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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33 views

Space related to the powers of the fractional Laplacian

Let $\Omega \subset \mathbb R^n$ be a smooth domain. Is it possible to find (non-zero) functions $f$ belonging to the space $$X:=\left\{u\in L^2(\Omega): \sum_{k \ge 0}|\lambda_k^K\langle u, \phi_k\...
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1answer
78 views

Build an explicit "small perturbation" of the identity satisfying some properties

How can I build (i.e. find an explicit formula) a smooth function $f_\epsilon: \mathbb R \to \mathbb R$ depending on a parameter $\epsilon >0$ which is "almost the identity" but constant ...
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121 views

Closed form series for reciprocal cubic function

consider a cubic of the form f(x)=$x^3-2x+z$ Is it possible to derive a power series of coefficients for the function $x^y/f(x)$, for some $y=0,1,2...$ that does not require the use of Faà di Bruno'...
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1answer
71 views

A proof for the existence of smooth solution of PDE in form $\Delta u=f(x, u)$ in Michael E. Taylor's book Partial Differential Equations III

This part is from page 107 in Michael E. Taylor's book Partial Differential Equations III. In this part, we want a proof for the existence of smooth solution of the PDE $\Delta u=f(x, u)$ on $U$ with ...
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87 views

General version of maximum modulus principle [closed]

Suppose $f$ is holomorphic on domain $D$ containing closed unit disk. We have also given $|f(0)| =|f(e^{i\theta})|$ for all $\theta\in [0,2\pi)$. Then by using the maximum modulus principle we can ...
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2answers
151 views

Asymptotics of a delay differential equation

Say we have the line segment $L(t) = [0,t]$, and randomly remove open intervals of length $1$ from $L(t)$ until no more open intervals of length $1$ remain. Define $u(t)$ as the expected measure of ...
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1answer
56 views

Inductive proof that $\dot{M}_{n+1}=-M_{n+1}+W^{(n+2)}(0)+vM_{n+2}$

The motivation for the following is to convert the integro-differential equation \begin{equation} \kappa\ddot x+\dot x=-kx+\beta\int_{-\infty}^t W'(x(t)-x(s))e^{s-t}ds, \end{equation} into a ...
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1answer
138 views

Criteria for operators to have infinitely many eigenvalues

Normal compact linear operators on Hilbert spaces have infinitely many (counting multiplicities) eigenvalues by the spectral theorem. For non-normal operators this no longer has to be true. There ...
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121 views

Legendre-Fenchel transform

Suppose $F:\mathbb R^n\to \mathbb R$ is a convex continuous function. Moreover, for any $x\in \mathbb R^n$, $$ \limsup_{\lambda\to\infty} \frac {|F(\lambda x)|}{\lambda}<\infty. $$ I would like to ...
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1answer
131 views

On the fundamental solution for elliptic PDE

In the well known paper by Littman-Weinberger-Stampacchia "Regular points for elliptic equations with discontinuous coefficients", the authors were able to prove the validity the following ...
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2answers
2k views

The source of the Integral

Wolfram alpha calculates the integral $$\int\limits_0^\infty \frac{x^2\ln{x}}{e^x-1}dx=2\zeta^\prime(3)+3\zeta(3)-2\gamma\zeta(3).$$ However, I need to cite the source of this identity (the table of ...
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1answer
90 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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110 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
504 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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297 views

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 (proof is on pages 6 – 10). For a two dimensional system, the following system ...
4
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1answer
63 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
173 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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1answer
222 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
304 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
116 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
165 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
134 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
209 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
140 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....
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0answers
33 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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167 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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56 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
89 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
72 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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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 ...
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2answers
226 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) ...
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1answer
134 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 ...
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1answer
143 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 $v'(0) = v'(L) = 0$ [closed]

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)\\\...
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1answer
172 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. ...
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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 \\&...
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1answer
135 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 $...
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2answers
193 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
51 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 $$\...
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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
130 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
610 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
133 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 ...
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2answers
399 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\...
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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:...
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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
63 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 ...
1
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1answer
140 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
123 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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