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

learn more… | top users | synonyms (2)

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
99 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
33 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
482 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
27 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 ...
6
votes
2answers
74 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
428 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
43 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
167 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
54 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
664 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}}} := ...
2
votes
1answer
69 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 ...
3
votes
1answer
125 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
121 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
248 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, ...
2
votes
1answer
126 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
109 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 ...
12
votes
0answers
186 views

Surprising approximate identity

While answering this MO question Connection between Bernoulli numbers and Riemann-Siegel theta function? Dan Romik found the following surprising approximate identity: $$\ln{8\pi}\approx \pi\left[ ...
2
votes
0answers
61 views

Optimal condition for the weak convergence of the jacobian determinant

Whenever $n<q,$ it is known that given a sequence $\{ u_{k} \}$ which is weakly convergent in $W^{1,q}(U)$ one has that the Jacobian determinants $\text{det} Du_{k}$ converge weakly in ...
0
votes
0answers
47 views

A question about the approximation of convex cones

I have the following question which maybe is too naive. Let $K$ be a convex cone on $\mathbf{R}^n$. Can we approximate $K$ by a sequence of polyhedral convex cone $K_i$ such that for any compactly ...
3
votes
2answers
167 views

Name for an orthogonal decomposition of $L^2 (\mathbb{R}^2; \mathbb{C})$

The space $L^2 (\mathbb{R}^2; \mathbb{C})$ can be decomposed as $$ L^2 (\mathbb{R}^2; \mathbb{C}) = \bigoplus_{k \in \mathbb{Z}} L^2_k (\mathbb{R}^2; \mathbb{C}), $$ where $$ L^2_k (\mathbb{R}^2; ...
5
votes
0answers
63 views

How to derive explicit bound for the solution of following equation?

Let's have equation $$ y''(t) + \left(\frac{3}{16t^{2}} + \frac{a}{t} -\frac{b}{t^{\frac{5}{4}}}cos(2t)\right)y(t) = 0, \quad t \in (1, \infty), \quad a, b > 0 $$ How to derive explicit upper bound ...
2
votes
0answers
182 views

Hartogs's extension theorem

Let $(P,H)$ be a Euclidean Hartogs figure in $\mathbb{C}^n$, and let $f:H\to \mathbb{C}^n$ be a holomorphic injective map. Then we know that $f$ extends holomorphically to the polydisc $P$, i.e. there ...
1
vote
0answers
61 views

Derivatives of $O$-regular varying functions are $O$-regular varying functions?

The Monotone Density Theorem for regularly varying functions says, in essence: Theorem (Monotone Density Theorem). Let $f$ be a differentiable regularly varying real-function of index $\rho$ ...
1
vote
0answers
33 views

How to treat equation with alternating square of frequency?

Let's have equation $$ \tag 1 \frac{d^{2}y(t)}{dt^{2}} +\omega^{2}(t)y(t) = 0, \quad t \in (t_{\text{in}}, \infty) $$ Here $$ \omega^{2}(t) = A(t) - B(t)cos(2t), $$ and functions $A(t), B(t)$ have ...
6
votes
1answer
484 views

An elementary lower bound on the number of primes

Recall the second Chebyshev function: $$\psi(x) = \sum_{p \leq x} \lfloor \log_p x \rfloor \log p$$ where $x$ is a positive integer, and $p$ runs over all primes $\leq x$. In a hunt for an ...
1
vote
1answer
137 views

Is this function concave or convex? [closed]

let $g_{n,\gamma}(\sigma)$ be the function defined as the following $$ g_{n,\gamma}(\sigma)= \left(\frac{(\sigma-1)^2 +\gamma^2}{\sigma^2 +\gamma^2} \right)^{n/2} T_n\left( \frac{\sigma(\sigma-1) ...
0
votes
0answers
40 views

Mathieu equation instability

Let's have Mathieu equation: $$ \tag 1 \frac{d^2y(x)}{dx^2} +(A -2q\times cos(2x))y(x) = 0, \quad q\geqslant 1, \quad A > 2q $$ This case is different from the famous case $q << 1, A> ...
6
votes
1answer
134 views

Asymptotic behaviour of an integral

For $k\in\mathbb{N}_{0}$ and $x\in\mathbb{R}$, define $$I_{k}(x):=\int_{0}^{\pi/2}\cos(xg(\theta))\sin^{2k}\theta\,\mathrm{d}\theta$$ where ...
3
votes
0answers
153 views

Dynamics of an inequality

The dynamics $D\ni(r_i,r_{i+1})\mapsto(r_{i+1},r_{i+2})\in D$ on the set $D:=\{(x,y)\in\mathbb{R}^2\colon x>0,y>x^2/2\}$ is given by the recurrence $$r_{i+2}=\frac{r_{i+1}^2}2+\frac1{r_{i+1}^3} ...
6
votes
2answers
186 views

differential equation of conics

Function $y(x)$ on the plane defines a conic (strictly speaking, at most second order algebraic curve) if and only if $\frac{d^3}{dx^3} (y'')^{-2/3}=0$. What is intuition behind this? How may I see ...
5
votes
0answers
93 views

Functions on [0,1] with positive fractional series coefficients and symmetric under x->1-x

Suppose a function $g(x)$ has a convergent power series expansion with real (not necessarily integer or rational) exponents on $[0,1]$ of the form $$ g(x)=\frac{1}{x^\delta}(1+\sum_{i=1}^\infty c_i ...
1
vote
1answer
57 views

A Statement from Brauer and Nohel's book on stability of time-depending linear systems

On page 158 (The qualitative theory of differential equations; an introduction) the authors cite a $2 \times 2$ counterexample by Vinograd to the system $y'=A(t)y$, where $$ A(t) = \left( ...
7
votes
2answers
412 views

The sum of a series, continued

In this question the OP asks whether the sum $$ f(q, \alpha) = \sum _{k=1}^{\infty } \frac{q^k \left(q^k-1\right)^\alpha}{(q;q)_k} $$ is ever zero. An experiment with Mathematica indicates, to any ...
12
votes
1answer
162 views

Interval arithmetic with different definitions of intervals

Interval arithmetic normally deals with intervals defined as $[a,b]$ with rules like $$[a,b]+[c,d]=[a+c,b+d]$$ I am interested in interval arithmetic with different interval definitions such as ...
4
votes
1answer
148 views

Hyperfunctions supported at a point

Is it true that the space of hyperfunctions on $\mathbb{R}^n$ supported at 0 coincides with the space of Schwartz distributions supported at 0? More explicitly, is it true that any hyperfunction ...
4
votes
0answers
65 views

Applications and main properties of hyperfunctions

I am trying to get familiar with hyperfunctions, and I do have some familiarity with the classical theory of distributions. I am wondering whether hyperfunctions have any advantages over ...
10
votes
5answers
615 views

Identities and inequalities in analysis and probability

Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of ...
1
vote
0answers
100 views

Algorithm for finding eigenfunctions

I have an $ L^2(\mathbb{R}) $ operator that looks like $$ \Omega = \int \partial\phi(a, b)\ \ |b, a\rangle \langle b, a |, $$ where $ \langle x | a, b \rangle = f_a(x - b) e^{x^2/2} $ and $ f_a \in ...
54
votes
6answers
2k views

Why does $d^n \exp(-x-x^{-1})/(dx)^n$ only have $n$ positive real zeroes?

Set $f(x) = \exp(-x-x^{-1})$. An easy induction shows that $$\frac{d^n}{(dx)^n} f(x) = \phi_n(x^{-1}) f(x)$$ for $\phi_n$ a polynomial of degree $2n$. Clearly, the roots of $\phi_n(x^{-1})$ are the ...
10
votes
2answers
650 views

What's a good introduction to category theory for someone doing analysis?

I do functional analysis, and diagrams are popping all over the place. It is about time I learned me some category theory. Any recommendations?
0
votes
1answer
87 views

Fit a system of linear ODEs from several experiments

Assume we are given several initial vectors $x^{(1)},\ldots,x^{(r)} \in \mathbb{R}^n$, where the dimension $n=6$ (in any event a number below 10) , and the number of initial vectors $r$ is in the ...
0
votes
0answers
33 views

Explicit solution for a variational inequality?

Assume $\eta_t \in \mathbb{R}$ is a given continuous function depending on time. It is known that if we look for a continuous solution $\zeta_t \in \mathbb{R}$ depending on time of the following ...
26
votes
5answers
998 views

How many rearrangements must fail to alter the value of a sum before you conclude that none do?

This will not be altogether unrelated to this earlier question. For which classes $C$ of bijections from $\{1,2,3,\ldots\}$ to itself is it the case that for all sequences $\{a_i\}_{i=1}^\infty$ of ...
1
vote
0answers
29 views

Error bounds for approximation with dyadic sums of polynomials

Are there any bounds known for approximating a genuine multidimensional polynomial function with a sum one-dimensional polynomials over the independent variables? In the 2-dimensional case the ...
0
votes
0answers
91 views

Integral inequality with Hardy's inequality

Let $f\in C_{0}^{\infty}((-1,1))$. Prove that for any $t\in (-1,1)$ we have $$(f(t))^4\le ...