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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57 views

Prove $\int_\Omega \left(\rho_{1} \ln \frac{\rho_{1}}{\rho_{2}}\right)dx dy \leq C\int_\Omega |\rho_1-\rho_2|dxdy$ for $0 \le \rho_1, \rho_2 \in L^1$

Let $\rho_1, \rho_2 \in L^1(\Omega;\mathbb R_+)$ such that $\int \rho_i|\ln \rho_i| < \infty$. Is it true that there exists a constant $C>0$ such that \begin{align*} \int_\Omega \left(\rho_{1} \...
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0answers
92 views

A kind of reflection formula for the logarithmic derivative of the zeta function

So I was messing around with Bernoulli numbers and values of $\zeta'$ at integers $-$ and suddenly I came about a non trivial identity which can be written in terms of the logarithmic derivative of ...
5
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1answer
456 views

Is this infinite product entire?

Let $(z_i)$ be a square-summable sequence which is even summable but not absolute summable, i.e. $\sum_{i=1}^{\infty} \vert z_i \vert = \infty$,$\sum_{i=1}^{\infty} \vert z_i \vert^2 < \infty$ and $...
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0answers
235 views
+50

On the solvability of a nonlinear differential system

A nonlinear formulation of differential Galois theory was discussed here and here for three dimensional nonlinear systems. For a two dimensional system, the following system of differential equations ...
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1answer
59 views

How to solve this minimax matrix optimazation problem?

Recently, I want to know how well can a $\ell_1$ ball be approximated by the image of a $\ell_2$ ball under a linear transform. I formulate this problem as the following optimization problem. \begin{...
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2answers
132 views

Does my construction always result in a stationary Poisson point process of intensity $1$? How so?

My construction is as follows: Let $X_k$ be a real-valued continuous random variable centered at $k$ (an integer), having distribution $F_k(x,s)$ where $k$ is the location parameter and $s$, a ...
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0answers
28 views

Normed spaces linear operators [closed]

I have this problem to solve that my professor gave me during class, We defined the operator norm as $\sup{\|Tv\|/ \|v\|}< \infty$ where the first norm is in $ W$ and the second one in $V$, $T:V\to ...
18
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1answer
216 views

Functions of $\mathbb{R}^d$ preserving convexity of sets

Consider a function $f : \mathbb{R}^d \to \mathbb{R}^d$, with $d\geq 2$, such that: $f$ is injective, For any convex set $A$ of $\mathbb{R}^d$, $f(A)$ is also convex. What can we say about $f$ ? In ...
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3answers
290 views

Generating function of product of binomial coefficients

Let $m, n\in \mathbb{N}$ and $|x| < 1$. I look for hints to derive an analytic formula for $$f_{m,n}(x) = \sum_{k \in \mathbb{N}} {n + k \choose k} {m + k \choose k} x^{k}. $$
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2answers
109 views

Extending a discrete singular kernel

Let $\{\phi(n)\}_{n\in\mathbb Z}$ be a sequence of complex numbers with the following properties: $\phi(0)=0$ and $|\phi(n)|\leq \frac{C_1}{|n|}$ for all $n\neq 0$ and $C_1>0$ is independent of $n....
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1answer
163 views

Volumes of unit balls in $S^2 \times\mathbb R$ and $H^2 \times\mathbb R$

The following integrals are equal to the volume of a unit ball in $S^2 \times \mathbb R$ and $H^2 \times\mathbb R$, respectively: $$8\pi\int_0^1\sin^2 \frac{\sqrt{1-h^2}}2 \, dh$$ $$8\pi\int_0^1\sinh^...
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1answer
132 views

Finding closed forms/related constants to a limit involving tetration

I was working on finding a series expression for a function $f: \mathbb{C} \rightarrow \mathbb{C}$ such that $f(x^y) = f(x)^{f(y)}$ along the way for construction of such a function I came across a ...
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1answer
201 views

Infinite series with inverse trigonometric functions

Consider the infinite series $$ F(s)=\sum_{k=1}^\infty \frac{1}{k^{2s+1} \sin(k \pi \sqrt{2})} $$ Prudnikov Vol.1 , gives a result for this sum in 5.4.16.13 for s=1 $$ F(1)=-\frac{13 \pi^3}{360 \sqrt{...
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1answer
138 views

Discrete singular integrals

Let $\{\phi(n)\}_{n\in\mathbb Z}$ be a sequence of complex numbers with the following properties: $\phi(0)=0$ and $|\phi(n)|\leq \frac{C_1}{|n|}$ for all $n\neq 0$ and $C_1>0$ is independent of $n....
3
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0answers
31 views

Reproducing kernel Hilbert space of Matérn kernels

I am trying to read a recent paper titled "Interpolation and learning with scale dependent kernels" by Pagliana, Ruidi, De Vito, and Rosasco. (The paper can be found on ArXiv) On the top of ...
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0answers
161 views

Singularity of inverse exponential integral function

The exponential integral function is defined by $$ Ei(z) = \int_{-\infty}^z dw \frac{e^w}{w} \,.$$ Away from the negative real axis the exponential integral function has a Taylor series about $z=0$: $$...
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55 views

A sub-logarithmic complexity in Analysis and N.Th

The question will be about complexity $\ \mathcal C(p)\ $ being positive and the same for all primes $\ p.$ Function $\ \mathcal Q\ $ is defined in the set of finite sequences of positive rational ...
4
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1answer
88 views

Long list of exactly solvable nonlinear SDEs

In P. E. Kloeden & E. Platen (1995). Numerical Solution of Stochastic Differential Equations. pg.118, they go over some special cases of nonlinear SDEs $dX_t=\alpha(t,X_t)\,dt+\sigma(t,X_t)\,dB_t$ ...
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1answer
68 views

Fundamental of a signal

Consider the space $S$ of real functions with the norm $$\|f\|^2 = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} e^{-x^2/2} f^2(x) ~\mathrm{d}x, $$ or any reasonable Euclidean norm such that bounded ...
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0answers
37 views

Explicit expression for the Poisson kernel solving the Dirichlet problem for geodesic balls

Let $X$ be a Riemannian manifold with the Laplace-Beltrami operator denoted by $\mathscr L$ and we look at its geodesic balls say $B$. Let $u$ be a continuous function on the geodesic sphere which is ...
3
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2answers
218 views

Change of variables in Riemann–Stieltjes integral

I want \begin{equation} \int_a^b f(g(x))dg=\int_{g(a)}^{g(b)} f(t)dt,\tag{$\heartsuit$}\label{heart} \end{equation} where $g$ is a continuous but not necessarily monotone (or bounded variation) ...
4
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1answer
124 views

Are there a geometry behind the singularity formation in solutions to nonlinear ODE's?

Disclaimer: This question was originally posted in math.stackexchange.com and, after 30 days with no answers, I followed the instructions of this topic. If we take two apparently simple first order ...
1
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1answer
139 views

What functions do we need to solve linear second order differential equations with polynomial coeficients?

Disclaimer: This question was originally posted in math.stackexchange.com. I also followed the instructions on this topic. I'm now trying to understand how can a ordinary differential equation be ...
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0answers
55 views

Function $v \in W^{1,2}(0,L)$ minimizing $\int_0^L f|u'|^2/\int_0^L f|u|^2$ and satisfying $\int_0^L fu = 0$ must satisfy $v'(0) = v'(L) = 0$

In this paper: https://arxiv.org/pdf/1110.2960.pdf on page 6, specifically in the proof of Lemma 3.1, the author states that a function $v$ achieving the infimum $$\lambda = \inf_{u \in W^{1,2}(0,L)\\\...
4
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1answer
169 views

On some convergence theorems by Felix E. Browder (1967)

I have been reading Felix E. Browder's Convergence Theorems for Sequence of Nonlinear Operators in Banach Space and I was hoping I could find answers to a couple of questions I have about the paper. ...
1
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2answers
52 views

Construct suitable cutoff function

Let $\bar x \in \mathbb R$. Is there a cut-off function such that $\phi_\epsilon \in C^\infty(\mathbb R)$, $0 \le \phi \le 1$, and $$\phi_\epsilon(x) = \begin{cases} 1 &\text{ if } |x - \bar x| \...
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0answers
59 views

Airy-type integrals (with different power $\neq 3$)

I am looking for integrals closely related to the Airy function \begin{eqnarray} && A_1= \int _0^\infty x \sin \Phi dx \nonumber \\ && A_2= \int _0^\infty \cos \Phi dx \nonumber \\&...
1
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1answer
134 views

Does the Implicit Function Theorem in Banach spaces holds if the differential is only one-to-one (not onto!)?

Is the Implicit Function Theorem in the following form correct: Let $V_1,V_2,W$ be Banach spaces, and $Ω⊂V_1×V_2$ an open subset containing $(x_0,y_0)$. Let consider a continuously differentiable map $...
0
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2answers
192 views

Asymptotic for eigenvalues for the following ode?

Consider the following Sturm-Liouville problem, $$(\sqrt{\sin \theta} Y')' + \lambda \sqrt{\sin \theta} Y =0$$ where $Y(\theta):[0,\pi] \to \mathbb{R}$ with boundary conditions $Y'(0)=Y'(\pi)=0.$ I ...
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1answer
50 views

Iterated integrations by parts using the fractional Laplacian

Let $u \in C^\infty_c(\mathbb{\Omega})$ and $\varphi$ be an eigenfunction of the fractional Laplacian $(-\Delta)^s$ in $\Omega$ with eigenvalue $\lambda$. In what sense, if any, is it true that $$\...
2
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1answer
58 views

How to compute this limit involving the associated Legendre function?

I am working on an eigenvalue problem whose general solutions involve the associated Legendre functions. Since the goal is to find bounded solutions, my question boils down to understanding the ...
3
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1answer
128 views

Strong convexity inequality w.r.t. infinity norm $\lVert\cdot\rVert_{\infty}$

Consider a twice differentiable 1-strongly convex function $f:\mathbb{R}^n \to \mathbb{R}$. Is it true that there exists $\alpha>0$ independent of $n$ such that, for all $x \in \mathbb{R}^n$: \...
4
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2answers
603 views

Is there a specific named function that is the inverse of $x+x^a$ for $x$ real?

This seems such a simple question that I fear I must have missed some elementary maths. I am looking for a way to solve $x+x^a = y$ by reference to an already defined function, $a,x,y > 0$ are real....
2
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0answers
132 views

Turán–Nazarov's lemma for algebraic polynomials?

Nazarov proved a version of Turán's lemma in Complete Version of Turan’s Lemma for Trigonometric Polynomials on the Unit Circumference, which is now known by the name Nazarov–Turán's lemma. A special ...
8
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2answers
398 views

Whitney extension theorem preserving monotonicity

This question is related to Monotone version of one-dimensional Whitney extension theorem. Let $m$ be a positive integer or $m=\infty$. Suppose that $E\subset\mathbb{R}$ is a closed set and $f:E\to\...
1
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1answer
167 views

2-dimensional Fourier transform

Let $k=(k_1,k_2) \in \mathbb{Z}^2$. Let $\lambda=(\lambda_1,\lambda_2)\in [0,2\pi]^2$ and $F(\lambda)$ be a bounded real function of $\lambda\in [0,2\pi]^2$. I am interested in the following equation:...
1
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1answer
42 views

Minimum norm polynomial subject to equality constraints

I had asked this question in the math stackexchange about a year ago. I did not get any response, so I am asking it here. Given distinct points $x_1,\dots,x_m \in [0,1]^n$ and real numbers $y_1,\dots,...
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0answers
62 views

Algebraic ode of exponential generating series

Let $G(z)$ be a rational function. So if we have a series $$S(x):=\sum_{n}a_n x^n $$ where $$ a_n = \prod_{i=1}^{n}G((i-1)h) $$ We can conclude that the series satisfies a Linear differential ...
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1answer
139 views

A continued J fraction for $a_n = \frac{1}{(n+1)^2}$?

The following is called a J continued fraction: $$\cfrac{\alpha_0}{1+a_0x-\cfrac{b_1x^2}{1+a_1x-\cfrac{b_2x^2}{1+a_2x-\cdots}}}$$ where the constants are real numbers. Let $\alpha_n= \frac{1}{(n+1)^2}$...
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1answer
97 views

Computing global maximum

For $\lambda\in\mathbb{R}$, I want to find the expression of $f(\lambda)$: $$f(\lambda)=\max_{E\in\mathbb{C}}arccosh{\frac{|E^2+i\lambda E|+|E^2+i\lambda E-4|}{4}}-arccosh{\frac{|E^2-i\lambda E|+|E^2-...
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0answers
82 views

A question on continued J-fraction

Consider the following two continued fractions $A$ and $B$: $$\frac{\alpha_0}{1+a_0x-\frac{b_1x^2}{1+a_1x-\frac{b_2x^2}{1+a_2x-\cdots}}}$$ $$\frac{\beta_0}{1+c_0x-\frac{d_1x}{1+c_1x-\frac{d_2x}{1+c_2x-...
2
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0answers
65 views

Stability test for LTV systems by differential Lyapunov inequalities

Consider a linear time-varying system: \begin{equation} \dot x(t) = A(t) x(t), \tag{$*$} \end{equation} where $A(t)$ is a time-varying block matrix defined as $$ A(t) = \begin{bmatrix} 0 & I\\ -\...
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0answers
66 views

Linear differential equation hypergeometric series

I am studying the following series $$\sum_{d\geq 1}\left(\frac{x^d}{d \, h^d} \sum_{k=1}^d (-1)^{d-k} \frac{1}{(k-1)! \, (d-k)!}\prod_{i=1}^d G((k-i)h)\right).\tag{*} $$ It's proven that (*) satisfies ...
1
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1answer
114 views

If $f \circ u \in BV$ and $f$ is strictly monotone, then is $u \in BV$?

Let $f: \mathbb R \to \mathbb R$ be a Lipschitz strictly monotone (so, in particular, invertible) function. Let $u: \mathbb R \to \mathbb R$. If $f \circ u \in BV$ can we conclude that $u \in BV$?
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218 views

Can one prove Rademacher’s theorem via the rising sun lemma?

The classical Rademacher’s theorem states that Lipschitz continuous functions on $\mathbb R^n$ are differentiable almost everywhere. In dimension one, a stronger result holds - it can be shown that ...
4
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0answers
111 views

Techniques for showing non-degeneracy results (PDE)

Motivation: Consider the equation, $$-\Delta u = u^p$$ in $\mathbb{R}^n$ for $n\geq 3$ and $p=2^*-1.$ Then we know that this equation has unique positive solutions given by functions of the form $U_{a,...
4
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1answer
113 views

Estimate on $C^1$-norm of solution of the Dirichlet problem for the Laplace equation

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with $C^\infty$-smooth boundary. Let $\phi\in C^\infty(\partial \Omega)$. Let $u$ be the solution of the Dirichlet problem of the Laplace equation \...
1
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1answer
96 views

Local dimension of measures

For a Borel prob measure $\mu$ in $\mathbb{R}^n$, define the local dimension of $\mu$ at $x$ by $$ {\rm dim}_*(\mu, x)=\liminf_{r\to 0} \frac{\log \mu(B(x,r))}{\log r}, {\rm dim}^*(\mu, x)=\limsup_{r\...
5
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1answer
179 views

Existence of genus 0 solution for linear ordinary differential equation

This question is about the linear differential equations with polynomial coefficients. I am interested in the necessary and sufficient conditions for the existence of genus 0 for linear differential ...
0
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0answers
67 views

How to understand the following limit exist: $\lim_{n\to \infty} \frac{1}{n} f(X)^Tf(X)=?$

I read one paper about Approximate Message Passing algorithm and I am confused about the notations. If I have a matrix $X\in\Bbb R^{n\times d}$ and a function $f: \Bbb R^{n\times d} \to \Bbb R^{n\...

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