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

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8
votes
2answers
479 views

Prove that the Dirichlet eta function is monotonic

Let us consider $\eta(p):= \sum\limits_{n=1}^\infty \frac{(-1)^{n+1}}{n^p}$ for $p>0$. Has anyone come along with an elementary proof that $\eta(x)$ is monotonically increasing on this set? By ...
3
votes
0answers
79 views

Operator theory of initial-value ODE problems

The theory of elliptic boundary value problems is usually treated from the perspective of functional analysis, and the theory of operators between Hilbert spaces. In contrast to that, the theory of ...
3
votes
1answer
396 views

the existence of a real polynomial satisfying the following property

It is easy to verify that $$ \frac{t}{2}\leq \frac{1}{4}+\frac{1}{4}t^2\leq \frac{t}{1-(1-t^2)^2}-\frac{t}{2} \quad \quad 0<t\leq1$$ I want to ask if there exist a real polynomial $h(t)$ such ...
1
vote
0answers
34 views

Does the following measurable Halmilton-Jacobian equation admit a Lipschitz solution?

I have the following question: Let $F:\Omega\times \mathbb{R}^n\to [0,\infty)$ be a convex Finsler norm, which means that $F(x,\cdot)$ is convex with respect to the second variable. $F(\cdot,v)$ ...
3
votes
2answers
228 views

Uniqueness of solutions to an ODE system

For each $i$ (up to infinity), let $u_i \in C^1(0,T)$ satisfy $$\frac{d}{dt}u_i(t) + \sum_{j=1}^\infty b(t;w_j,w_i)u_j(t) = 0$$ $$u_i(0) = u_i(T)$$ where $b(t;\cdot,\cdot)$ is an inner product on some ...
12
votes
3answers
809 views

Poincare lemma for non-smooth differentiable forms

The Poincare lemma is almost always formulated for differential forms with smooth coefficients (or sometimes for currents that have distributional coefficients). I would like to have it for ...
9
votes
3answers
317 views

$L^p$ norm means

Consider the unit sphere $S_p^{n-1}$ of an $L^p$ normin $\mathbb{R}^n.$ The question is: what is the expected value of the $L^q$ norm on $S_p^{n-1}?$ Since (I assume) this is intractable in closed ...
0
votes
2answers
187 views

A book about almost periodic functions [closed]

Can anyone give me suggestions for new books about Besicovitch's almost periodic functions? Thanks a lot.
1
vote
0answers
162 views

Solving polynomials of arbitrary degree

Is there any analogue of the method for expressing roots of polynomials of degree 5 with elliptic and η-functions that generalizes to polynomials of degrees n>5? More specifically: does there exist ...
0
votes
1answer
57 views

Extending point-wise bound to uniform bound

Suppose $f(t,x)\in \mathcal{C}^0([0,1]\times \mathbb{R}^n)$. Further suppose that for each $t$ $$ C(t):= \sup_{x\in\mathbb{R}^n} |f(t,x)|<\infty \, .$$ Does it follow that $f$ is bounded? Note ...
11
votes
2answers
1k views

How much can one say about this differential equation?

Consider the ODE $y^{\prime \prime}(x) = \cos(x) y(x)$ with boundary value conditions $y(0)=1$, $y(1)=2$. Solving it results in a linear combination of Mathieu functions, but what I find more ...
7
votes
1answer
239 views

A conjecture about the measure estimates of a trigonometric polynomial

Formulation of the Conjecture Let $\Omega =(0,\pi)\times (0,2\pi)\subset\mathbb R^2$ and let $\psi:\Omega\to \mathbb{R}$ defined by $$\psi(x,t)=\sum_{k\in S \,j\in S'} \sin(kx)\left( ...
4
votes
1answer
121 views

Are there superexponential Pfaffian functions?

This question is motivated by model theory, but it's really an analysis question (which means it may have an easy analysis answer that I just don't have the background for). Here's the main question, ...
0
votes
0answers
63 views

The trivility of Besov space for large parameter

For all $s>0$, $1\le p<+\infty$ and $u\in L^p(\mathrm{d}x)$, we define $$D_{s,p}(u)=\sup_{r>0}\frac{1}{r^{sp+n}}\int_{\mathbb{R}^n}\int_{B(x,r)}|u(y)-u(x)|^p\mathrm{d}y\mathrm{d}x$$ and ...
5
votes
0answers
194 views

Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?

Preliminary Definitions Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
1
vote
0answers
87 views

Laplacian mapping on various function spaces

I have a question related to a certain elliptic operator on $R^N$ but I think i can clarify my confusion if I just consider the Laplacian $\Delta$ on the unit ball in $R^N$. If $ 1 <p< ...
1
vote
1answer
89 views

Controling mixed derivatives

This is a cross-post from Math.SE since the question got nothing (but upvotes) even after offering a decent bounty. If it is too trivial or in other ways not suited for this site, please let me know ...
2
votes
2answers
123 views

Caratheodory equations

Ok, I am reading Fillipov book on discontinuous right hand side differential equations (the red book). He states the next lemma: " Let the function $f(t,x)$ satisfy the Caratheodory conditions and ...
2
votes
2answers
124 views

Some Questions from Reading on Wave Front Set from Hormander's Linear PDE Vol. 1

In Hormander's Linear PDE Vol. 1 (pg 252-253, before the definition of wave front set is introduced), Lemma $8.1.1$ says that if $\phi \in C_{0}^{\infty}$ and $v \in \mathcal{E}^{\prime}$, then ...
1
vote
0answers
76 views

Estimating convolutions of powers

I would like an asymptotic estimate of $$ \sum_{y \in \mathbb{Z}^d} \frac{1}{|y-a_1|^{d-1} \ldots |y-a_n|^{d-1}} $$ that does not involve any infinite summation. In order to lighten the notation, I ...
1
vote
0answers
172 views

Is there an asymptotic bound for this oscillatory integral?

I have an oscillatory integral: $$ \int u(x,y) e^{i\lambda f(x,y)} dx $$ with $f(x,y)\in \mathbb{C}^{\infty}$ a complex-valued function in a neighborhood of $(0,0)$ satisfying: $$ \text{Im} f \geq ...
1
vote
1answer
97 views

Solution of General Parametric Oscillator

I am wondering if there is a general solution for this ODE $\ddot X +2\gamma \alpha \dot X + (\alpha+S(t)) X = \beta $ the dot represents time derivative, and $\gamma>1$, so it is in the ...
6
votes
1answer
222 views

Spectrum of this ODE

I noticed something interesting studying this Sturm-Liouville Problem: $$ \frac{d}{dx}\left(\sqrt{(1-x^{2})}\frac{df}{dx} \right)+\frac{\left(n \alpha x+\alpha^2 x^{2} + ...
7
votes
1answer
313 views

Prove that …, f(x-2), f(x-1), f(x), f(x+1), f(x+2),… is algebraically linearly independent without the Fourier transform

Suppose that $f$ is in $L^2(\mathbb{R})$ and consider the set of integer translates of this function, $V=\{f(x-k):k\in\mathbb{Z}\}$. This set is linearly independent: taking the Fourier transform of ...
0
votes
0answers
62 views

A solution of a q-difference equation

Is it possible to find a solution of the $q$-difference equation $$f(q^{-1}x)-f(qx)=x(a-x)f(x),$$ with $f(0)=1$, (perhaps) in terms of basic hypergeometric series? Or in another rather explicit form? ...
5
votes
1answer
114 views

Is this graph of reciprocal power means always convex?

Let $$ p = (p_1, \ldots, p_n) $$ be a finite probability distribution, which for convenience I'll assume to have no zeroes: thus, $p_i > 0$ for all $i$ and $\sum_i p_i = 1$. Is the function ...
-3
votes
1answer
161 views

Derivatives of infinite order [closed]

Is there any sense of taking an infinite number of derivatives? Is it discussed in the literature? For example, can one make sense of $$\frac{\partial^{\infty}f(x_1,x_2,\cdots)}{\partial x_1 ...
3
votes
0answers
84 views

Variational Principle for a System of Differential Equations

I am studying a differential operator of the form $$ L\left(\begin{array}{c} u \\ v \end{array}\right) = -\Delta \left(\begin{array}{c} u \\ v \end{array}\right) + V(x)\left(\begin{array}{c} u \\ v ...
1
vote
1answer
92 views

Positive Definiteness of a certain function

Recall that a function $f: \mathbb{R}^d \longrightarrow \mathbb{C}$ is positive definite, iff for all numbers $N$ and $x_1, \dots, x_N \in \mathbb{R}^d$, the matrix $(a_{ij})$ with entries $$a_{ij} = ...
2
votes
1answer
89 views

M-Wright function asymptotics

Let $M(z;\nu):= \frac{1}{\pi}\sum_{n=1}^{\infty} \frac{(-z)^{n-1}}{(n-1)!}\Gamma(\nu n)\sin(\nu n\pi)=\frac{1}{2\pi i}\int_{\text{H}_a}\exp(\sigma -z\sigma^{\nu})/\sigma^{1-\nu} d\sigma$, ...
1
vote
0answers
31 views

Backlund transformation related to two NL differential equations

I'm looking for a Backlund transformation linking the following two nonlinear differential equations for real $t$: $$\dfrac{d^2}{dt^2}f(t)=\cos\left[f(t)\right]$$ ...
0
votes
0answers
79 views

Integral of Bessel function of 1st kind with complex exponential

Does someone know the solution (simple closed form) of one of theses integrals: $$\int_0^t J_l(s) e^{-iA(t-s)}ds$$ $$\int_0^t \frac{J_l(s)}{s} e^{-iA(t-s)}ds$$ with $l>0$, $t>0$, $\Re(A)>0$, ...
0
votes
0answers
90 views

elliptic regularity when right hand side in weak $L^p$

I am interested in the following question (whose answer i assume is well known) but just not by me. Suppose $u,f$ are smooth functions defined on $B_1$ and $ \Delta u = f$ in $B_1$ with $u=0$ on $ ...
9
votes
2answers
575 views

Functions that Calculate their $L_p$ Norm

are there any examples of functions $f:x\in\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ and intervals $(a,b), 0\le a \lt b \le \infty$ , for which $$\Big(\int_a^b{|f(x)|^p dx}\Big)^\frac{1}{p} = f(p)$$ ...
2
votes
1answer
115 views

A question on existence of solutions of a linear ODE system

I am working on a problem of harmonic functions on surfaces, and in one step I got the following system of ODEs with prescribed asymptotes. I was wondering what methods could give us the existence or ...
2
votes
0answers
60 views

Holder continuity of Poisson equation with divergence free drift

I am interested in the following PDE. Suppose $u_m$ is a smooth solution of a elliptic equation of the form $$ -\Delta u_m(x) + a_m(x) \cdot \nabla u_m(x) = f_m(x) \qquad B_1 $$ with $ u_m=0 $ on ...
1
vote
1answer
83 views

A Non-homogeneous, Linear (Matrix) System of ODEs: What's Known About it? [closed]

Consider the following system of ODEs $$ Y^{'}(t) = - \left[ A Y(t) + Y(t) A \right] + B(t) , $$ where $Y(t)$,$A$,$B(t)$ are all matrices, with the properties $A=A^T$, $Y=Y^T$. $Y(t)$ is the matrix ...
1
vote
1answer
167 views

All solutions to a set of integral equations

I would like a better understanding of the set of pairs $(f_1,f_2)$ of functions $[0,1] \times [0,1] \to [0,1]$ which satisfy the following conditions: For all $y \in [0,1]$, $f_1(x,y) \geq ...
10
votes
0answers
203 views

A multiple integral

Let us consider the multiple integral $$I_{n}=\int_{-\infty }^{\infty }ds_{1}\int_{-\infty}^{s_{1}}ds_{2}\cdots \int_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos {(s_{1}^{2}-s_{2}^{2})}\;\cdots \cos ...
1
vote
0answers
66 views

Convergence and summation of power towers or Ackermann functions

I deleted this at StackExchange as it seems advanced or not very relevant for there and I want to ask here. Let z be a complex number, $Im(z)\neq 0$ or $Re(z)<4$, m be a natural number, let ...
3
votes
1answer
82 views

Estimating number of zeros of solutions of linear 2nd order ode

Let $y''+fy=0$ be a second-order linear ode on $y$, where $f(x)>0$, and $I=\left[ a,b \right)$ be an interval. Suppose we want to estimate the number of zeros of a (not identically zero) solution ...
3
votes
2answers
299 views

If two functions are equal to their Newton series, is their composition also equal to its Newton series?

Suppose we have two real-valued functions $f(x)$ and $g(x)$, both equal to their Newton series expansion: $$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)$$ $$g(x) = \sum_{k=0}^\infty ...
2
votes
0answers
100 views

proving quasi convexity of multivariable function

Given an arbitrary $(N \times N)$ square matrix ${\bf X}$ a positive definite $(M\times M)$ matrix ${\bf T}$ a $(Q\times MN), Q< MN$ matrix ${\bf Z}$ consisting of only 1s and 0s where there is ...
51
votes
2answers
2k views

Is it possible to express $\int\sqrt{x+\sqrt{x+\sqrt{x+1}}}dx$ in elementary functions?

I asked a question at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here: Is it possible to express ...
3
votes
0answers
83 views

Existence and smoothness of convolutions of distributions in Sobolev spaces

Let $f\in H^{s_1}(\mathbb{R}^n)$ and $g\in H^{s_2}(\mathbb{R}^n)$, where $s_1, s_2 \in \mathbb{R}$ and can be positive or negative. It is easy to show that $f *g$ is defined pointwise when ...
2
votes
1answer
246 views

Differentiability: Partially Defined Functions

These ideas came to my mind while reading Lee's Introduction to Smooth Manifolds. (Cf. discussion on p. 45.) Definition Let $E$ and $F$ be two Banach spaces together with a plain subset ...
7
votes
2answers
500 views

Survey of the history of calculus?

Boyer 1939 is a nice readable survey of the history of the calculus, but it's showing its age. Discussing the notion of instantaneous velocity, he has: Mathematics knows no minimum interval of ...
2
votes
0answers
130 views

A strong form of implicit function theorem (what happens when the derivative is degenerate?)

(this can be considered as some ad) Consider the system of equations $F(x,y)=0$. (Here $x$, $y$ are multi-variables. The equations are over a local ring. e.g. polynomial/analytic/formal/$C^\infty$ ...
9
votes
1answer
265 views

Is a Bessel function larger than all other Bessel functions when evaluated at its first maximum?

Let $\mathcal{J}_{n+1/2}$ be the Bessel function of order $n+1/2$. Let $x_n$ denote the first zero of its derivative, which is also the location of the first maximum of $\mathcal{J}_{n+1/2}$. My ...
4
votes
0answers
109 views

Carleman estimates on monotonicity formulas

I am trying to derive a monotonicity formula for a certain Dirichlet critical point (or even maybe a minimizer) of an energy of the type, say for simplicity, an energy of the from $$\int_{B_r} ...