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

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4
votes
1answer
179 views

Density argument with Schwartz functions?

I was wondering whether the Schwartz functions are also dense in $$\{f \in L^2(\mathbb{R}^n); \int_{\mathbb{R}^n} |x|^2 |f(x)|^2 dx + \int_{\mathbb{R}^n}|\xi|^2 |\hat{f}(\xi)|^2 d \xi < \infty\}$$ ...
-4
votes
2answers
180 views

Continuous map from $\mathbb R^2$ to $\mathbb R$? [closed]

There must be a map from $\mathbb R^2$ to $\mathbb R$, since they are the same cardinality. But is there a construction for a continuous map from $\mathbb R^2$ to $\mathbb R$? I guess what I mean is ...
1
vote
2answers
244 views

How can we obtain the $-\frac{4\pi}3\mu(x)$ term?

Given the expression $$K_{ik} := \frac{\partial}{\partial x_k} \int_{\mathscr X} \frac{y_i-x_i}{|y-x|^3} \mu(y) dy,$$ where $\mathscr X=\mathbb R^3$, how does one derive the expression \begin{align} ...
1
vote
1answer
99 views

Is there a way to solve this integral equation?

I have ran into the following integral equation as part of my phd research project, trying to enforce a boundary condition of a parabolic pde problem. For $\xi = (\alpha\theta)^{1/\alpha}$ and for ...
0
votes
0answers
53 views

Bounds on the positive roots of a bivariate polynomial

It is well known that various real root isolation methods are based on computing, first, the bounds on the values of the positive real roots of a polynomial equation. For the univariate case such ...
10
votes
2answers
839 views

Witness to a failure of Fubini/Tonelli

Is it provable in ZFC that there is a subset of the plane all of whose vertical cross sections have Lebesgue measure zero and all of whose horizontal cross sections are complements of sets of Lebesgue ...
12
votes
0answers
170 views

Concept associated to the Eudoxus reals

I am aware of three different constructions of the field of real numbers : The Cauchy sequence construction : in this case, we see the field $\mathbb{Q}$ as a metric space and $\mathbb{R}$ is the ...
0
votes
0answers
55 views

Fourier analytic estimate

The following question arises naturally from applications to the image processing. Let $\alpha\in [0,1]$ and assume that for infinitely many $n\ge 1$ we have $$\sum_{k=1}^n\frac{1-|\cos(2\pi k\alpha)|...
2
votes
0answers
37 views

Points of continuity of a lower semicontinuous function have non empty interior

Let $X\subset \mathbb R^d$ having non empty interior and let $f:X\to\mathbb{R}$ be lower semicontinuous. I know that the set of discontinuities of such a function is contained in a meager set, and ...
-2
votes
1answer
50 views

Is every implicit function reparametrized? [closed]

Consider a continously differentiable non-constant function $f:\mathbb{R}^2\to\mathbb{R}$. Define $$ K=\{x\in\mathbb{R}^2|f(x)=0\}. $$ I wish to know whether there is a continuously differentiable ...
6
votes
2answers
234 views

Can a nowhere differentiable function preserve measurability?

I want to know whether a continuous nowhere differentiable function $f: \mathbf{R} \to \mathbf{R}$ can map Lebesgue measurable sets to Lebesgue measurable sets. More generally I'm interested to know ...
2
votes
1answer
148 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
67 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 $\mathbb{P}^1(\...
2
votes
0answers
51 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
34 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, $$\lvert\lambda_1-...
0
votes
0answers
106 views

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 ...
1
vote
0answers
49 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 j}\{\Phi[\sqrt{w}(v-u_k)]\}...
0
votes
1answer
53 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 ...
6
votes
1answer
130 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
52 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
23 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
134 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
48 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
187 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 $f(x)=\...
0
votes
0answers
33 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 ($g(w;x):\...
5
votes
1answer
211 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
78 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
2answers
288 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
51 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
81 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 \overline{z}}-z\frac{\...
2
votes
1answer
102 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
152 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
98 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), j=1,...
1
vote
0answers
41 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
88 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 [0,\...
2
votes
3answers
241 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
63 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
190 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
53 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, http://www.fen.bilkent.edu.tr/~gurses/...
1
vote
1answer
57 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 n\...
2
votes
1answer
68 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
74 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
62 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 math....
14
votes
2answers
519 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
118 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
57 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
58 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 $\textit{...
2
votes
0answers
58 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
269 views

Two similar integrals

Let $n$ be a given even positive integer. We have the following integral \begin{align} \int_0^{\infty}\cdots\int_0^{\infty}e^{-(x_1+\cdots+x_n+y_1+\cdots+y_n)}\prod\limits_{i=1}^n\prod\limits_{j=1}^n(...