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

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10 views

A question for uniqueness of configuration thoerem

Recently I am reading a book of Katok and Hasselblatt. I was confused by the proof for the following theorem: If $f:R^n\times R^n\rightarrow\mathbb{R^n}$ is C^2, and for any $M>0$, there exists ...
3
votes
1answer
80 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
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0answers
55 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 ...
3
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0answers
209 views

Symmetry on a sphere

Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every nonempty connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at ...
2
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0answers
87 views

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

Preliminary Definition We define the Zygmund spaces $C^r_{*}$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are allowed to have values in $\mathbb{R}^m$, via working with ...
1
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0answers
76 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
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1answer
71 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 ...
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2answers
113 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 ...
1
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0answers
70 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
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0answers
150 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
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1answer
62 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
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1answer
186 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
297 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 ...
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0answers
52 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
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1answer
107 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
139 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
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0answers
75 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
87 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
78 views

M-Wright function asymptotics

Let $M(z;\nu):= \frac{1}{\pi}\sum_{n=1}^{\infty} \frac{(-z)^{n-1}}{(n-1)!}\Gamma(\nu n)\sin(\nu n\pi)=\frac{1}{2\pi i}\int_{\text{H}_a}\exp(\sigma -z\sigma^{\nu})/\sigma^{1-\nu} d\sigma$, ...
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0answers
28 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 $t$: $$\dfrac{d^2}{dt^2}f(t)=\cos\left[f(t)\right]$$ ...
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0answers
63 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 $l>0$, $t>0$, $\Re(A)>0$, ...
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0answers
85 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
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2answers
564 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
106 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
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0answers
54 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
75 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
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1answer
139 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 ...
9
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173 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
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0answers
47 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
79 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
286 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
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0answers
65 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 ...
45
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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
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0answers
80 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
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1answer
189 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
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2answers
459 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
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0answers
104 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
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1answer
245 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
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0answers
101 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} ...
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0answers
79 views

Resources about integral maximization problem

I am looking at the following problem. Given an interval I, and a function f over that interval, find sub-intervals for which: The sum of the length of the sub-intervals is < k; The sub-intervals ...
2
votes
1answer
225 views

What does this ODE have to do with the associated Legendre polynomials?

I am currently struggeling with the following differential equation: $$(t^2-1)f''(t)+tf'(t)(1-8a+8at^2)-4(a+a^2-2at^2+\phi (-a+2at^2))f(t)= 4\lambda f(t),$$ where $a \in \mathbb{R}$ constant, $\phi ...
3
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2answers
208 views

Existence of an integral equation (Faedo-Galerkin, Banach fixed point, Picard-Lindelof)

Let $m \geq 2$ and let $m'$ be its conjugate. Let $w_j$ for $j=1, ..., k$ be a basis of $H_1 \cap L^{m'}$. The task is to show that there is a $u(t) \in \text{span}(w_1, ..., w_k)=:A$ such that ...
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0answers
66 views

Linear dynamical system with discontinuous coefficients

I am solving a linear dynamical system $X'=A(t) X$, where $t$ is the independent variable and $A(t)$ is a square matrix. Some of the coefficients of $A(t)$ have a discontinuity at a certain value of ...
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2answers
352 views

Spectrum of Mathieu equation

I have the differential equation $-f''(x)-q \cos(x) f(x) = \lambda f(x)$ and I want to find all the eigenvalues of this equation analytically on $[0,2\pi]$ that satisfy the boundary condition $f(0) = ...
6
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1answer
201 views

Shortcut from discrete Fourier transform F{x} to zero-padded F{x:0…0}

Summary: Given $X$ (the discrete Fourier transform of some unknown vector $x$ of length $N$), is there any shortcut to computing $X'$ (the Fourier transform of $x$ after padding it with $N$ zeros)? ...
12
votes
1answer
269 views

Cesaro(?)/Euler(?) - summation of the $s(p)=\sum_{k=0}^\infty (-1)^{H(k)} (1+k)^p$ for $p=1,2,3,…$ (where $H(k)$ is the Hamming-weight)

In another thread (in MO) there was a question about a series where the signs at the terms alternate with the "Hamming-weight", that means according to the number of bits in the binary representation ...
4
votes
3answers
289 views

Sum of series $a^{i^2}$

Is there any closed form known for the expression $\sum_{i=1}^\infty a^{i^2}$ where $|a|<1$? Thanks!
1
vote
2answers
244 views

What is known about $\displaystyle \sum_k{a^{b^k}}$?

What is known about $\displaystyle \sum_k{a^{b^k}}$? I am very interested in the possible applications of this series. I have asked about this on Mathematics Stack Exchange here. I'm wondering if ...
0
votes
0answers
56 views

Trying to solve for total derivatives at a stationary point (maybe using the implicit function theorem)

Suppose we have a function $F(q) \in \mathbb{R}$, where $q=(q_1, \dots, q_n) \in [0,1]^n$, at least thrice differentiable in $(0,1)^n$. We fix the value of one variable $q_i \in (0,1)$, then maximize ...
1
vote
1answer
356 views

Monge–Ampère type equation

Let $B(x,y) \geq 0$ be a function defined for $x, y \geq 0$ such that $B(x,0)=B(0,y)=0$ and $B''_{xx}\leq 0, B''_{yy}\leq 0$ (i.e. it is bicocncave function). I am looking for the solutions among of ...