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

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2
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1answer
118 views

Convolution vanishes on an interval

Fix a "test" function $f(x)=x\exp(-x^2)$, which is nonzero except $x=0$. Suppose that $g$ is a function with some necessary regularity. Consider the convolution. $$ (f\ast g ...
3
votes
1answer
292 views

How can I show that “almost all function” have property P?

The following is cross-posted from http://math.stackexchange.com/questions/1391293/is-almost-all-function-a-well-defined-concept since I didn't (yet) get an answer there. (I hope that's okay?) ...
2
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0answers
19 views

Is this function $u\in SBV(\Omega)$ also belongs to $L^\infty(\Omega)$?

Some early discussion can be found here. My question: Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Also $\mathcal H^{N-1}(\partial\Omega))<\infty$. Let $u\in ...
5
votes
0answers
183 views

Version of Stone Weierstrass for functions not vanishing at infinity

I am trying to see what is known about uniform density of function spaces in $C(\mathbb{R}^n)$ or $C_b(\mathbb{R}^n)$ (bounded continuous functions on $\mathbb{R}^n$). By uniform density, I mean ...
0
votes
1answer
191 views

Is this set of function belongs to $L^\infty$?

Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $u\in SBV\cap L^\infty(\Omega)$ be given. We write $$ Du = \nabla u\lfloor \mathcal L^N + (u^+-u^-)\otimes \nu_u\mathcal ...
0
votes
2answers
69 views

Smoothness of a power of smooth non-negative function [closed]

Let $f$ be a non-negative infinitely smooth function on the real line. Is it true that for any constant $\alpha$ the function $f^\alpha$ is infinitely smooth?
0
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0answers
90 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 ...
0
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0answers
41 views

Is the heat kernel satisfies the heat equation in viscosity sense?

Let us see the heat kernel \begin{equation} k(x,t)= \begin{cases} (4\pi t)^{-\frac{n}{2}}\exp\{-\frac{1}{4}t^{-1}|x|^{2}\},t>0\\ 0,t\leq0. \end{cases} \end{equation} It is easy to see that $k\in ...
1
vote
1answer
67 views

Image of a Jordan compact set under a degenerate map

This is crossposted from MSE, I hope this is suitable here, since there is no reaction there. I need this lemma for teaching, and I would appreciate any help. Briefly: Is the image of a Jordan ...
6
votes
1answer
165 views

Brownian motion, exists $c < \infty$?

Suppose $B_t$ is a standard Brownian motion. Does there exist $c < \infty$ such that with probability one$$\limsup_{t \to \infty} {{B_t}\over{\sqrt{t \log t}}} \le c?$$I need to know whether or not ...
0
votes
0answers
36 views

Uniform convergence problem of the iterative function series

A process $\{\theta_{t}\}_{t=1}^{\infty}$ with finitely continuous state space $\mathcal{S}=[\underline{\theta},\bar{\theta}]$.The transition density is $\phi(\theta_{t},\theta_{t+1})$.I have known ...
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0answers
45 views

The property reservation conditions in the functional iteration process

Given a integral equation: $$K(y,p)=\int_{a}^{b}f(x,y)K(x,p)dx$$ Using the iteration method,choosing an arbitrary inition function $K^{0}(x,p)$: $$K^{1}(y,p)=\int_{a}^{b}f(x,y)K^{0}(x,p)dx$$ ...
0
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0answers
102 views

Norm inequalities between difference operators

Assume $(v_i)$ to be a sequence in $\ell_\infty(\mathbb{R})$ for $i=1,\dotsc,N$. Define the difference operator as $\Delta v_i:=v_{i+1}-v_i$ and $\Delta^n v_i:=\Delta(\Delta^{n-1} v_i)$. Then, how can ...
0
votes
2answers
84 views

Expected summation of dropped intervals?

For each $n\in\mathbb{N}$, let $I_n$ be an interval of length $1/2^{n}$. We drop each $I_n$ onto the interval $[0,1]$ uniformly at random (so that there is "wraparound" if need be). What is the ...
1
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0answers
50 views

The best constant in Poincare-liked inequality in $BV$ and $BD$ space

This question has been posted on Math Stack exchange for a while and received no response. So I decide to move it here to get more attention. Let $\Omega\subset \mathbb R^N$ be open, bounded and with ...
2
votes
1answer
478 views

What is the rate of convergence? [closed]

How quickly does the series defined by $$x_0 = 0, \ x_{n+1} = \frac{x_n^2+1}{2}$$ converge to $1$?
1
vote
1answer
76 views

question about $TGV^2$ space

Let us just stay in $\mathbb R^1$. The space $TGV^k$ is defined as the function $u\in L^1(I)$ and $$ TGV^k(u,I):=\sup\left\{\int_I u\,\phi^{(k)}\,d\mu, \,\phi\in ...
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0answers
117 views

Is this has anything to do with Riesz representation?

The Riesz representation is very useful in study BV space. There is a lot of version of it and one of the good one can be found in this book, page 49. Here I come up with a question which has similar ...
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0answers
199 views

Existence of topology on the space of continuous functions

Let $C:=C([0,1],\mathbb{R})$ be the space of real-valued continuous functions defined on $[0,1]$. Could we find a topological vector space topology $\pi$ on $C$ such that the following two conditions ...
4
votes
1answer
242 views

What is the importance of convergence of variation of Fourier reconstruction to that of variation of the function?

Let $f$ be a periodic function of bounded variation which jumps at a point $x_0\in\mathbb{R}$. Let $S_{N}[f]$ denote the partial Fourier sum of $f$ and let $C_{N}[f]$ denote the Cesaro partial sum. It ...
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0answers
51 views

How to modify a SBV convergence sequence to obtain uniform integrability?

Given $\Omega\subset \mathbb R^N$ is open bounded with smooth boundary. Assume $(u_n)\subset SBV(\Omega)$ a sequence of functions such that $u_n\to u_0$ weakly in $SBV$ for some function $u_0\in ...
4
votes
0answers
136 views

Equidistribution of spheres in $\mathbb{R^2}/\mathbb{Z^2}$

Let $\mathbb{H^2}$ be the hyperbolic upper half place, and let $\Gamma$ be a lattice in $SL(2,\mathbb{R})$ acting on $\mathbb{H^2}$. A proof of the equidistribution of spheres on $\mathbb{H^2/\Gamma}$ ...
5
votes
2answers
178 views

Is taking the product of signed measures weakly continuous?

For a Polish space $X$, let $C_b(X)$ denote the real Banach space of bounded continuous real-valued functions on $X$. Let $M(X)$ denote the space of all finite signed Borel measures on $X$, equipped ...
0
votes
1answer
117 views

Square Integrable Harmonic Functions in an Infinite Strip

Suppose $S= \left\{x \in \mathbb{R}^3 : a <x_1< b \right\} $ is an infinite strip the three dimensional Euclidean Space. Is it true that the only $L^2$ harmonic function in this strip is the ...
0
votes
1answer
195 views

On Cantor sets every map is $C^{\infty}$ [closed]

For a fixed Cantor set $K\subset [0,1]$ and a continuous function $g:[0,1]\to \mathbb R.$ Is it always possible to find a $C^{\infty}$ map $f:[0,1]\to \mathbb R$ such that $g$ and $f$ coincide in $K?$ ...
0
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0answers
42 views

Construction of a path of quadratic variation

This question has been posted to Stack Exchange earlier, and no answer is available yet. Consider a path $x: [0,1] \to \mathbb R$. it's $p$-variation on an interval is defined by $$V_{p}(x, [a, b]) ...
18
votes
1answer
537 views

Minimum value of $|p(1)|^2+|p(2)|^2 +…+ |p(n+3)|^2$ over all monic polynomials $p$

Let $n$ be a positive integer. Determine the smallest possible value of $|p(1)|^2+|p(2)|^2 +...+ |p(n+3)|^2$ over all monic polynomials $p$ of degree $n$. This question was proposed (problem ...
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vote
2answers
172 views

Lower bound for $ \sum_{i=1}^n x_i f(x_i)$ when $\sum_{i=1}^{n}x_i = K$

Considering, the set of all n dim. vectors $\{x_i\}_{i=1,...,n} $ such that $x_i \geq 0 $ and $\sum_{i=1}^{n}x_i = K$ Any continuous and strictly increasing function $f^+(x)$ : $ \mathbb R^+ \to ...
0
votes
1answer
144 views

About weak derivatives [closed]

I have a question about weak derivatives. Let $u,v \in L^{1}_{loc}(U)$ (the space of locally integrable functions) for some open set $\emptyset \neq U \in \mathbb{R}^{n}$. We often say that $v$ is ...
0
votes
1answer
99 views

Reproducing Kernel of a RKHS of continuous functions may not be continuous in two variables together

Let $\mathcal{K}$ be a Hilbert Space of continuous functions on some topological space, where point evaluations are continuous linear functional on $\mathcal{K}$. That is $\mathcal{K}$ is RKHS, ...
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0answers
107 views

Positivity of alternating series

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $\limsup_n \frac{1}{n}\log a_n=-\infty$. Then $$ f(x)=\sum_{n\geq 0}a_n x^n $$ converges absolutely for all $x$. Under ...
4
votes
2answers
214 views

Sets $X,Y \subset [0,1]$, stronger than being measure $0$, such that $X+Y = [0,2]$

A set $X\subset \mathbb{R}$ is called nice if for every $\epsilon > 0$ there are a positive integer $k$ and $k$ bounded intervals $I_1,I_2,...,I_k$ such that $X \subset I_1 \cup I_2 \cup ...
2
votes
1answer
196 views

An elementary functional inequality

Let $g$ be a $C^1$ function with $g(0)=0$ and $g(t)>0$ for all $t>0$. I am surprised that for all such $g$ the following seems to hold $\frac{\int_0^t(g'(s))^2ds}{g^2(t)}\geq \frac{1}{t}$ for ...
1
vote
1answer
84 views

Convergence of sequence of polynomials defined by boundary conditions

I'm sorry if my question sounds trivial, but analysis is not my field. Consider the interval $[a,b]\subset \mathbb{R}$. On $[a,b]$, for every $n\in\mathbb{N}$, $n\ge 3$, I define the polynomials ...
1
vote
0answers
34 views

Condition for maximizer of convex combination to be expansion mapping

I have $\Pi_n:\mathbb R^{n+1}\rightarrow \mathbb R$ and $F_n:\mathbb R^2\rightarrow \mathbb R$ with $$F_n(x,a)=\Pi_n(x,...,x,a)$$ $$f_n(x)=\operatorname{ArgMax}_{a\in\mathbb R}\{F_n(x,a)\} $$ such ...
3
votes
2answers
144 views

What is the identity of this shift operator-like infinite series?

I just ran across the following expression and would like to know if anyone can identify it: $\sum\limits_{n=1}^\infty \frac{(-1)^n}{n!}\frac{d^n f(x)}{dx^n}\frac{d^n g(y)}{dy^n}$. It almost looks ...
3
votes
0answers
39 views

Limit Behavior of Iterated Curvature-Function

What can happen, if one defines an infinite sequence of functions as follows $f_0\in C^\infty: x\in\mathbb{R}\mapsto y\in\mathbb{R}$ $f_{n+1}: \int_0^x ...
2
votes
2answers
136 views

Are all mixtures of these unimodal functions unimodal?

Let us say that a function $F\colon(0,\infty)\to\mathbb{R}$ is increasing-decreasing if, for some $c\in[0,\infty]$, $F$ is non-decreasing on $(0,c]$ and non-increasing on $[c,\infty)$. Is it true that ...
0
votes
1answer
326 views

Collection of graduate research projects in Real Analysis [closed]

While there are many open problems in Real Analysis like Khabibullin's conjecture or Lehmer's conjecture, those are big enough to take an expert's life for several years, let alone some graduate ...
0
votes
1answer
141 views

Sobolev type embedding

Consider a compact manifold $M$ and a point $q \in M$. Let us say that that the following inequality holds: $$ \Vert \varphi u\Vert_{L^p} \leq C\Vert \varphi u\Vert_{H^1},$$ where $\varphi \in ...
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0answers
65 views

Assymptotics of a Selberg type integral

Does any one know some references/ ideas on how to study the assymptotics as $N$ goes to $\infty$ of the following Selberg type integral $$\int _{\mathbb R^N} e^{-|x|^2}\ \prod_{1\le i<j\le N} ...
0
votes
1answer
129 views

Unimodality of a certain parametric integral

Suppose $f: [0,1] \to [0,\infty)$ is a smooth, concave and strictly increasing function satisfying $f(0)=0$. Is it true that the map $$ F(y) = \int_0^1 \frac{y^{3/2}}{(y+f(x))^2} dx $$ has exactly one ...
1
vote
1answer
154 views

Extension of a smooth function from a convex set

Let $C\subset \mathbb{R}^{n}$, $C'\subset\mathbb{R}^{m}$ be two convex sets with a non-empty interior. A function $F\: : \: C\to C'$ is said to be differentiable at $x\in C$ if there exists a linear ...
7
votes
1answer
3k views

Eliminating Gibbs phenomenon, and approximating with jumping functions in Fourier Analysis : An attempt and a question in this regard

This problem seems like a nightmare to me. I tried to expand $K_{\omega}^f(t)$, but I am clueless of getting some kind of a closed form or some kernel like structure. If I try to take ...
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0answers
99 views

Inverse Fourier transform of $\frac{1}{\sqrt{\xi_1} + \xi_2}$

Consider the inverse Fourier transform of $\frac{1}{\sqrt{\xi_1} + \xi_2}$. My question is, how can we conclude about the decay properties, support and smoothness of the inverse Fourier transform? I ...
2
votes
1answer
202 views

$BMO$-property via a John-Nirenberg type estimate?

Let $\Omega \subset \mathbb R^d, d\ge 2$, be bounded and denote a ball in $\mathbb R^d$ by $B$. Denote also $$ f_B:= \frac1{|B|}\int_B f \, dx. $$ Suppose $f \in L_{\rm loc}^p(\Omega)$ for all ...
4
votes
0answers
164 views

Evaluate a multiple integral

I want to compute this integral and I would appreciate any help: $N\geq 1$ is fixed. $$I_N=\int_{0\le r_n\le r_{n-1}\le\cdots\le r_1} e^{-(r_1^2+\cdots+r_n^2)} \prod_{i<j} \sinh(r_i-r_j) ...
1
vote
0answers
73 views

Does the Total variation of the Fourier partial sum of a bv function with jumps converge to TV of the function as $N\to\infty$

Does the total variation of the Fourier partial sum of a piecewise continuous bv function converge to the total variation of the function as $N\to\infty$. To explain briefly, Let $f$ be a periodic ...
1
vote
1answer
152 views

First mean value theorem for integration and Lebesgue measureability

According to first mean value theorem for integration, if $G \ : \ [a,b] \to \mathbb{R}$ is a continuous function, there exists $x \in (a,b)$ such that $$\int_a^b G(t) dt = G(x)(b-a)$$ Assume $G$ is ...
8
votes
1answer
260 views

Exceptional values of real-valued functions on [0,1]

Given a continuous real-valued function $f$ from $[0,1]$ to itself with $f(0)=0$ and $f(1)=1$ such that $f^{-1}(c)$ is finite for all $c$ in $[0,1]$, let $E(f)$ be the set of $c$ in $[0,1]$ such that ...