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

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-3
votes
0answers
33 views

What is correlation kernel and compare with gaussian kernel [on hold]

I read a paper that said about correlation kernel that defined: $$W(x-y)=(α/1+d(|y − x|))$$ where $α =  (\int(1+d(y − x)dy)^{-1}$, $(d(|y − x|))$ is spatial Euclidean distance from the central ...
3
votes
0answers
53 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 ...
0
votes
0answers
49 views

A Littlewood-Paley inequality

Littlewood-Paley theory affirms that, for $1 < p < \infty$, $$ \Vert Sf \Vert_p \sim \Vert f \Vert_p, $$ where $$ Sf := (P_k f)_{k \in \mathbb{Z}}, \quad \vert Sf \vert := (\sum_{k\in ...
1
vote
1answer
81 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
66 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
26 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
55 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
71 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
540 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
102 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
51 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
69 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 ...
0
votes
1answer
118 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
votes
0answers
155 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 ...
-2
votes
0answers
23 views

How can I find a tranformation matrix/Mathematical relation between two 5th degree polynomial curves in space? [migrated]

I have the equation of two 5th degree polynomials which they don't intersect with each other. Each curve is made of 100 points and these two curves look similar but there are small differences. I am ...
1
vote
0answers
35 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 ...
0
votes
0answers
70 views

Convergent's numerators of the continued fraction for $\pi$ [migrated]

Call $C_{\pi} = \{ 1,3,22,333,355,… \}$, it´s the sequence of the numerators of convergents of the continued fraction for $\pi$, its OEIS' A002485, http://oeis.org/A002485. Let $n \in C_{\pi}$, such ...
2
votes
1answer
61 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
274 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
55 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 ...
43
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 ...
2
votes
0answers
76 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
139 views

Differentiability of partially defined functions?

This problem arose while reading Lee's Introduction to Smooth Manifolds. Given Banach space: $E,F$ with a plain subset: $A\subseteq E$ and a function: $f:A\to F$ Call a function differentiable at ...
2
votes
1answer
242 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 ...
1
vote
0answers
82 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
240 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
96 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} ...
0
votes
0answers
78 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
215 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
votes
2answers
198 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 ...
1
vote
0answers
63 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 ...
1
vote
2answers
227 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
votes
1answer
194 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
248 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 ...
3
votes
3answers
277 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
232 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
50 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
340 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 ...
3
votes
1answer
78 views

Taylor Series Expansion [closed]

I am reading an article and came across this expression and would appreciate some explanation. "We have a function $$u(F)=2\Big[\frac{F-L}{L}-ln\frac{F}{L}\Big]$$ Since $u''(F)>0$ a Taylor ...
3
votes
1answer
188 views

Can I approximate Schwartz functions which integrate to zero by $C_0^\infty$ functions which integrate to zero?

Let $X$ be the closed subspace of Schwartz space $\mathcal{S}(\mathbb{R}^N)$ defined by \begin{equation*} X=\left\{f\in\mathcal{S}(\mathbb{R}^N):\quad \int f\; dx=0\right\}. \end{equation*} My ...
3
votes
0answers
70 views

Monotonicity of solution of very simple integral equation [closed]

Given p>1 denote by x=x(p) the solution of $$ \int_0^x\frac{dt}{1+t^p}=1. $$ Prove that $p\to x(p)$ is a decreasing function of $p$ on $(1,\infty)$
10
votes
1answer
228 views

smooth Luzin theorem

For measurable functions $f(x)$, $g(x)$ on $[0,1]$ define the distance $\rho(f,g)$ as a Lebesgue measure of the set $\{x:f(x)\ne g(x)\}$. Then Luzin's famous theorem states that $C[0,1]$ is dense with ...
3
votes
1answer
123 views

Generalization of the Hermite-Bielher-Kakeya Theorem

A stable polynomial is one with zeros in the upper half plane or lower half plane. Interlacing polynomials are polynomials with only real zeros, where between every two zeros of one polynomial lies a ...
1
vote
0answers
72 views

Estimating decay of certain trigonometric polynomials

For $p=0,1,2,\dots$ and $n=0,1,2,\dots,$, let $f_{n,p}(z)=\sum_{k=0}^n k^p z^k$ be a sequence of polynomials. Restricted to the unit circle, the functions $g_{n,p}(t):=f_{n,p}(e^{it})$ are ...
2
votes
0answers
166 views

More information on Kruskal's treatment of Surreal numbers as an asymptotic behavior of a real valued function

The only way that I could think about Surreal numbers is how Conway defined them inductively, with the two axioms and so on. I can't find any information about Kruskal's point of view and would very ...
-2
votes
1answer
101 views

a question regarding the interchange the order of finite summation with finite integration [closed]

Question (1) What are the conditions the complex function $f_n(t)$ and real parameter $B>1$ and positive integer $N>1$ need to satisfy such that the interchange of the finite summation with ...
2
votes
0answers
78 views

Diffusion equation on mixing of diffusing particles

I am trying to study mixing of diffusing particles like it was done by E. Ben-Naim On the Mixing of Diffusing Particles. The picture below shows the idea how permutations and inversion numbers reflect ...
4
votes
2answers
267 views

Abstract ODE; PDE; uniqueness of solution

I have a somewhat vague question regarding an abstract ODE in a Banach space. Suppose $A:D(A) \subset X \rightarrow X$ is some linear operator (let's assume it's closed) and maybe add some other ...
17
votes
2answers
589 views

What can be said about the Fourier transforms of characteristic functions?

What can be said about the Fourier transform of the characteristic function $1_A$, where $A\subset \mathbb{R}^n$ is of finite Lebesgue measure? In particular, What properties are common to ...
0
votes
1answer
199 views

Theorem with an example [closed]

i have this theorem in the paper they gives an example: but here $H_1$ is not satisfied ! How to correct it please?