## New answers tagged ca.classical-analysis-and-odes

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,
...

- 81.9k

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 ...

- 85.3k

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-...

- 18.2k

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
\...

- 85.3k

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 $...

- 1,123

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 ...

- 81.9k

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 ...

- 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 ...

- 24k

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 ...

- 81.9k

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^{\...

- 85.3k

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 ...

- 85.3k

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|<...

- 81.9k

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)...

- 85.3k

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\...

- 81.9k

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