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13 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, ...
18 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, ...
  • 6,354
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 ...
  • 1,687
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 $a(0)=3$, $...
  • 5,377
0 votes

Existence for a singular Sturm-Liouville "eigenvalue" problem with non-homogeneous boundary condition

Write your differential equation as $Ph = \lambda h$, the differential operator is unbounded and symmetric on the Hilbert space $\mathcal{L}_N$, with an appropriate domain. As analyzed by Amadori-...
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 \...
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 $...
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
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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 ...
  • 5,377
3 votes

Second order differentiability of convex functions

The following argument is used in my recent paper with D. Azagra and A. Cappello (work in progress). The answer is yes. I asked about whether the proof of Theorem 1 can be modified so to cover Theorem ...
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 ...
4 votes
Accepted

Does gravity constant affect boundedness of solution?

$\newcommand\la\lambda$No. E.g., if $n=1$, $x_0\ne0$, and $f(u)=-u^2/2$ for all real $u$, then the solution $$x(t)=\frac{x_0}{2} \, \Big(\frac{e^{\la_+ t}-e^{\la_- t}}{\sqrt{4 g+1}}+e^{\la_- t}+e^{\...
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. ...
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)^{...
  • 54.4k
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)...
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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