Part 1 follows from the Cauchy-Schwartz inequality, applied to the two vectors $(R(x), I(x))$, $(\nabla R(x), \nabla I(x))$. Part 2 follows from the simple inequality $|\partial_j R(x)| \leq \sqrt {\...
  • 1,729
0 votes

Uniqueness of solutions of Young differential equations

For such small $\beta$, we need to use Rough path theory to make sense of the integral and so below I go over that. (Indeed for $\beta\in (\frac{1}{2},1]$, there is an ODE theory for Young integrals ...
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2 votes
Accepted

Reference request: "Tangent relation" in metric spaces

One reference: Elisabeth Burroni and Jacques Penon, A metric tangential calculus. Theory and Applications of Categories 23 (2010), 199–220. The first sentence of the paper says that a fuller account ...
  • 26.1k
2 votes

A ball with slit at the radius is not $W^{1,1}$-extension domain

If $\Omega$ was a $W^{1,1}$-extension domain, then restrictions of $C_c^\infty(\mathbb{R}^n)$ functions to $\Omega$ would be dense in $W^{1,1}(\Omega)$ since they are dense in $W^{1,1}(\mathbb{R}^n)$. ...
  • 1,808
1 vote

Elementary convexity example

Here is a much simpler proof, actually of the more general fact that $$f(x):=x^p(1+\ln^+ x)^s$$ is convex in $x\ge0$ for any real $p\ge1$ and $s\ge0$, where $\ln^+ x:=\ln\max(1,x)$. For the left and ...
4 votes
Accepted

On existence of a concave function

Such a function doesn't exist for some choices of $a$. For notational purposes I will change the unit interval by $[-2,2]$. Consider a $C^\infty$ function $a:[-2,2]\to\mathbb{R}$ such that $a(0)=3$, $...
  • 5,357
20 votes
Accepted

Ruling out the existence of a strange polynomial II

Just a slight modification of the previous example: $$f(x,y):=\left(5 y^2+5 y+1\right) \left(x^2+y^2\right)+\left(10 y^2+10 y+1\right) y^2.$$ Your conditions $f(0,0) = 0$ and $$f(a,b) > 0 \text{ ...
50 votes
Accepted

Ruling out the existence of a strange polynomial

The polynomial $f(x,y)=(x^2+1)(5y^2+5y+1)\in\mathbb{Z}[x,y]$ is an example. Note that $5y^2+5y+1>0$ for $y\in\mathbb{Z}$, but $5y^2+5y+1<0$ at $y=-\frac{1}{2}$.
6 votes

For which Sheaf topoi is Brouwer's fan theorem true?

The classic 1979 paper Fourman & Hyland, Sheaf models for analysis, has a number of results along these lines. They show in particular in Theorem 3.2 that every spatial topos, i.e. topos of ...
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2 votes
Accepted

Property of sets of positive Lebesgue measure in $\mathbb{R}^2$

Firstly, a set $P$ of positive measure need not contain anything of the form $A\times B$, for example consider for some $k\in\mathbb{R}\setminus\{0\}$ the set $P=\{(x,y)\in\mathbb{R}^2;x-ky\not\in\...
  • 5,357
2 votes
Accepted

Inequality with decreasing rearrangement and non-decreasing function

$\newcommand\la\lambda\newcommand{\R}{\mathbb R}\newcommand{\de}{\delta}$The answer to this question is yes. Indeed, let $h:=f^*/g$. Then $h$ is a nonincreasing function and the inequality in question ...
21 votes
Accepted

The $9$th tetration of $-\sqrt2$

This is not a huge coincidence: the idea is that the sequence $a_n={}^{n}(-\sqrt{2})$ has small norm until $n=6$, then it gets out of hand for $n=7$ ($a_7\sim-33+29i$), so that $a_8=e^{a_7\ln(-\sqrt{2}...
  • 5,357
3 votes

Status of the fundamental theorem of algebra for the locale of real numbers

I can see two issues of non-constructivity here, but I feel that they are both noise rather than central to the mathematics: The leading coefficient of the polynomial could fail to be apart from zero, ...
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6 votes
Accepted

Inequality with decreasing rearrangement function

No (if $c$ cannot depend on $f^*$ or $g$). Indeed, let $h:=f^*/g$. Then $h$ can be any positive function and the inequality in question can be rewritten as $$lhs:=\Big(\int_0^\infty h(s)^{p'}ds\Big)^{...
3 votes
Accepted

What is the optimal asymptotic behavior of this integral over the sphere?

$\newcommand\la\lambda\renewcommand{\S}{\mathbb S}\newcommand{\si}{\sigma}$Let us show that \begin{equation*} J_\la=e^{-\la(m+o(1))} \tag{1}\label{1} \end{equation*} (as $\la\to\infty$), where \...
1 vote

The field structure on the locale of real numbers

Richard Dedekind's original paper does not include a construction of multiplication. In The Dedekind Reals in Abstract Stone Duality, Andrej Bauer and I go into considerable detail about all of the ...
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2 votes
Accepted

On partial absolute continuity

The notation $L^2$ for Lebesgue measure is confusing. I denote $\lambda$ and $\lambda_2$ the Lebesgue measure on $\mathbb{R}$ and $\mathbb{R}^2$, respectively. The answer is no. Fix a discrete measure ...
1 vote

Can the Riemann integral be defined through a closure/completion process?

The process described in this comment actually works, and defines Riemann-Loomis integration for functions valued in Banach spaces. When the Banach space is finite-dimensional, a function is Riemann-...
  • 32.5k
3 votes

Integrability of derivatives

The function $0\neq x\mapsto x^2\cos\frac1{x^2}$, $0=x\mapsto0$, is everywhere differentiable with a derivative that is not Riemann integrable, not Lebesegue integrable, but Denjoy--Perron--Luzin--...
2 votes

Reference request: "Tangent relation" in metric spaces

I don't know whether there are "good" references to what you are actually asking but at least in some kind of implicit sense this kind of "tangency" was already considered by ...
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16 votes

Does there exist a continuous function $f(x)$ such that $f(0)=0$ and $0<\lim_{n\to\infty}\prod_{k=1}^n f(k/n)<\infty$?

If you do not require monotonicity of $f$, the construction is pretty simple and is a combination of a few facts we normally (should) teach in elementary number theory and Fourier analysis classes. ...
  • 54.4k
2 votes

Does there exist a continuous function $f(x)$ such that $f(0)=0$ and $0<\lim_{n\to\infty}\prod_{k=1}^n f(k/n)<\infty$?

Possible way to find such a function $f$ Since $$\prod_{k=1}^n \frac{ke}{n} = n!\Big(\frac{e}{n}\Big)^n \sim \sqrt{2\pi n} \text{ as } n \to +\infty,$$ it is sufficient to find a continuous function $...
4 votes
Accepted

The field structure on the locale of real numbers

There are several (equivalent) way to go about it: You can start form the fields operation on $\mathbb{Q}$ and use that they are "locally uniformly continuous" to extend them by continuity ...
  • 35.1k
1 vote

The field structure on the locale of real numbers

In constructive mathematics its common to use formal topologies rather than locales, so you might have more luck searching the literature for "formal topology" rather than "locale."...
  • 3,271
1 vote

Does there exist a continuous function $f(x)$ such that $f(0)=0$ and $0<\lim_{n\to\infty}\prod_{k=1}^n f(k/n)<\infty$?

Comment Let $S_n = \prod_{k=1}^n f(k/n)$ and $g(x) = \log f(x)$. Then $$ \frac{1}{n} \log S_n = \frac{1}{n}\sum_{k=1}^n g\left(\frac{k}{n}\right) $$ which should [improper integral, so not certain] ...
  • 38.5k
3 votes
Accepted

Quantitative analytic continuation estimate for functions small except on a small set

This conjecture is correct. Take $K=e$, and let $\gamma\leq 1/4$; we will fix $\gamma$ later. First we give a crude estimate of $c_0$. Let $g(z)=\sum_{1}^\infty c_nz^n.$ Since $|c_n|\leq e^n$, we ...
1 vote
Accepted

Continuity of a reaching time of a submanifold

If $V$ is closed in $\mathcal{O}$, then $\tau^V$ is continuous at $x_0$. If $V$ is not closed, then it is not hard to find counterexamples (e.g. imagine $\mathcal{O}=\mathbb{R}^3$, $V$ is an open disk ...
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0 votes
Accepted

Smooth extension of functions at corners

$\newcommand\ep\varepsilon$Take any $\ep\in(0,1/2)$. Let $$A:=A_0\cup A_1\cup F,$$ where $A_0:=[0,\ep)\times B$, $A_1:=(1-\ep,1]\times B$, $F:=[0,1]\times\{0\}$, and $B$ is the unit ball (you did not ...
4 votes
Accepted

Entire function with almost periodic boundary condition?

The answer seems to be negative. Suppose that an entire function $f$ satisfies $f(z+v_i)=e^{A_iz}f(z)$, where $v_1$ and $v_2$ generate a lattice. Let $\Pi$ be the fundamental parallelogram of this ...
5 votes

Quantitative analytic continuation estimate for a function small on a set of positive measure

$\newcommand\ep\varepsilon\newcommand{\de}{\delta}\newcommand{\R}{\mathbb R}$As shown in the post by Alexandre Eremenko, the answer to this question is yes if $C=1$ (and hence if $C\le1$). On the ...
2 votes
Accepted

Construction of the Lipschitz function with a given Lipschitz constant, given two values and with small Lipschitz norm

I can achieve $L(f - g) \leq (\frac{1}{2} + \frac{\pi}{4})\epsilon = (1.285\ldots)\epsilon$. Two reductions: (1) we can assume $|f(t)| < c$ for all $t \in (a,b)$ and (2) we can take $\epsilon = 1$. ...
  • 39.2k
2 votes

General version of Weyl's lemma

unfortunately Weyl lemma cannot be generalized to any divergence form elliptic operator. The problem is given by the smoothness of the coefficients $a_{ij}$. There is a huge literature on the subject. ...
22 votes
Accepted

Is there an analytic non-linear function that maps rational numbers to rational numbers and it maps irrational numbers to irrational numbers?

Answering a question of Erdos, Barth and Schneider proved that for every countable dense sets $A$ and $B$ in the complex plane, there exists an entire function such that $f(z)\in B$ if and only if $z\...
16 votes

Is there an analytic non-linear function that maps rational numbers to rational numbers and it maps irrational numbers to irrational numbers?

Let $q_n$ be a numbering of the rationals, with $q_1=0$. We can define a non linear, analytic function $F:\mathbb{R}\to\mathbb{R}$ which is strictly increasing and that maps $\mathbb{Q}$ surjectively ...
  • 5,357
7 votes

Quantitative analytic continuation estimate for a function small on a set of positive measure

The answer depends on $C$. For example, for $C=1$ it is positive. Your estimate $|f^{(m)}(0)|\leq m!$ implies that $|f_n(z)|\leq 1/(1-|z|).$ Take $|z|=1/2$, you conclude that $|f_n(z)|\leq 2,\; |z|<...
11 votes
Accepted

Quantitative analytic continuation estimate for a function small on a set of positive measure

Unfortunately, no, as requested: Take any sequence $\delta_j\in(0,1)$ decaying to $0$, choose small $\mu_j>0$ such that $\prod_j \delta_j^{\mu_j}=e^{-1}$ and put $f_n(z)=e^n\prod_j B_{\delta_j}(z)^{...
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2 votes
Accepted

Construction of the Lipschitz function with a given Lipschitz constant and given two values

$\newcommand\ep\varepsilon$Yes, it is easy to construct a counterexample here. Indeed, if $g_\ep$ is such a function for each given real $\ep>0$ (so that $|g_\ep| \geq c$, $g_\ep(a)=f(a)$, $g_\ep(b)...
2 votes

Discontinuous functions without removable discontinuities

I claim that one can always get rid of all removable discontinuities of a function to obtain a function without removable discontinuities. For generality, we shall work in the framework of general ...
4 votes
Accepted

Bounds on zeros of rational function

Let us drop the assumption $x_j\in[1,2]$, it is not needed. Proving the result by contradiction, denote our function by $f_N$, suppose that $f_N(z_N)=-i$, and $\mathrm{Im}\ z_N= 1/(N^2R_N)$ where $R_N\...

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