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.

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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?
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2 votes
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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
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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(...
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1 vote
1 answer
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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 $...
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3 votes
1 answer
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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 \...
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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. ...
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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,...
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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
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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
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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 $b_1 < b_2 < \dotsb < b_N \...
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6 votes
2 answers
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$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{...
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1 answer
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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}...
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1 answer
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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 ...
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1 vote
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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(...
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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(...
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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 $C^2$-...
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2 votes
1 answer
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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 ...
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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'...
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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 ...
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3 votes
1 answer
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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,\...
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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
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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 ...
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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 ...
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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}, \...
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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 $M$...
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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.
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1 vote
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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
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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 ...
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0 answers
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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 ...
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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} \...
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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{...
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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 ...
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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 $d\sigma$ ...
  • 411
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 ...
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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}(...
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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 ...

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