Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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-3
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0answers
26 views

Condition to be Fourier coefficients for an periodic function [on hold]

It is well known that Dirichlet's theorem can be a sufficient condition for a question whether a periodic function $f(t)$ has its Fourier series expansion. Inversely, I wonder what condition is ...
1
vote
1answer
61 views

Ask for a special function related to the error function

I am wondering whether anyone knows the following integration has a named special function or a reference $$ F_{a,b}(z) :=\frac{2}{\sqrt{\pi}} \int_0^z \text{erf}(a+b y)\: e^{-y^2} \text{d}y $$ for ...
4
votes
1answer
53 views

Integral Expression in Complex Dynamics

Let $\phi\in \mathbb{C}(z)$ be a degree $d\geq 2$ rational map, which we can write as $\phi = \frac{f}{g}$ for $f,g\in \mathbb{C}[z]$. Let $\omega_{FS}$ denote the Fubini-Study form on ...
5
votes
0answers
39 views

Calculation of characteristic strip, solutions of a geometric PDE [on hold]

Say I have a family of spheres of radius $1$ with centers in the $xy$-plane$$u = G(x, y, \lambda, \mu) = \sqrt{1 - (x - \lambda)^2 - (y - \mu)^2}.$$I've found that $u_x = -(x - \lambda)u^{-1}$ and ...
1
vote
0answers
41 views

A sufficient condition of the following integral to be positive

Suppose $G=(A,B,E)$ is a bipartite graph whose partition has the parts $U=\{u_1,\cdots,u_m\}$ and $V=\{v_1,\cdots,v_n\}$. Consider the following integral ...
2
votes
0answers
42 views

On two functions with isodirectional gradients

Let $U\subset \mathbb{R}^n$ be open and $f,g:U \to \mathbb{R}$ be two $C^1$ functions whose gradients are always in the same direction, i.e. $\forall i,j \in \left\{1,...,n\right\}$ \begin{equation} ...
0
votes
0answers
23 views

Joint distribution of eigenvalue and matrix entry

Let $X=(X_{n,m})_{n,m=1}^N$ be a $N\times N$ GUE random matrix, and let $\lambda_1,\dots,\lambda_N$ denote its unordered eigenvalues. What can be said about the distribution of, say, ...
-3
votes
0answers
35 views

the hessian and the gradient [on hold]

$A\in{\mathbb{R^{m\times n}}}$, $b\in{R^m}$. For $ x\in{\mathbb{R^n} }$, we define q(x)=f(Ax+b), with $f:\mathbb{R^m}\longrightarrow{\mathbb{R}}$. Calculate the hessian matrix and the gradient of f .
-2
votes
0answers
65 views

Prove a function is entire

Let $f$ function such that $\int_{R^m} (1+|y|)^N|f(y)|dy <\infty$. Consider a function $g(z) = \int h(y,z) f(y) dy,$ where $|h(y,z)| \leq C e^{|z||y|}$ with $z\in C^m$ and ...
0
votes
0answers
70 views
+50

Closed-Form solution for system of simple nonlinear equations

I am interested in analytical solutions for a system of nonlinear equations. (The question was first asked at math.SE, where (after 1months and one rounds of bounty) there is only interesting ...
0
votes
0answers
36 views

Limit probability of a complete bipartite random graph $G(n,n,p)$ is connected [on hold]

I need to calculate the following probability limit for a complete bipartite random graph $G(n,n,p)$ in the Erdos-Renyi model: \begin{equation} \lim_{n\rightarrow\infty}\mathbb{P}[G(n,n,p) \text{ is ...
1
vote
0answers
45 views

Show this function is strictly concave

Please help me show that $f(w)$ is strictly concave in $w\in[0,\infty)$: $f(w)=\sum_{j=1}^N P_j (w)\cdot u_j $ where $P_j (w)=\sqrt{w}\int _{-\infty}^{\infty}\Pi_{k\neq ...
0
votes
1answer
49 views

Request for references about computing or estimating Rademacher complexity

Is Rademacher complexity defined for any space of functions? Or are there restrictions on the function space over which this can be defined? For example is the Rademacher complexity defined or has ...
5
votes
1answer
75 views

Density of the linear span of products of harmonic polymomials

Let $\mathcal{H}$ denote the space of all harmonic polynomials with complex coefficients in $n$ variables $x_1,\ldots, x_n$ in $\mathbb{R}^n$. I'm trying to show that the linear span of the set ...
2
votes
0answers
47 views

How much must a curve bend to intersect another curve twice?

Suppose $c_1$ and $c_2$ are segments of smooth plane curves. To be concrete, say $c_1$ and $c_2$ are graphs of smooth functions $f_i:[a_i,b_i]\to \mathbb R$, $i=1,2$. If the curves were lines, then ...
0
votes
0answers
16 views

Comparison of lengths: Theodorus spiral and Archimedes' Spiral

It is known that the Spiral of Theodorus approximates the Archimedean spiral. Let's consider the arc-length of the Spiral of Theodorus $L_T(\phi)$ and the Archimedean one: $L_A(\phi)$. Can we ...
0
votes
0answers
88 views

local analyticity of volumes of slices of semi-algebraic sets

I would like a reference and/or a simple proof using well-known results of the following, which I think is true. (If it's false, I'd like to know that as well of course -- and ideally a way to modify ...
1
vote
0answers
44 views

Dirichlet series decomposition of arbitrary function

Originally asked on MSE here: http://math.stackexchange.com/q/1780149/52694 Analytic functions can be decomposed into a Taylor series, and furthermore the Taylor series converges back to the original ...
2
votes
1answer
169 views

Extension of a function from almost everywhere to everywhere

The informal general question is: let $f$ be a "sufficiently nice" function, defined "almost everywhere". Can we develop a method to uniquely extend $f$ to the "remaining" points? Example: Let ...
1
vote
0answers
27 views

Optimizing sum of approximate and exact functions

This is a research question that I had asked in Math.SE about a month ago, but even after putting a bounty on it, I did not get any answers. I have two real values functions, where one ...
5
votes
1answer
180 views

Fourier transform surjective on $L^p(\mathbb{R}^n)$ for $p \in (1,2)$?

I know that $F_2:L^2 \rightarrow L^2$ is of course unitary, whereas $F_1:L^1 \rightarrow C_0$ is injective but not surjective. This can be seen by looking at the dual map. Riesz-Thorin gives us that ...
3
votes
1answer
76 views

Algorithm for definite integral of rational functions of x and exp(-x)

I'd like to find the class of functions that may appear as definite integrals of a rational function of $x$ and $\exp(-x)$ from $0$ to $\infty$. For that I imagine there might be an algorithm to ...
-3
votes
0answers
71 views

Bernoulli's Inequality when $-2≤x<-1$ [migrated]

Why is it that Bernoulli inequality $(1+x)^r>1+rx$ is said to be true for every integer $r≥0$ and every real $x≥-1$; why the range $-2≤x<-1$ is not included? It seems that, by induction (or ...
3
votes
2answers
277 views

A possible norm on a subspace of $C^\infty([0,1])$?

I have posted the following question (with minimal differences) on MSE some days ago, without receiving a satisfactory answer, so let me try here again. Take the vector space of infinitely ...
1
vote
0answers
43 views

Quotient of two smooth functions extension

Assume we are given smooth functions $f, g: U \to \mathbb{C}$, where $0 \in U \subset \mathbb{R}^n$ is open and $0 \in g^{-1}(0) \subset \{x_n = 0\}$. Furthermore, suppose that $\nabla g \neq 0$ on ...
1
vote
0answers
76 views

extension for a complex operator

Let be $\lambda>0$. Put $$ L_{\lambda}=\Big[-\frac{\partial^{2}}{\partial z \partial \overline{z}}+\lambda^{2}|z|^{2} +\lambda\Big(\overline{z}\frac{\partial}{ \partial ...
2
votes
1answer
95 views

Can there be a numerical system in which logarithms can be expressed in terms of exponentials in closed form?

The invention of complex numbers allowed to express trigonometric functions through hyperbolic ones in closed form. Is there possible an extension of real/complex numbers in which logarithms and ...
5
votes
0answers
148 views

Complex Stone-Weierstrass Type problem

I have come across this problem which resembles complex Stone-Weierstrass theorem except for a problem that the conjugate of the functions are not necessarily in the sub algebra. Suppose $\Omega$ is ...
2
votes
1answer
89 views

Are compactly supported continuous functions dense in the Continuous functions of Sobolev space? [closed]

I have a question about Sobolev space. Let $\Omega$ be an open subset of $\mathbb{R}^{d}$, we consider the Sobolev space $H^{1}(\Omega):=\left\{ u \in L^{2}(\Omega) : D_{j}u \in L^{2}(\Omega), ...
1
vote
0answers
37 views

The decay rate of the spectrum of the Gaussian kernel on compact manifolds

It seems that the $k^{th}$ largest eigenvalue of the intergral operator induced on $S^n$ by the Gaussian kernel, $e^{-\frac{\vert \vec{x} - \vec{y} \vert _2^2}{2\sigma^2}}$ decays as $k^{-k}$. This is ...
2
votes
1answer
78 views

A continuity/bootstrap argument

I am trying to understand how one can prove the following assertion using a continuity argument: Let $0<\epsilon<\epsilon_0$. Let $I=[t_0,R]$ be a compact interval. Suppose that $S:I\to ...
2
votes
3answers
235 views

Nonzero solutions of an infinite product

Let $-\frac{1}{2}\le a \le\frac{1}{2}$ and $b\in[0,\infty)$. Definitions: $$f_k(a;b):=\frac{(2k+\frac{1}{2}+a)^2+b}{(2k+\frac{1}{2}-a)^2+b}(\frac{k}{k+1})^{2a},$$ $$f(a;b):=\prod\limits_{k=1}^\infty ...
1
vote
0answers
59 views

Multimodal property of polynomial logistic distribution

Let $P(x)$ be a polynomial (of an odd degree $n$) strictly increasing on $(-\infty, +\infty).$ Then $F(x)=\displaystyle \frac{1}{1+\exp\{-P(x)\}}$ is a distribution function of a polynomial logistic ...
3
votes
1answer
180 views

Fréchet L-Spaces

According to the paper The emergence of open sets, closed sets, and limit points in analysis and topology famous mathematician Maurice Fréchet who introduced the concept of metric spaces has also ...
0
votes
0answers
48 views

Bounds on the the spherical harmonics on $S^{p-1}$

The only reference I could find in this regard is for upper bounding the n-homogeneous spherical harmonics on $S^{p-1}$ as in equation 4.29 here, ...
1
vote
1answer
56 views

linear recurrence inequality of positive terms

This is a follow up on my previous linear recurrence inequality question. I have some matrices which satisfy a linear recurrence formula of the form $$ A_{n+1} = \alpha A_{n} + \beta A_{n-1},\qquad ...
2
votes
1answer
65 views

linear recurrence inequality

Given two real analytic functions, $g(x)$ and $f(x)$, on an open interval $I\subset \mathbb{R}$, it is obvious that $g(x) \leq f(x)$ does not imply $g_n \leq f_n$ (here $g_n = [x^n] g(x)$ denotes the ...
2
votes
1answer
73 views

On cluster points of a particular sequence

This is the sequel of a previous question. Let us consider the sequence $$ \xi_n = 2n \{n\xi\}-n, $$ where $\xi>0$ is a given real irrational number and $\{\cdot\}$ is the fractional part. Do ...
0
votes
0answers
60 views

Functions $f \in S(\mathbb{R})$ with slow decay rate?

I came across this question by thinking about whether there are functions $f \in S(\mathbb{R})$ that satisfy for all(!) $a>0$ that $f e^{a|.|^a} \notin L^{\infty},$ as somebody on ...
13
votes
2answers
482 views

Is there a $C_c^{\infty}( \mathbb{R}^d)$ function whose Fourier transform we can explicitly write down?

I noticed that although $C_c^{\infty}$-functions are dense in some quite large spaces and well understood (especially their Fourier transform) I have never encountered an explicit example of a ...
1
vote
0answers
18 views

Concavity of maxima [closed]

Suppose we have the following optimization problem : $\min\limits_x kf(x) + g(x)$ where $f$ is a decreasing convex function in $x$ and $g$ is an increasing convex function. Can we say that $x^*$ is ...
2
votes
0answers
116 views

Conditions for continuity of non-simple eigenvectors

Here, http://math.stackexchange.com/a/1146455, it is noted that eigenprojections are continuous, but eigenvectors are not. Are there any conditions where the eigenvalues are not simple, but the ...
1
vote
0answers
52 views

A problem on Markov chains and Dirichlet forms

Let $X$ be a countable set. Let $c:X\times X\to[0,+\infty)$ satisfy $$c(x,y)=c(y,x)\text{ for all }x,y\in X,$$ $$m(x)=\sum_{y\in X}c(x,y)\in (0,+\infty)\text{ for all }x\in X,$$ $$c(x,x)=0\text{ for ...
0
votes
1answer
55 views

Long term behavior of a certain discrete time dynamical system on graphs

Consider the graph $(V,E)$ with vertex set $V=\{v_1,...,v_n\}$ and edge set $E\subset V\times V$. Further, assume that $\forall v_i\in V, (v_i,v_i)\in E$. Assume that each vertex has an ...
2
votes
0answers
45 views

Harmonic functions in tempered distribution sense

Suppose $g$ is a metric on $\mathbb{R}^3$ and $\Omega \subset\subset \mathbb{R}^3$. We assume that $g$ is euclidean outside $\Omega$. My question concerns solutions to $\triangle_g u =0$ that are say ...
4
votes
1answer
264 views

Two similar integrals

Let $n$ be a given even positive integer. We have the following integral \begin{align} ...
2
votes
1answer
147 views

Simplify proof for rapidly decaying functions

I want to show the following theorem in a lecture: Let $F \in C^{\infty}(\mathbb{C}^{k}, \mathbb{C})$ such that $F(0)=0.$ Let $G: \mathbb{R}^n \rightarrow \mathbb{C}^{k}$, $x \mapsto ...
3
votes
1answer
172 views

Is the number of intervals you get by slicing the closed region under a nice graph continuous from below?

Let $I$ be a closed interval in $\mathbb{R}$, and let $f:I \to \mathbb{R}$ be a bounded function, smooth except at finitely many points (piece-wise smooth). There are also only finitely many critical ...
1
vote
1answer
63 views

Partition of Real Number [closed]

Can the set of Real numbers be partioned into two parts such that both are uncountable,dense and have empty interior and any closed interval intersects both at uncountably many points?
0
votes
0answers
88 views

Does this set of (structured) equations always have a solution?

Let $r_1,\ldots,r_K$ be arbitrary positive numbers. Does $$|\mathcal{A}|\log\left(1+\frac{1}{|\mathcal{A}|}\left(\sum_{n\in \mathcal{A}} \sqrt{x_n(\exp(r_n)-1)}\right)^2\right)\leq \sum_{n\in ...