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

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-3
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0answers
75 views

Is a surjective mapping of R2 to itself with full rank derivative everywhere injective? [on hold]

Let $f:\mathbb R^2 \rightarrow \mathbb R^2$, and $rank(\frac{df}{dx}) = 2$ everywhere. If $f$ is surjective $f$ necessarily injective? Also, what if $f$ maps $\mathbb R^{2+}$ (i.e. $\{x_1>0, x_2&...
0
votes
0answers
67 views

some strange sums ramanujan type

i found sum as $$\sum _{k=0}^{\infty } \frac{e^{k x} \left(-\frac{1}{y}\right)^k}{p-e^{k x}}=\sum _{n=1}^{\infty } \left(\frac{-1+\, _2F_1\left(1,\frac{2 i n \pi -\log \left(-\frac{1}{y}\right)}{x};\...
2
votes
0answers
89 views

Decay of the Fourier transform of a surface area measure

Let $\mu$ be a surface area measure of a manifold $M\hookrightarrow\mathbb{R}^{n+1}$. If $M$ is the unit sphere $S^n$, it's known that surprisingly the Fourier transform of $\mu$ decays: $$|\hat\mu(\...
0
votes
0answers
48 views

Min-max theorem - Subspaces over $H_0^1$ [closed]

I am trying to understand what is going on (in using the Minimax Principle; Weyl's law) in the Dirichlet eigenvalue problem $$\left\{ \begin{aligned} -u''(x)&=\lambda u(x)\quad 0<x<\...
1
vote
0answers
70 views

Can we always extract a proper Hausdorff measurable subset from a Hausdorff measurable set?

I also put this question on MSE here Let $\Gamma\subset \Omega\subset \mathbb R^N$ be such that $\mathcal H^{N-1}(\Gamma)<+\infty$ (this also implise that $\Gamma$ is Hausdorff measurable). Let $\...
0
votes
1answer
201 views

The spherical harmonics are the EIGENVECTORS of Beltrami operator [closed]

In the well-known book "THE PRINCETON COMPANION TO MATHEMATICS" page 296, it is indicated that the spherical harmonics are the EIGENVECTORS of the Beltrami operator. In the document Spectral Geometry ...
5
votes
1answer
62 views

Optimal constant for a Sobolev-type inequality

Let $\overset{\circ}{H^s}(\mathbb T)$, where $s\ge 0$, be the space zero average of $2\pi$-periodic functions $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx},$ such that $$ \lvert u\rvert_s = \...
1
vote
0answers
45 views

How much can the integrability at zero tell about the decay rate around zero? [migrated]

Suppose that $g$ is a continuous, nonincreasing and nonnegative function on $(0,1)$. The question is whether one can characterize the integrability of such functions at zero by their decay rates at ...
2
votes
0answers
121 views

Parametric normalized Fermat curves (Fermat functions)

The normalized Fermat curve is $X^n+Y^n=1$. We have of course infinite possibilities for parametrisation, but the periodicity is a special characteristic here. E.g. $\cos^2x+\sin^2x=1$ has ...
13
votes
1answer
416 views

Why sum of three squares of real polynomials is a sum of two squares?

If $f(x),g(x)$ are real polynomials, then $f^2+g^2+1$ is a sum of two squares of polynomials. This easily follows from Fundamental Theorem of Algebra, but is there an argument avoiding it? What are ...
0
votes
1answer
39 views

characterization of normality by selection theorem

The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $...
3
votes
0answers
49 views

Riesz basis of Paley-Wiener space

Let us consider the Paley-Wiener space: $$PW_\pi:=\{f\in L^2(\mathbb R)\cap C(\mathbb R),\ \operatorname{supp}\hat f\subset [-\pi,\pi] \}.$$ Let $\{\lambda_n\}_{n\in\mathbb Z}$ be a sequence of real ...
6
votes
1answer
122 views

The “Peano phenomenon” for differential equations

Consider the following statement: If $f:\mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, for the autonomous equation $$x' = f (x)$$ the "Peano phenomenon" can arise only at those values ...
5
votes
3answers
275 views

Nonlinear ODE: $y'=(1+axy)/(1+bxy)$

Consider the first order nonlinear ODE problem: $$ y'(x)=\frac{1+ay(x)x}{1+by(x)x}, \quad x>0 $$ where $a, b>0$ are some constants. I would like to know if these kind of equations were ...
3
votes
2answers
352 views

An integral identity evaluating the gamma function

While reading a number theory paper I encountered the identity $$ \int_{- \infty}^{\infty} (1 + x^2)^{ - \frac{z}{2} - 1} dx = \sqrt{\pi} \frac{ \Gamma(\frac{z + 1}{2}) }{\Gamma(\frac{z}{2} + 1)},$$ ...
3
votes
0answers
81 views

Optimal Kakeya Maximal Bound for Bushes

Let $\{T_{\alpha}\}$ be a collection of $1\times\cdots\times 1\times N$ tubes, where $N\gg 1$, with maximal $1/N$-separated directions, which all are centered at the origin (i.e. they form a bush). In ...
0
votes
1answer
76 views

Behavior of a Solution of a Nonlinear ODE

In my work, I encountered the following equation: $$ (a'(x)+1)^2+k^2(x) a^2(x)=1,\;\;k(x)=2 {\mbox{sech}}(x). $$ I would like to know as much as possible about the solution. More particularly, I would ...
14
votes
2answers
307 views

Needing proof of convergence for a sequence

Let $\left\{u_i\right\}_{i=1}^\infty$ be a sequence of real vectors (i.e. $u_i\in R^n, i=1,2,... $) and $m$ an integer large enough such that $\sum_{i=1}^m u_i u_i^T$ is a positive definite matrix. ...
5
votes
1answer
150 views

Domain of Laplacian

Let $L$ be an operator on $C^2(\mathbb R)$, defined by $$L \phi (x) = \int_{|y|<1} (\phi(x+y) - \phi(x) - \phi'(x) \ y)\ \nu(dy), \text{ for all } x\in \mathbb R$$ for a measure $\nu(dy) = |y|^{-2} ...
1
vote
1answer
96 views

Is there a way to solve this integral equation?

I have ran into the following integral equation as part of my phd research project, trying to enforce a boundary condition of a parabolic pde problem. For $\xi = (\alpha\theta)^{1/\alpha}$ and for ...
0
votes
0answers
24 views

diffusion and potentials in several dimensions

In a Lagrangian field theory there are conserved quantities, like total energy. This allows geometric arguments to be made about the behaviour of a system with known potential. (E.g. on asymptotic ...
2
votes
1answer
82 views

Smooth dependence on the initial condition of the integral of an ODE

I am considering an ODE $\dot{x}=f(x)$, with $x\in\mathbb{R}^d$ and $d<\infty$. $f$ is a $C^k$ function and I denote by $\Phi_t x$ be the value of the solution at time $t$. I assume that my ODE ...
2
votes
1answer
223 views

Generalizations of the Euler-Maclaurin Summation Formula

I'm using the Euler-Maclaurin formula in a research I'm working on. However brilliant is the elementary proof found here, I need and want to know more about it. Namely Generalizations to broader ...
2
votes
1answer
177 views

Is there a matrix that converts the gradient of every possible function to gradient of other function?

I have already asked this question on math.stackexchange.com http://math.stackexchange.com/questions/1789476/is-there-a-matrix-that-converts-the-gradient-of-any-function-to-gradient-of-othe Now I ...
6
votes
1answer
214 views

Is the exponent $2$ sharp in the Balog-Szemerédi-Gowers Theorem?

The Balog-Szemerédi-Gowers theorem can be stated in the following form: let $A,B$ be subsets of $\mathbb{Z}/n\mathbb{Z}$ (say) with equal cardinality, such that $$ \|1_A*1_B\|_2 \ge K^{-1} \|1_A\|_1 \|...
2
votes
1answer
150 views

Ramanujan-type sum

Could you show that $$\sum _{k=0}^{\infty } \frac{k}{e^{\frac{\pi k}{2}}+1}=\frac{7 \pi ^2+6 \left(\psi _{e^{2 \pi }}^{(1)}(1)+\psi _{e^{2 \pi }}^{(1)}\left(1-\frac{i}{2}\right)-\psi _{e^{2 \pi }}^{(...
1
vote
0answers
103 views

Ramanujan sum type

I try to show $$\sum _{k=1}^{\infty } \frac{e^{-2 k} k}{e^{-2 k}+1}=\frac{\pi ^2}{48}-\frac{\pi ^2-6 \left(\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}(1)+\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}\left(\frac{-2 i+\pi }...
1
vote
0answers
45 views

Dirichlet series decomposition of arbitrary function

Originally asked on MSE here: http://math.stackexchange.com/q/1780149/52694 Analytic functions can be decomposed into a Taylor series, and furthermore the Taylor series converges back to the original ...
1
vote
0answers
56 views

History of Cauchy-Euler Equations

As I teach a class in ODE, and following this post and Rota's paper, I wandered what is the history of the research of - $\sum\limits_{k=0}^n a_k x^k y^{(k)}(x) = g(x),\quad \forall k=0,\...
3
votes
1answer
76 views

Algorithm for definite integral of rational functions of x and exp(-x)

I'd like to find the class of functions that may appear as definite integrals of a rational function of $x$ and $\exp(-x)$ from $0$ to $\infty$. For that I imagine there might be an algorithm to ...
3
votes
0answers
60 views

Factoring Bessel functions into an amplitude and a phase

Take some $\nu>0$. Let $J_\nu(x)$ be the Bessel function of the first kind. Let's restrict its domain to $\mathbb R^+$. Is it possible to find a pair of functions $A_\nu(x), \phi_\nu(x):\mathbb R^+\...
7
votes
1answer
150 views

$C^{k,\alpha}$ diffeomorphisms and vector fields

This feels like something I should know, but I have a hard time finding a definite reference. Let $M$ be a compact (Riemannian) manifold, $k\ge 1$ be an integer and $\alpha\in(0,1)$. When v is a $C^k$...
0
votes
1answer
77 views

Question on Littlewood-Paley trichotomy

In proving the product estimate, we need the Littlewood-Paley trichotomy. See http://www.math.ucla.edu/~tao/254a.1.01w/notes3.ps. In the decomposition $$P_k (fg)=\sum_{k',k''\in Z} P_k (P_{k'} f P_{k'...
0
votes
2answers
59 views

Functions with scalar times orthogonal Jacobian [duplicate]

I am interested in understanding functions $f:\mathbb{R}^d \rightarrow \mathbb{R}^d $ whose Jacobian at every point $x \in \mathbb{R}^d$ is a scalar times an orthogonal matrix. I've seen a similar ...
0
votes
1answer
82 views

Branches of the tetration function

Letting $\eta = e^{1/e}$ where $e$ is Euler's constant, there exists a function $F(z)=\, ^z \eta$ with the following relevant properties. (I won't bother showing the existence of this function, or the ...
7
votes
1answer
287 views

Rotation invariance of an integral

Consider the integral depending on 2 parameters $$f(\tau,x):=\int_{-\infty}^{+\infty}\frac{dp}{\sqrt{p^2+1}}e^{-\sqrt{p^2+1}\tau+ipx},$$ where $\tau >0,x\in \mathbb{R}$. This integral absolutely ...
3
votes
2answers
236 views

Nice way to express $H^{-1}(\mathbb{S}^1)$

I am looking for a good way to write down what $H^{-1}(\mathbb{S}^1)$ is. What do I mean by this: Well, certainly one can define this space via charts that map from the sphere into $\mathbb{R}^2$ and ...
3
votes
0answers
172 views

Modified Jacobi’s theta function

Be $t\in\mathbb{R}_0^+$. Jacobi’s theta function is $$\Theta(t):=\sum\limits_{k=-\infty}^{+\infty} e^{-\pi k^2 t}$$ with $$\Theta(\frac{1}{t})=\sqrt{t}\Theta(t)$$ Therefore $$\sum\limits_{k=1}^\infty ...
1
vote
0answers
34 views

convergence of ODE [closed]

I have 2 coupled linear ODEs. I used Mathematica to solve for analytical solution. But the analytical solution looks too complicated. I only need to derive some monotonicity property of the solution. ...
2
votes
3answers
240 views

Nonzero solutions of an infinite product

Let $-\frac{1}{2}\le a \le\frac{1}{2}$ and $b\in[0,\infty)$. Definitions: $$f_k(a;b):=\frac{(2k+\frac{1}{2}+a)^2+b}{(2k+\frac{1}{2}-a)^2+b}(\frac{k}{k+1})^{2a},$$ $$f(a;b):=\prod\limits_{k=1}^\infty ...
2
votes
2answers
90 views

Analytic continuation of a specific integral with respect to a parameter

The following integral absolutely converges for $Re(z)<0$ and is analytic in this domain: $$F(z):=\int_{0}^{+\infty}\frac{\sin r}{r}e^{\sqrt{r^2+m^2}z}dr ,$$ where $m>0$ is fixed. Question. To ...
1
vote
1answer
144 views

What are the best known bounds on the Hermite polynomials?

The best I could find on the net is this paper, http://arxiv.org/pdf/math/0401310.pdf Has this been improved?
2
votes
0answers
51 views

Harmonic functions in tempered distribution sense

Suppose $g$ is a metric on $\mathbb{R}^3$ and $\Omega \subset\subset \mathbb{R}^3$. We assume that $g$ is euclidean outside $\Omega$. My question concerns solutions to $\triangle_g u =0$ that are say ...
0
votes
0answers
28 views

Uniqueness and Properties of Nonlinear 2nd Order ODE with Asymptotically Constant Coefficients

I have the following differential equation: $$ V(z) = e^z [1-\varphi(\gamma)]+(\gamma-g)V'(z)+\frac{1}{2}\kappa^2 V''(z)$$ with $$ e^z \varphi'(\gamma) = V'(z)$$ where $\varphi(\cdot)$ is a ...
2
votes
0answers
42 views

1D inhomogeneous linear Schrodinger equation

I have the following problem: $iu_t - u_{xx} = f$ on the interval $[0,L]$ with $u(0,t)=u(L,t)=0$ and $u(x,0)=0$. I can show that $\|u\|_{L^2(x,t)}$ (space-time) is controlled by the norm $\|f\|_{L^2(...
2
votes
0answers
28 views

A question about Carleman linearization

Carleman linearization is a technique used to embed a finite dimensional system of analytic ordinary differential equations into an infinite system of linear differential equations:¹⁻² Let $f$ be ...
5
votes
4answers
407 views

How do the roots of a polynomial change when another polynomial is added?

I need to obtain an analytical solution to an equation of the following form: $$ (x-a)(x-b)(x-c)=d(x-e)(x-f), $$ where $a$, $b$, $c$, $d$, $e$, and $f$ are known numbers and $x$ is the variable. Of ...
1
vote
1answer
158 views

find solution of complex number recurrence equation

I have the following recurrence equation: $$(\mu\ n + \nu) f_{n} + J\Phi^{*} \sqrt{n+1}f_{n+1} + J\Phi\ \sqrt{n}f_{n-1} = 0$$ for complex numbers $f_{n}$ where $n = 0,1,2,3,...,\infty$ and complex $\...
2
votes
0answers
78 views

Is a one-dimensional unstable manifold of an ODE a union of the associated equilibrium point and two full orbits? [closed]

Consider an ordinary differential equation (ODE) system \begin{align} \frac{dx}{dt} = f(x) \end{align} where $x \in \mathbb{R}^n$ ($n \geq 2$) and the vector field $f$ is defined on an open subset $X$ ...
2
votes
1answer
142 views

Proof of Euler's reflection formula via rapidly decreasing Fourier series

Story I want to prove Euler's reflection formula by showing that \begin{equation*} f(s) = \sin(\pi s) \Gamma(s) \Gamma(1 - s) \end{equation*} is constant, where $s = \sigma + it$. It's easy to see ...