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

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5
votes
0answers
160 views

Faà di Bruno formula

Let $f,g$ be smooth functions from $\mathbb R$ to $\mathbb R$. Then $$ \frac{(g\circ f)^{(n)}}{n!}=\sum_{1\le r\le n}\frac{(g^{(r)}\circ f)}{r!} \sum_{\substack{(n_1,\dots, n_r)\in {\mathbb N^*}^r\\...
2
votes
0answers
100 views

Log-concave polynomial is a log-concave function?

A polynomial $\sum\limits_{k=0}^n a_kx^k$ is log-concave if $a_0,\ldots,a_n$ constitute a log-concave sequence. I wonder whether the log-concave polynomial is also a log-concave function with respect ...
3
votes
2answers
197 views

functions with orthogonal Jacobian

I'm working on a model that would require to use vectorial functions of $\mathbb{R}^n \rightarrow \mathbb{R}^n$, such that $\forall x, y \in \mathbb{R}^n$, $\lVert \frac{df(x)}{dx}(y) \lVert_2 = \...
8
votes
2answers
300 views

Is the Fourier transform of $e^{-|x|^n}$ positive?

Let $$\Phi(x) = \int_{\mathbf{R}^n} e^{-|y|^n +i (x,y)} dy.$$ Is $\Phi$ positive everywhere in $\mathbf{R}^n$? Could someone helps me answer this question or gives a reference for it? Thanks.
4
votes
1answer
110 views

Hellinger integral for the Student/Cauchy family

Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral is $H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$. Let now $p$ be ...
1
vote
0answers
77 views

A limsup representation for the upper Buck density

The upper Buck density, introduced by R.C.Buck in 1946, is defined to be the function $$ \mathfrak{b}^\star: \mathcal{P}(\mathbf{N}^+) \to \mathbf{R}: X\mapsto \inf_{X\subseteq A:A\in \mathscr{A}}\...
1
vote
0answers
45 views

Bounds on $\sum_{j=1}^m\frac{\pi^j}{\Gamma(j)(x^2+(j+1/4)^2)}$

During our search of real rooted entire function approximations to Riemann $\Xi$ function, we need to calculate the upper and lower bounds of $$f_m(x):=\sum_{j=1}^m b_j(x):=\sum_{j=1}^m\frac{\pi^j}{\...
3
votes
1answer
58 views

Redundancy in transseries representation of functions?

"Transseries" are a kind of generalized power series that allow things like fractional exponents and exponentials (with another transseries as the exponent). I know very little about them but I have ...
1
vote
1answer
61 views

ODE estimate for boundary value problem

Let $X$ be a solution to the boundary value problem $$ X^{\prime\prime}(s) = A(s)X(s), \quad X(0) = X_0, ~~X(t) = X_1,$$ where $0 < t \leq 1$, $A$ is some matrix-valued function, defined on $[0, 1]$...
-1
votes
1answer
54 views

Idempotent solutions to the implict function theorem other than the identity?

I am interested in the following problem. Assume that an (anti)symmetric function $g:\mathbb{R}^2 \to \mathbb{R}$ satisfies the implicit function theorem. That is, $g(x,y) = \pm g(y,x)$ and $g(x,y)=0$ ...
9
votes
2answers
748 views

Algebraic independance of exponentials

First of all, a happy new year. Be it better than 2015, healthy, wealthy, fruitful and cross-fertilizing for you, familly and friends. In order to cope with families of solutions of evolution ...
8
votes
1answer
143 views

How to eliminate secular terms for perturbed non-oscillatory equations?

Even in a linear second order equation like $x''+x'+\epsilon x=0$ the standard asymptotic expansion has a secular term already in the first order of $\epsilon$, namely $$x(t)=a_0+b_0e^{-t}+\epsilon(...
9
votes
1answer
368 views

Number of real roots of an exponential polynomial

Let $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_n$ be real numbers, and assume that $\{a_i\} \neq \{b_i\}$. Can the equation $$ e^{a_1 x} + e^{a_2 x} + \dots + e^{a_n x} = e^{b_1 x} + e^{b_2 x} + \...
3
votes
2answers
166 views

Roots of the Chebyshev polynomials of the second kind

It is known that the roots of Chebyshev polynomials of the second kind, denote it by $U_n(x)$, are in the interval $(-1,1)$. I have noticed that, by looking at the low values of $n$, the roots of $(...
1
vote
2answers
131 views

Convergence of Sobolev functions near the boundary

Let $B_0(1)$ be the unit ball in $\mathbb R^n$, $n\geq2$. Let $f\in W_0^{1,2}(B_0(1))$, and $W^{1,2}(B_0(1))\ni f_i\to f$ in the sense of $L^2(B_0(1))$-norm, as $i\to \infty$. Question 1: Can we ...
1
vote
0answers
49 views

“Harmonic oscillator” with $p$-Laplacian

I wonder if there is any literature on the eigenvalue problem for the "$p$-harmonic oscillator" $$-(|u'|^{p-2}u')'(x)+(x^2-\lambda) |u(x)|^{p-2} u(x)=0$$ in $L^p(\mathbb R)$, $p\in(1,\infty)$. Are ...
6
votes
1answer
111 views

Are the extremal points of a certain set of functions $\mathcal P(\mathbf N) \to \bf R$ weakly additive?

Let an upper density (on $\mathbf N$) be a (set) function $f: \mathcal P(\mathbf N) \to \mathbf R$ such that, for all $X, Y \subseteq \bf N$ and $h,k \in \mathbf N^+$, the following hold: (F1) $f(\...
9
votes
3answers
293 views

Decay of real continuous algebraic functions at infinity

Let $f$ be a real valued continuous algebraic function on $\mathbb R^n$. Suppose the zero set of $f$ is bounded, i.e., if $|x|$ is large enough, $f(x)\neq 0$. Is there any estimate of the sort $|f(x)|\...
2
votes
0answers
59 views

Reference request on a notion of independence for families of [real-valued] functions

This is basically another reference request. Let $X$ be a set, and $\mathscr{F} = (f_i)_{i \in I}$ an indexed family of functions $X \to \bf R$. If $\preceq$ is a partial order on $I$, we say that ...
12
votes
1answer
550 views

How to prove that a monotone function is differentiable at some point?

This fact, which eventually belongs to Lebesgue, is usually proved with some measure theory (and we prove that the function is differentiable a.e.). Is there a significantly different approach? Let me ...
4
votes
3answers
554 views

Exact formula for $\sum_{n=0}^{+\infty}\frac1{n^2+1}$

Actually, as the corresponding integral $\frac{\ln(\cos x)}{1-x}$ or something like that) cannot be expressed in closed form by Liouville theorem, this shouldn't exist, but I believe I have seen it ...
5
votes
0answers
70 views

Some questions about the Lévy monoid of certain densities

Let $\bf H$ be a set, $f: \mathcal P({\bf H}) \rightharpoonup \bf R$ a partial function, and $\mathcal{D}$ the domain of $f$. Next, denote by $\mathcal M(f)$ the set of all (total) functions $\theta: ...
1
vote
0answers
53 views

Complexity theory and closed form formulas in analysis

My question concerns definitions of "closed form" solutions. In hamiltonian systems this is closely related to complete integrability. In this context closed form can refers to having $(q(t),p(t))$ ...
2
votes
1answer
189 views

General solution to system of stochastic linear differential equations

Assume we are given the system of linear stochastic differential equations $$dx_i = \sum_{j=1}^n a_{ij}(t) \cdot x_j \cdot dt + \sum_{j=1}^n \sigma_{ij}(t) \cdot x_j \cdot dB_{ij,t} + b_j(t)\cdot dt+\...
1
vote
0answers
72 views

Asymptotics of a elliptic pde when exponent gets large

I am interested in the following pde $$ -\Delta w_p + \left( \frac{1}{p-2} +1 \right) \frac{ | \nabla w_p|^2}{w_p} + \epsilon(p) \left( \frac{1}{w_p} \right)^{(p-2)} = (p-2) w_p $$ in the unit ball $...
1
vote
0answers
98 views

Integrating a series expansion of $\mbox{frac}(x)\lfloor x\rfloor$ coming from Fourier series of sawtooth function

Let me preface this question by saying that I am not exactly sure it counts as research level. It is crossposted on mathstackexchange: http://math.stackexchange.com/questions/1519724/integrating-a-...
2
votes
0answers
72 views

Non-existence for a sort of probability measures

We suppose $X$ solves our SDE $dX_{t}=-X_{t}dt+dW_{t}$ for $t\geq0$ with initial condition $X_{0}=0$ w.r.t to our measure $P$ on $(\Omega,\mathcal{F})$. $W_{t}$ is standard Wiener. This solution is ...
2
votes
0answers
127 views

An alternative to the Euler--Maclaurin formula: Approximating sums by integrals only

The Euler--MacLaurin summation formula can be written as $$ \sum_{i=0}^{n-1} f(k)\approx \int^{n-1}_0f(x)\,dx + \frac{f(n-1) + f(0)}2 + \sum_{j=1}^m\frac{B_{2j}}{(2j)!}[f^{(2j - 1)}(n-1)...
6
votes
1answer
345 views

A property of the derivatives of a function

Suppose that $f,g_1,g_2,\dots$ are functions from $\mathbb{R}$ to $\mathbb{R}$ such that $f'=f\,g_1$ and $g'_j=g_j^2-g_j g_{j+1}$. Here and in what follows, $j$ is any natural number. Then, by ...
1
vote
0answers
68 views

Recurrence sequence

Is it possible to find a Recurrence sequence that Satisfying the following inequality $ d_{n+k}\geq \alpha ^k d_n +‎\beta‎‎^k ‎\delta‎(A,B),$‎‎ ‎‎where $0<‎\alpha‎<1, \alpha ^k+‎\beta‎‎^k\geq ...
2
votes
1answer
103 views

Lower bounds from Fourier dimension?

According to Mattila, Geometry of sets and measures in Euclidean spaces, p. 168, the Fourier dimension $\text{dim}_F(A)$ of $A\subseteq \mathbb R^n$ is the unique number in $[0,n]$ such that for any $...
2
votes
0answers
38 views

Asymptotic expansion of Mellin transform of products of modified Bessel function K

Let $n\ge 1$ be an integer, let $$F(x,y)=\int_0^\infty u^{n(x+y)} (K_{x-y}(u))^n du$$ for $x,y\ge 0$. When $n=1$, this is just Mellin transform of the Bessel K function. When $n=2$, $F(x,y)$ has ...
10
votes
1answer
491 views

A tricky tractrix question about vertical tangents

This is raised by a recent question occurring in combinatorial geometry. It is about a sort of tractrix, but instead of a line, the pulling end moves along a circle of radius $r>\frac12$ (...
2
votes
0answers
28 views

Uniform convergence of the best $L_1$ approximations by polynomials

Let $P_n$ be the vector space of real multivariate polynomials in $d$ variables of total degree no more than $n$ and let $f \in C(X)$, where $X \subset \mathbb{R}^d$ is compact. Let $(p_n)_{n=1}^\...
6
votes
2answers
75 views

Closed field lines in the plane

A dipole in the plane consists of a positive charge P and an equal and opposite negative charge N separated by a fixed distance . Almost all of the resulting electric field lines (which fill the ...
7
votes
2answers
439 views

A result of Sierpiński on non-atomic measures

There is a classical result commonly attributed to W. Sierpiński that reads as follows: Theorem 1. If $f: \Sigma \to \bf R$ is a non-atomic (*) measure on a set $S$, then for every $X \in \Sigma$ ...
1
vote
0answers
44 views

Triangle inequality for nonconvex functions of singular value vector

For a nonconvex function $p_\lambda(\cdot)$, with hyper-parameter $\lambda$, it can be decomposed into two parts that $p_\lambda(t) = \lambda |t| + q_\lambda(t)$, where $q_\lambda(t)$ is the concave ...
8
votes
2answers
259 views

A specific linear differential equation on $\mathbb{C}-\{0,1\}$ whose monodromy group represents the fundamental group of $\mathbb{C}-\{0,1\}$

Is there a linear differential equation on $\mathbb{C}$ with singularities at $0$ and $1$ whose monodromy group represents the fundamental group of $\mathbb{C}-\{0,1\}$? If so, can someone give a ...
0
votes
1answer
115 views

growth of derivative

(Alexandre Eremenko gave the answer to the question I posted earlier. Then I realised the question didn't give what I wanted. Here is the updated question.) Suppose that $f:[a,\infty)\to \mathbb{R}$ ...
2
votes
1answer
172 views

comparing Laplacian and gradient of function on boundary

Consider $ E(x)$ some smooth function on $ \Omega$ (some smooth bounded domain in $ R^N$) and suppose $E=0$ on $ \partial \Omega$. Suppose one knows that there is some $C_1,C_2 \in R$ such that $ x ...
1
vote
0answers
57 views

Log convexity for the norm of a vector-valued function

Log convexity of various functions defined on the space of Hermitian matrices plays an important role in matrix analysis and probability theory. Given $v \in \mathbb{C}^n$, $D$ a diagonal matrix with ...
1
vote
1answer
102 views

Can a (non-measurable) autonomous flow have a non-trivial periodic orbit without a minimal period?

Given a set $X$, a function $x \colon \mathbb{R} \to X$ is periodic if there exists $\tau>0$ such that $x(t+\tau)=x(t)$ for all $t \in \mathbb{R}$; and if $\tau$ is the smallest positive number ...
16
votes
1answer
666 views

Are the algebraic numbers dense everywhere on the boundary of the Mandelbrot set?

Let $\mathcal{B}$ denote the boundary of the Mandelbrot set, and let $\overline{\mathbb{Q}}$ denote the algebraic closure of the rationals. Further put $\mathcal{B}_{\overline{\mathbb{Q}}} := \mathcal{...
2
votes
1answer
71 views

Challenging problems concerning Jacobian elliptic functions with complex modulus

I study some qualitative properties of Jacobian elliptic functions. Consider, for example, function $sn(u,k)$. In most applications, modulus $k\in(0,1)$ and then everything is very clear, since $sn(u,...
3
votes
1answer
126 views

Generalized Theorem of Laguerre

There is known theorem of Laguerre, that every linear ordinary differential equation of second order $$y''+A(t)y'+B(t)y=0$$ by point transformation could be mapped into $$y'' = 0,$$ that in few words ...
2
votes
1answer
124 views

An extreme of Jacobi elliptic function on an interval

Consider the Jacobi elliptic function $sn(\cdot,k)$ restricted to the interval $(0,2K)$, where $K=K(k)$ is complete elliptic integral of the first kind. If $0<k<1$, then it is well known the ...
10
votes
1answer
272 views

Normal approximation of tail probability in binomial distribution

My problem: From the Berry--Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|=O\left(\frac 1{\sqrt n}\right),$$ where $B_n$ has the standardized binomial distribution and $\Phi$ ...
4
votes
1answer
71 views

Getting out a system of linear ODEs by knowing the Magnus expansion

Assume we are given for a transition between two time points $t_0 = 0$ and $t_1$ a matrix relationship, eventually describing the solution of a system of linear with non-constant coefficients, $$Y(t_1)...
2
votes
1answer
130 views

A question about viscosity solutions

Let $A:=[a,b]$ be a closed interval in $\mathbb{R}$. Let $F(x,p,q,r)$ be a function from $[a,b]\times \mathbb{R} \times \mathbb{R} \times\mathbb{R} \rightarrow \mathbb{R}$ describing a second order ...
2
votes
1answer
112 views

Rate of convergence of ergodic averages related to irrational rotation

Let $\alpha$ be an irrational number, consider the basic dynamical system $T^{n}(0) = \{n \alpha\}$ where $\{.\}$ denotes the fractional part. Let $a < b$ be two numbers in $[0, 1]$. Then by ...