# Questions tagged [ca.classical-analysis-and-odes]

Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.

3,187
questions

0
votes

1
answer

40
views

### ODE with conditions within the interval

Can anyone please recommend some publications related to ODEs with non-initial, non-boundary conditions, but conditions for points inside the interval, on which the ODE is defined?

2
votes

0
answers

36
views

### Complex interpolation of Fourier-Lebesgue spaces and Lebesgue spaces

The complex interpolation of Lebesgue spaces $L^{p_0}$ and $L^{p_1}$ are well-known, that is,
$[L^{p_0}, L^{p_1}]_\theta = L^p$, where $(1-\theta)/p_0+1/p_1 = 1/p$ for some $\theta \in [0,1]$.
Now I ...

-1
votes

0
answers

52
views

### Commutativity of convolution and pointwise application of functions

Question:
is it possible to have for non-trivial functions $f,g,h$ examples of $f(\lbrace g*h\rbrace(t))\equiv \lbrace g*f\rbrace(h(t))$, i.e. that e.g. applying a function $f(x)$ to a time-series $h(...

1
vote

1
answer

74
views

### Why does failure of boundedness of this operator for $p<q$ implies its failure for $p>q^{\prime}$?

I am reading the paper "P.Sjolin, Convolution with Oscillating Kernels, Indiana University Mathematics Journal Vol. 30, No. 1 (1981), pp. 47-55" where $L^p-L^p$ boundedness of the operator
$...

3
votes

1
answer

323
views

### Curious infinite product, convergence, connection to prime numbers

I have been playing with the following function:
$$
f(x)=\frac{\pi x (1-x^2)}{\sin\pi x}\prod_{k=2}^\infty \frac{\sin(\pi x/k)}{\pi x/k}
$$
It is hard to get correct numerical values. I'll start with ...

4
votes

1
answer

228
views

+100

### Sum over Bessel functions: what is $\sum_{n=1}^\infty J_n(u)J_n(v)\sin(n\alpha)$?

By Neumann's addition theorem, we know that the following identity holds (including for complex $\alpha$):
$$J_0(u)J_0(v)+2\sum_{n=1}^\infty J_n(u)J_n(v) \cos(n\alpha) = J_0 \left( \sqrt{u^2+v^2-2uv \...

0
votes

0
answers

39
views

### A question on the paper "On the low-rank approach for semidefinite programs arising in synchronization and community detection"

I got stuck at theorem 15 when reading this paper "On the low-rank approach for semidefinite programs arising in
synchronization and community detection" by Bandeira, Boumal, Voroninski.
...

0
votes

0
answers

71
views

### Closed formula for iterated Fourier series

I'm trying to obtain a closed formula for the following integral.
\begin{align}
I_n = {} & \int_0^h \Bigr[\sum_{r_1=1}^\infty a_{1,r} \cos\left(\frac{2\pi}{h} r_1t_1\right) \\[6pt]
& {}+ b_{1,...

0
votes

1
answer

138
views

### Looking for English version of a paper of Jean Ginibre

I am in serious need of an English translation for the following paper:
Séminaire Bourbaki by Jean Ginibre, Le problème de Cauchy pour des EDP semi-linéaires
périodiques en variables d’espace, d'après ...

4
votes

1
answer

131
views

### How to compute the asymptotics of this oscillatory integral?

I posted this on Stackexchange but got no responses or comments.
Consider the following integral, for $\epsilon\ne 0:$
$$\displaystyle\frac{1}{(2\pi)^2\epsilon^4}\int_{\Omega}yb\,e^{\frac{i}{\epsilon}[...

1
vote

0
answers

64
views

### Functions $f: \mathbb R \to \mathbb R$ such that $\det [f(a_j-b_k)]_{j,k} \neq 0$ for all $a_1,b_1, \dots, a_N,b_N$ and all $N \in \mathbb N$

A function $f: \mathbb R \to \mathbb R$ is called totally positive if for every $N \in \mathbb N$, every $a_1< a_2< \dotsb < a_N \in \mathbb R$ and every $b_1 < b_2 < \dotsb < b_N \...

6
votes

2
answers

272
views

### $L^p-L^q$ boundedness of this simple singular oscillatory integral operator

Let $0<\alpha<1$ and define
$$Tf(x):=\int e^{\dot{\imath} x y} \frac{f(y)}{|x-y|^{\alpha}}dy.$$
The Hardy-Littlewood-Sobolev inequality characterizes $L^p-L^q$ boundedness of
$Hf(x):=\int \frac{...

0
votes

1
answer

58
views

### SDE with non-degenerate diffusion visits every point

I am asking an extension of the question here for SDEs of the Ito form.
Consider the SDE $dX_t =\sigma(X_t) dW_t$, where $W$ is a $d$-dimensional Brownian motion and $\sigma:\mathbb{R}^n\to \mathbb{R}...

0
votes

1
answer

51
views

### Given positive $\epsilon$ and $c$, find a density $\phi$ such that $t\phi(\epsilon/t) \ge c \|\phi'\|_\infty$ for all positive $t$

A nice density (on $\mathbb R$) is function $\phi:\mathbb R \to \mathbb R$ such that
(1) $\phi(x) \ge 0$ for all $x \in \mathbb R$,
(2) $\int_{-\infty}^\infty \phi(x) \mathrm{d}x = 1$,
(3) $\phi$ is ...

1
vote

0
answers

45
views

### Necessary and sufficient conditions so that $\sum_{k\ge 0}\frac{x^k}{\Gamma(1+a(k))}\sim\int_0^\infty \frac{x^t}{\Gamma(1+a(t))}\,dt$ as $x\to\infty$?

First, some notation. I'll write $f(x)=o(g(x))$ if $\lim_{x\to\infty} \left|\frac{f(x)}{g(x)}\right|=0$. I'll also write $g(x)=\omega(f(x))$ if $f(x)=o(g(x))$, i.e. $\limsup_{x\to\infty} \left|\frac{g(...

1
vote

1
answer

63
views

### When is it true that $\sum_{k\ge 0}\frac{x^k}{\Gamma(1+a(k))}\sim\int_0^\infty \frac{x^t}{\Gamma(1+a(t))}\,dt$ as $x\to\infty$?

First, some notation. I'll write $f(x)=o(g(x))$ if $\lim_{x\to\infty} \left|\frac{f(x)}{g(x)}\right|=0$. I'll also write $g(x)=\omega(f(x))$ if $f(x)=o(g(x))$, i.e. $\limsup_{x\to\infty} \left|\frac{g(...

2
votes

1
answer

109
views

### A possible characterization of subharmonic functions

Let $\Omega\subset \mathbb{R}^n$ be an open subset. Let $u\colon \Omega\to [-\infty,+\infty)$ be an upper semi-continuous function.
If $u$ is subharmonic then for any point $x\in \Omega$ and any $C^2$-...

2
votes

1
answer

148
views

### Exponential sum vs. exponential integral via Poisson summation

When we want to estimate an exponential sum
$$
\sum_{M<m\le M'}e(f(m))
\quad\text{with}\quad
1\le M\le M'\le 2M
\quad\text{and}\quad
e(x):=\exp(2\pi ix)
$$
where $e(x):=\exp(2\pi ix)$
and the phase ...

0
votes

0
answers

82
views

### Polynomial / quadratic autonomous system of ODEs – proving monotonicity / convexity

Problem:
Consider the autonomous ODE system
\begin{align*}
\dot{x} &= (1-x) (z-xy)\\
\dot{y} &= \tfrac 1 2 y^2 - (a+xy)(1-y) \\
\dot{z} &= \tfrac 1 2 z^2 - \tfrac 1 2 y^2 + (a+xy)z
\end{...

12
votes

2
answers

430
views

### Asymptotics of $\int_0^\infty \frac{x^{2z}}{\Gamma(1+z)}\,dz$ for large $x$

I'm interested in the asymptotics of
$$\int_0^\infty \frac{x^{2z}}{\Gamma(1+z)}\,dz$$
as $x\to\infty$. I expect the results to behave similarly to $e^{x^2}=\sum_{k\ge 0}\frac{x^{2k}}{k!}$. However, I'...

1
vote

1
answer

202
views

### Where or what is the general formula for the $n$th derivative of the power-exponential function $x^x$?

It is well-known that the power-exponential function $x^x$ and its first few derivatives are often taught in calculus.
Does the general formula for the $n$th derivative of the power-exponential ...

3
votes

1
answer

66
views

### A type of singular limit for systems of differential equations

Suppose I have a system of differential equations for the unknowns $(x_1,v_1,\ldots,x_N,v_N)$ (interpreted as the positions & velocities of $N$ labeled particles),
$$\begin{cases}\dot{x}_{i,\...

0
votes

0
answers

73
views

### Question about symmetric bilinear form and convex geometry

Consider a Finsler norm $\varphi$ in $R^4$, and a positive definite symmetric bilinear form $S$, there also exists a vector-valued symmetric linear form $Q$, such that if $S(v, w) = 0$, then $Q(v, w)$ ...

4
votes

0
answers

132
views

### Proximity of zeroes of Bessel functions

I have been running into a question for which I found no reference in the litterature. I do not have a strong background in number theory ; for me this is motivated by a question in PDEs (how close ...

4
votes

0
answers

211
views

### If a derivative is defined everywhere and $\pm1$ almost everywhere, is it constant?

Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that the set $A:=\{x\in\mathbb{R};f'(x)\not\in\{1,-1\}\}$ has measure $0$. Does this imply that $f'$ is constant?
Context: I was ...

3
votes

2
answers

100
views

### Floquet coefficients under time change

Let's consider two ODEs $\tag{1}\frac{du}{dt}=\gamma(u(t))\ F(u(t))$ and $\tag{2}\frac{dv}{d\tau}=F(v(\tau))$ where $f\in C^\infty(\mathbb R^n,\mathbb R^n)$ and $\gamma\in C^\infty(\mathbb R^{n}, \...

0
votes

0
answers

121
views

### Is $p^{-s}$ transcendental if $\zeta(s)=0$?

Let $K$ be a number field and $\mathcal{O}_{K}$ its ring of integers.
Let
$$\zeta_{K}(s)=\prod_{\mathfrak{m}}\frac{1}{1-\#(\mathcal{O}_{K}/\mathfrak{m})^{-s}}$$
be the $\zeta$ function associated to $...

7
votes

3
answers

483
views

### Rigorous estimates on roots of function

We consider the function
$$f(x) = 1- \frac{1}{N} \sum_{i=1}^N \frac{\sin\left(\tfrac{\pi i}{N}\right)^2}{1+\sin\left(\tfrac{\pi i}{2N}\right)^2-x}.$$
The arguments of the two sines differ by a factor ...

1
vote

1
answer

79
views

### $C^1$-extension theorem on cones preserving boundedness of derivatives

Let $K\subset \mathbb{R}^n$ be a closed convex cone with a nonempty interior. Let $f:K\to \mathbb{R}$ be a continuously differentiable function satisfying $\|\nabla f\|_\infty<\infty$ (if needed, ...

3
votes

2
answers

123
views

### Equivalence between two Sobolev norms on manifolds

On a compact Riemannian manifold $(M,g)$ without boundary, there are two ways to define a Sobolev norm on $M$. Assume that $f\in C^\infty(M)$ in the following.
Use pseudo-differential operators on $M$...

1
vote

2
answers

168
views

### What is the easiest way to prove the correctness of this inequality

I have the following inequality for some $0<x<0.1$: $$x^{1/10}-(1-(1-x)^{x^{-0.5}}) \geq 0$$
Is there an easy way to prove the correctness of such inequality? Thanks!

2
votes

1
answer

129
views

### Roots of rational function

Sorry, I asked a similar question yesterday which contained a mistake in the question posed, here is the real question.
Let $(x_n)_{n=1}^N$ be a sequence taking values in $[1,2]$ with the property ...

4
votes

0
answers

39
views

### Getting analytic center manifolds

The center manifold of a degenerate zero of an analytic vector field need not be unique nor analytic. But say I want it analytic. Does anyone know of additional conditions to be imposed on the ...

4
votes

1
answer

105
views

### Compositional inverse of Bessel function

Was ever studied a function $f$ which solves $J_0(f(x))=x$? Integral representations, natural domains of existence and whatever.

1
vote

0
answers

91
views

### Fourier transform of functions mapping manifolds, is there a definition?

$\DeclareMathOperator\SO{SO}$I have a problem which boils down to the analysis of functions of the form
$$
f : \mathbb{R} \to \SO(3)^n
$$
Since $\SO(3)$ is a compact group so is $\SO(3)^n$.
Now if ...

0
votes

1
answer

146
views

### Verifying the proof of a bilinear estimate in $L^2$

$\newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\ds}{\displaystyle} \newcommand{\Lpn}[2]{\left\lVert#1\right\rVert_{L^{#2}}}$
$\newcommand{\Lptxy}[3]{\left\lVert#1\right\rVert_{L^{...

4
votes

1
answer

135
views

### An algebraic inequality in three real variables

Is it true that
$$(v-u)^2+(w-u)^2+(w-v)^2 \\
+\left(\sqrt{\frac{1+u^2}{1+v^2}}
+\sqrt{\frac{1+v^2}{1+u^2}}\right) (w-u)(w-v) \\
-\left(\sqrt{\frac{1+u^2}{1+w^2}}+\sqrt{\frac{1+w^2}{1+u^2}}\right) (w-...

25
votes

2
answers

2k
views

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

Typical courses on real integration spend a lot of time defining the Lebesgue measure and then spend another lot of time defining the integral with respect to a measure. This is sometimes criticized ...

0
votes

0
answers

157
views

### How does one make sense of singular solutions to constant mean curvature equation?

Background:
Consider the following ODE:
$$\left(\frac{r^2 \dot{f}}{\sqrt{1+r^2(\dot{f})^2}}\right)' = c r$$
where $c$ is some positive constant (Lagrange multiplier), $f:[0,\infty)\to [0,\pi]$ is the ...

5
votes

1
answer

126
views

### On existence of a concave function

Let $a$ be a strictly positive $C^\infty$ smooth function on the unit interval. Does there exist a strictly positive $C^\infty$ smooth function $f$ on $I$ such that
$$ f’’(x) \leq 0\quad \text{and} \...

3
votes

0
answers

37
views

### What is the supremum of 1-dim Hausdorff measure of the nodal set of Neumann eigenfunction $u$ for planar convex domain

All descriptions of this question are limited to 2-dimension for simplicity. Recently, I read some papers on the nodal set of Laplacian eigenfunctions. Denote $\Sigma=\{u(x)=0\}$ be the nodal set of a ...

3
votes

0
answers

99
views

### A new arranging of discrete sine transform

Let $n$ be even and consider the discrete sine transform of type 5 which is the matrix
$$S=\left(\sin(k+1)(l+1)\frac{\pi}{n+\frac12}\right)_{k,l=0}^{n-1}$$
Let us denote by $s_{-,l}$ the $l^{\text{...

2
votes

0
answers

50
views

### Coupled 1-dimensional Allen-Cahn system

Suppose we solve Allen-Cahn on the interval $[-1,1]$
$$\epsilon^2 u_{xx} = u(u^2 - 1)$$
$$u(-1) = 0, \qquad u(1) = 0$$
For small $\epsilon$, such a solution is unique and can be chosen to be positive ...

0
votes

0
answers

59
views

### Stability of a special singular perturbation problem

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a lower bounded smooth function, i.e., $\inf_{x\in\mathbb{R}^n} f(x)>-\infty$. Consider the following singular perturbation problem:
$$\begin{cases}\dot{...

3
votes

1
answer

106
views

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

Let $k_{1},\dots, k_{d}>1$ be integers and consider the integral
$$J_{\lambda }=\int_{\mathbb{S}^{d-1}}e^{-\lambda \left(x^{2k_{1}}_{1}+\dots+ x^{2k_{d}}_{d}\right)} d\sigma(x)$$
where $d\sigma$ ...

24
votes

3
answers

1k
views

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

Does there exist a continuous function $f(x)$ such that $f(0)=0$ and $0<\lim\limits_{n\to\infty}\prod\limits_{k=1}^n f(\frac{k}{n})<\infty$ ?
I do not see any reason why such a function could ...

4
votes

1
answer

106
views

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

This question arises as a variation of this question, which was helpfully answered in the negative. It turns out that for my application, a substantially weaker conjecture suffices, which fails to be ...

1
vote

1
answer

120
views

### How to reduce the compact support to the case of small diameters in Tao's "A sharp bilinear restriction estimate for paraboloids"

I am reading Terence Tao's paper "A sharp bilinear restriction estimate for paraboloids"
to prove the bilinear restriction estimate on paraboloids. In Section 3, he assumes that $\text{diam}(...

3
votes

1
answer

143
views

### Does gravity constant affect boundedness of solution?

Consider a second order gradient-like system with linear damping
$$\ddot{x}+\dot{x}+\nabla f(x)=0, \quad x(0)=x_0,\quad\dot{x}(0)=0$$
Suppose $f\in C^2(\mathbb{R}^n)$ and $\inf_{x\in\mathbb{R}^n}f(x)&...

3
votes

1
answer

131
views

### Entire function with almost periodic boundary condition?

Let $v_1 =\lambda_1 \zeta_1$ and $v_2 = \lambda_2 \zeta_2$ with $\zeta_1 = \frac{4\pi i\omega}{3}$ and $\zeta_2 = \frac{4\pi i\omega^2}{3}$ where $\omega = e^{2\pi i/3}$ is the third root of unity and ...