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

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7
votes
2answers
188 views

Radial limit does not exist almost everywhere

Problem 4 in Chapter 4 of Stein's book "Real Analysis" says $\sum_{n\geqslant 0}z^{2^n}$ doesn't have radial limit as $z$ approaches the unit circle from inside almost everywhere. It's fairly easy ...
3
votes
0answers
47 views

Reference for a lemma on the asymptotic upper density of special sets with large gaps and intervals

Update. Based on Anthony Quas' comment below, the proof can be made sensibly shorter and the lemma can be slightly generalized by weakening the old assumption (iii). In a joint paper that I am ...
2
votes
1answer
138 views

A question on the Frechet derivative

Suppose the derivative of a functional is given by \begin{equation*} \int_{\Omega}(\vec{v}.\nabla u)|\nabla u|^{p-2} \phi=\int_{\Omega}\nabla.(u\vec{v})|\nabla u|^{p-2} \phi,~\phi\in ...
3
votes
1answer
68 views

Hyperelliptic generalization of Euler's formula

Are there any hyperelliptic generalizations of the following formula, first proved by Euler in 1782, ...
6
votes
1answer
130 views

On the independence of lower and upper asymptotic and Banach densities

Given a set $X \subseteq \mathbf N^+$, denote by $\mathsf{d}_\ast(X)$ and $\mathsf{d}^\ast(X)$, respectively, the lower and upper asymptotic (or natural) density of $X$, viz. $$\mathsf{d}_\ast(X) := ...
9
votes
1answer
240 views

Sard's Theorem For Banach Spaces

Given a smooth map from $\phi: B \rightarrow M$ where $B$ is a Banach Space and $M$ is a finite dimensional smooth manifold (for example, the end point map for a control system), what is the strongest ...
0
votes
0answers
42 views

Closed form for convolution of two-dimensional Gaussian with characteristic function of a disk

Is there a closed form expression for the convolution of a two-dimensional (elliptical) Gaussian function with the characteristic function of the interior of an ellipse? The motivation is that I have ...
3
votes
1answer
195 views

An integral equation

I have a Fredholm integral equation of second kind $$\frac{-1}{2\pi \omega'^2}+\int_{-\infty}^{\infty}\frac{1}{\pi ...
0
votes
0answers
54 views

Derivative of a conjugation of matrices

Let $\mathcal{M}_n$ be the space of complex $n\times n$ matrices. Let $\Phi\colon \mathbb{D}\to \mathcal{M}_n$ and $\psi \colon \mathbb{D}\to \mathcal{M}_n$ be holomorphic functions. Consider the ...
1
vote
0answers
22 views

gradient flow in simple settings

I have to solve an equation of the type $\partial_t X = \nabla U (X)$ in $\Omega \subset\mathbb{R}^2$ with an initial condition I know that under some conditions, $X(t)$ will converge to a critical ...
0
votes
0answers
36 views

help with an asymptotic estimate for a certain product

(I apologize in advance if this question is not suitable for Math Overflow, but it came up in a research problem and thought perhaps I could find some help here.) I'm having difficulty finding an ...
5
votes
2answers
197 views

A (likely) positivity property of the Lerch zeta-function

The problem is to show that $\Re L(b/2,1/2,p+1)>0$ for all real $b\ne0$ and all real $p>-1$, where $$L(\lambda,c,s):=\sum_{k=0}^\infty\frac{\exp(2\pi i\lambda k)}{(k+c)^s}$$ is the Lerch ...
4
votes
1answer
93 views

Uniform boundedness of eigenfunctions of an elementary differential operator

Consider a differential operator $L=(-1)^ma(x)\frac{d^{2m}}{dx^{2m}}$, with boundary value conditions $u^{(j)}(0)=u^{(j)}(1)=0, j=m,m+1,\ldots,2m-1$, where $m\ge1$ is an integer and $a(x)>0$ is ...
2
votes
1answer
262 views

Does the following type of Gronwall inequality hold?

Let $I=[0,b)$, $b< \infty$. Suppose $u$ is a positive bounded measurable function on $I$. $v(s)$ is a positive, smooth function on $I$. Note that $u(b),v(b)$ may be $0$. Suppose that $$ u(t) ...
-1
votes
0answers
76 views

Interpolation between strongly convex functions

Definition: We call $f:\mathbb{R}^n\rightarrow \mathbb{R}$ a $\lambda$-strongly convex function iff for every $x,y\in \mathbb{R}^d$ and $t\in [0,1]$ it follows $$ f(tx+(1-t)y) \leq t f(x) + (1-t) f(y) ...
1
vote
1answer
72 views

Derivative of a time evolution operator w.r.t. a parameter

Let $N\geq1$ be an integer and let $H:[0,1]^2\to\mathbb C^{N\times N}$ be a pointwise hermitean matrix valued function. For $y\in[0,1]$ and $0\leq a\leq b\leq 1$, let $U_y(b,a)$ be the time evolution ...
5
votes
0answers
90 views

behaviour of first eigenfunction near the boundary

Consider $\Omega$ a smooth bounded domain in $R^N$ and suppose $ \phi_1(x)>0$ is the first eigenfunction of $ -\Delta$ in $H_0^1(\Omega)$ normalized however one chooses. My interest is in how $ ...
6
votes
2answers
203 views

Existence of a measure-preserving bijection

Let $f, g \, \colon \mathbb{R}^n \rightarrow \mathbb{R}$ be two Borel-measurable functions such that $f$ is non negative and g is radially symmetric, the function $ (0, \infty )\ni t \mapsto g ...
2
votes
0answers
82 views

Motivation for the existence of periodic solutions [closed]

I have been reading the book Critical Point Theory and Hamiltonian System by Mawhin and Willem, as well as several other papers on the existence of periodic solutions for equations of the form ...
1
vote
1answer
202 views

Is there an analytic solution for this partial differential equation?

The Fokker-Planck equation for a probability distribution $P(\theta,t)$: \begin{align} \frac{\partial P(\theta,t)}{\partial ...
0
votes
1answer
98 views

methods for situations where well-posedness criteria hold but global solutions do not exist

I have been learning PDEs (more specifically, nonlinear dispersive equations (Schrödinger/wave/ Klein-Gordan equations etc...)) through the harmonic analysis methods. And I have read a couple of ...
0
votes
0answers
37 views

Orthogonality relation for associated Legendre functions

The associated Legendre Polynomials $P_l^m(x)$ obey orthogonality relations for fixed $m$ and fixed $l$: $$ \begin{align} \int_0^\pi P_k^m(\cos\theta)P_l^m(\cos\theta)\sin\theta d\theta ...
1
vote
0answers
42 views

A fundamental lemma involving a certain exponential kernel

Let $h \in L^1(\mathbb R^n, \mathbb R)$ be a scalar field and let $\Psi_t: \mathbb R^n \to \mathbb R$ be smooth mappings, parameterized by $t \in \mathbb R$. Suppose that we are given data $$D(v,t) = ...
9
votes
1answer
120 views

Nonconventional ergodic averages for commuting transformations

Let $S$ and $T$ be commuting measure-preserving transformations of a standard probability space $(X,\mu)$, so $S$ and $T$ define an action of $\mathbb{Z}^2$ on $(X,\mu)$. I am wondering about ...
1
vote
0answers
50 views

persistence of regularity for nonlinear Klein-Gordon equation

I have been reading the paper on nonlinear Klein-Gordon equation(NLKG) for initial data in modulation space: For detail please see the paper "Klein-Gordon Equations on Modulation Spaces (2014)" ...
2
votes
0answers
36 views

Global error estimates for numerical solutions of ODEs in Matlab or Mathematica [closed]

I need to find the first zero (smallest positive root) of the solution of the initial value problem $ry''+y'+f(r)y=0, \ \ y(0)=y'(0)=1$ for certain $f \in C^{\infty}(R)$. One can easily use ...
2
votes
1answer
114 views

Is every set of small measure contained in an open set of small measure with null boundary?

Let $\lambda( \cdot )$ denote Lebesgue measure on $[0,1]$. Let $(A_n)_{n=1}^\infty$ be a decreasing sequence of Borel subsets of $[0,1]$ such that $\bigcap_{n=1}^\infty A_n = \emptyset$. Given ...
2
votes
2answers
167 views

“C^0 estimate for solutions to $\Delta(u)+e^{-u} \geq 0$”

Let $u: \mathbb{R}^2 \to \mathbb{R}.$ Suppose I have a solution to the equation $$\Delta(u)+e^{-u} \geq 0$$ on $\mathbb{R}^2$. Let r be the radial coordinate on $\mathbb{R}^2$. Suppose that $$lim_{r ...
6
votes
2answers
359 views

Texts about Dwork's work

I want to ask about references to papers, that probably exist, which explain the articles of Bernard Dwork starting from "The rationality of the zeta function of an algebraic variety" to "On the ...
2
votes
0answers
73 views

Oscillation aspects of two-way infinite alternating series (a followup from the MO-question “functions that eventually oscillate”)

In the recent question on "eventually oscillating function" I had a heuristic for the function $d(x)$ that its amplitude is constant, but could not further describe that function. I just found a ...
0
votes
0answers
49 views

What can we say about variational energies here?

Let $V_{ij}^{lk}$ be any $nm \times nm$ real symmetric matrix, $\forall i,j,k,l$ \begin{equation} V_{ij}^{kl}=V_{ji}^{lk} \end{equation} (So for the indices we have $1 \leq k,l \leq m$ and $1 \leq i,j ...
6
votes
2answers
317 views

Interesting triple integral

Some time ago I stumbled on an alleged identity $$\int\limits_0^\infty \frac{dx}{x} \int\limits_0^x \frac{dy}{y} \int\limits_0^y \frac{dz}{z} [\sin{x}+\sin{(x-y)}-\sin{(x-z)}-\sin{(x-y+z)}]= ...
0
votes
0answers
105 views

Solving the transcendental equation $Li_{3}(e^{-kx}) + x\, Li_{2}(e^{-kx}) = k\, x^3$

I need to solve the following equation: $Li_{3}(e^{-kx}) + x\, Li_{2}(e^{-kx}) = k\, x^3$ for $x\in\mathbb{R}^{\ast}$ and where $k\in\mathbb{R}^{+}$. Here $Li_{3}$ and $Li_{2}$ are the third and ...
5
votes
0answers
97 views

Hyperbolic toral automorphisms, and maximizing over orbits the minimum along an orbit of a function

Setup: Let $\phi\colon T^2 \to T^2$ be a hyperbolic toral automorphism. Let $f\colon T^2 \to \mathbb{R}$ be a continuous function. For $x \in T^2$, let $\underline{f}(x) = \inf_{n \in \mathbb{Z}} ...
4
votes
1answer
82 views

Reference for higher order Campanato Lemmas, e.g. `Sufficiently fast L^2 decay on balls to affine functions implies C^{1,\alpha}'

Whence can I reference the following fact (I have seen it quoted as `standard' in respectable places, so I hope it is so)?: Let $f : B_2(0) \to \mathbb{R}$, say $f \in L^2(B_2(0))$ . Suppose that ...
1
vote
1answer
44 views

Generating function for products of laguerre polynomials?

In a quantum physics context, I would like to evaluate $S=\sum_{n=0}^\infty z^n\cos(\pi L_n(x))$ for $z<1$. I found generating functions for squares of Laguerre polynomials but not for any higher ...
3
votes
2answers
310 views

Summing cosines

Suppose I want to compute $$\sum_{n=1}^\infty \cos(n x)/n^{2k}.$$ Now, For any fixed $k$ this is, at least in principle, doable (see the excellent answer to my math.SE question a while back), but the ...
0
votes
1answer
115 views

Equality cannot hold unless $x \in \{-1,1\}$ and/or Wronskian is not zero [closed]

By playing around with assoc. Legendre polynomials, I arrived at $$((l+1)+m) (P_l^m(x))^2+((l+1)-m)(P_{l+1}^m(x))^2 = 2(l+1)x P_l^m(x)P_{l+1}^m(x).$$ Now, I want to show that we don't have equality ...
1
vote
2answers
343 views

More recently published comprehensive reference on inequalities in the spirit of Hardy-Littlewood-Pólya

Is there a comprehensive reference book on inequalities in the spirit of the one written by G.H. Hardy, J.E. Littlewood, and G. Pólya(*), but more up-to-date (i.e., published in more recent years and ...
4
votes
1answer
125 views

Symmetric inequality on generalized means

Do there exist two functions $f$ and $g$ continuous and strictly increasing $[0,1] \to \mathbf{R}$ such that $$ f^{-1}\left(\frac{1}{3} f(x) + \frac{2}{3} f(y)\right)<g^{-1}\left(\frac{1}{3} g(x) + ...
2
votes
0answers
64 views

Global existence for infinite dimensional ODE

Let us consider the ODE $\hskip3pt \dot x=F(t,x)\hskip3pt $ in an infinite-dimensional Banach space $E$, where the flux $F$ is defined and continous from the whole $\mathbb R\times E$ into $E$. ...
2
votes
0answers
40 views

Trigonometric multiple integral identity

How this alleged multiple integral identity can be proved? $$\int\limits_{-\infty }^{\infty }ds_{1}\int\limits_{-\infty}^{s_{1}}ds_{2}\cdots \int\limits_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos { ...
2
votes
0answers
89 views

Uniqueness of solution of elliptic equation with exponential nonlinearity

Consider the following equation $$\Delta v + p(r)e^v = 0$$ on $\mathbb{R}^n$ where $p(r)$ is a polynomial in $r = |(x_1,..., x_n)|$. I want to understand when equations like these have unique ...
4
votes
1answer
114 views

Infinite dimensional Cauchy-Lipschitz theorem [duplicate]

From the answers to Mathoverflow Question "Continuity in Banach space for non-linear maps", it is possible to infer that the assumption of the Cauchy-Lipschitz theorem for the autonomous equation $$ ...
1
vote
0answers
19 views

Renormalization for Transport Equations with SBD velocity field

In the paper Traces and fine properties of a $BD$ class of vector fields and applications by Ambrosio, Crippa and Maniglia (to be found here)the authors prove a chain rule for vector fields $B\in ...
34
votes
6answers
2k views

On an example of an eventually oscillating function

For $x\in(0,1)$, put $$f(x):=\sum_{n=0}^{\infty}(-1)^{n}x^{2^{n}}.$$ This function possesses interesting properties. It grows monotonically from $0$ up to certain point. Then it starts to oscillate ...
2
votes
1answer
164 views

Multivariate Maximal Hilbert Transform

One way to define the maximal Hilbert transform of a function, $f$, is by $$\mathcal{H}[f](x):=\underset{\varepsilon>0}{\sup}\;\left|\int_{|x-t|\geq\varepsilon}\dfrac{f(t)}{x-t}dt\right|,\quad ...
0
votes
0answers
43 views

Searching for conditions?

I have this operator $$Au(t)=\int_0^1 G(t,s) f(s,u(s)) ds$$defined from $H^1_{0}$ to $H_0^1$ and satisfy the problem: $$\begin{cases} -(Au)''(t)=f(t,u(t)), t\in[0,1]\\Au(0)=Au(1)=0\end{cases}$$ Where ...
5
votes
1answer
216 views

Asymptotic expansion of $\zeta(s \mid a,b)= \sum_{n=1}^{\infty} \frac{1}{(n+a)^{s}(n+b)}$

I'm interested in an asymptotic expansion of the following Riemann zeta-type function $$ \begin{align} \displaystyle \zeta(s \mid a,b) := \sum_{n=1}^{\infty} \frac{1}{(n+a)^{s}(n+b)}, \quad \Re a ...
16
votes
1answer
549 views

Interesting integral

Numerical evidence shows the validity of the following identity $$\int\limits_0^z\frac{xdx}{\sin{x}\sqrt{\sin^2{z}-\sin^2{x}}}=\frac{\pi}{4\sin{z}}\ln{\frac{1+\sin{z}}{1-\sin{z}}},\tag{1}$$ if $0< ...