All Questions
5,629 questions
5
votes
0
answers
258
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 $\mathbb{H^2/\Gamma}$ ...
- 528
8
votes
2
answers
785
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 ...
CommunityBot
- 1
1
vote
1
answer
390
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 ...
- 54.2k
20
votes
1
answer
755
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 A.611)
...
CommunityBot
- 1
0
votes
1
answer
274
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 $K?$
...
CommunityBot
- 1
0
votes
0
answers
82
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
$$V_{p}(x, [a, b]) =...
- 1,399
0
votes
1
answer
731
views
Reproducing Kernel of a RKHS of continuous functions may not be continuous in two variables together
Let $\mathcal{K}$ be a Hilbert Space of continuous functions on some topological space, where point evaluations are continuous linear functional on $\mathcal{K}$.
That is $\mathcal{K}$ is RKHS, ...
- 3,085
0
votes
1
answer
770
views
About weak derivatives [closed]
I have a question about weak derivatives.
Let $u,v \in L^{1}_{loc}(U)$ (the space of locally integrable functions) for some
open set $\emptyset \neq U \in \mathbb{R}^{n}$. We often say that $v$ is ...
- 8,086
1
vote
0
answers
150
views
Positivity of alternating series
Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $\limsup_n \frac{1}{n}\log a_n=-\infty$. Then
$$
f(x)=\sum_{n\geq 0}a_n x^n
$$
converges absolutely for all $x$. Under ...
CommunityBot
- 1
4
votes
2
answers
256
views
Sets $X,Y \subset [0,1]$, stronger than being measure $0$, such that $X+Y = [0,2]$
A set $X\subset \mathbb{R}$ is called nice if for every $\epsilon > 0$ there are a
positive integer $k$ and $k$ bounded intervals $I_1,I_2,...,I_k$ such that
$X \subset I_1 \cup I_2 \cup \...
CommunityBot
- 1
2
votes
1
answer
260
views
An elementary functional inequality
Let $g$ be a $C^1$ function with $g(0)=0$ and $g(t)>0$ for all $t>0$. I am surprised that for all such $g$ the following seems to hold
$\frac{\int_0^t(g'(s))^2ds}{g^2(t)}\geq \frac{1}{t}$ for ...
- 810
4
votes
1
answer
262
views
Convergence of sequence of polynomials defined by boundary conditions
I'm sorry if my question sounds trivial, but analysis is not my field.
Consider the interval $[a,b]\subset \mathbb{R}$. On $[a,b]$, for every $n\in\mathbb{N}$, $n\ge 3$, I define the polynomials $P_n:...
- 13k
1
vote
0
answers
52
views
Condition for maximizer of convex combination to be expansion mapping
I have $\Pi_n:\mathbb R^{n+1}\rightarrow \mathbb R$ and $F_n:\mathbb R^2\rightarrow \mathbb R$ with $$F_n(x,a)=\Pi_n(x,...,x,a)$$
$$f_n(x)=\operatorname{ArgMax}_{a\in\mathbb R}\{F_n(x,a)\} $$
such ...
- 43
3
votes
2
answers
181
views
What is the identity of this shift operator-like infinite series?
I just ran across the following expression and would like to know if anyone can identify it:
$\sum\limits_{n=1}^\infty \frac{(-1)^n}{n!}\frac{d^n f(x)}{dx^n}\frac{d^n g(y)}{dy^n}$.
It almost looks ...
CommunityBot
- 1
3
votes
0
answers
47
views
Limit Behavior of Iterated Curvature-Function
What can happen, if one defines an infinite sequence of functions as follows
$f_0\in C^\infty: x\in\mathbb{R}\mapsto y\in\mathbb{R}$
$f_{n+1}: \int_0^x \sqrt{1+(f_n'(t))^2}dt\mapsto\frac{f_n''(x)}{\...
- 13.2k
3
votes
2
answers
259
views
Are all mixtures of these unimodal functions unimodal?
Let us say that a function $F\colon(0,\infty)\to\mathbb{R}$ is increasing-decreasing if, for some $c\in[0,\infty]$, $F$ is non-decreasing on $(0,c]$ and non-increasing on $[c,\infty)$. Is it true that ...
CommunityBot
- 1
0
votes
1
answer
169
views
Unimodality of a certain parametric integral
Suppose $f: [0,1] \to [0,\infty)$ is a smooth, concave and strictly increasing function satisfying $f(0)=0$.
Is it true that the map
$$
F(y) = \int_0^1 \frac{y^{3/2}}{(y+f(x))^2} dx
$$
has exactly one ...
- 128k
1
vote
1
answer
200
views
Is regularity closed under products?
Let $G \colon [0,1] \to [0,1]$ be a differentiable cumulative distribution function (monotonically non-decreasing function with $G(0) = 0$ and $G(1) = 1$). We say that $G$ is regular if $$ x - \frac{1-...
- 128k
0
votes
1
answer
154
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 C^\...
- 16.2k
0
votes
1
answer
2k
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 ...
1
vote
1
answer
980
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 ...
- 1,424
11
votes
0
answers
137
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
$$\int _{\mathbb R^N} e^{-|x|^2}\ \prod_{1\le i<j\le N} \...
- 111
2
votes
1
answer
396
views
$BMO$-property via a John-Nirenberg type estimate?
Let $\Omega \subset \mathbb R^d, d\ge 2$, be bounded and denote a ball in $\mathbb R^d$ by $B$. Denote also
$$
f_B:= \frac1{|B|}\int_B f \, dx.
$$
Suppose $f \in L_{\rm loc}^p(\Omega)$ for all $1<p&...
- 632
0
votes
0
answers
808
views
Inverse Fourier transform of $\frac{1}{\sqrt{\xi_1} + \xi_2}$
Consider the inverse Fourier transform of $\frac{1}{\sqrt{\xi_1} + \xi_2}$. My question is, how can we conclude about the decay properties, support and smoothness of the inverse Fourier transform? I ...
1
vote
0
answers
189
views
Does the Total variation of the Fourier partial sum of a bv function with jumps converge to TV of the function as $N\to\infty$
Does the total variation of the Fourier partial sum of a piecewise continuous bv function converge to the total variation of the function as $N\to\infty$. To explain briefly,
Let $f$ be a periodic ...
- 698
4
votes
0
answers
188
views
Evaluate a multiple integral
I want to compute this integral and I would appreciate any help: $N\geq 1$ is fixed.
$$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) dr_1\...
- 41
1
vote
1
answer
1k
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 ...
- 29.8k
8
votes
1
answer
357
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 $...
- 24.2k
7
votes
1
answer
503
views
Poincaré inequality for curl-integrable functions
Let $B=B(r)$ denote a ball of radius $r$ in $\Omega \subset \mathbb R^d$ and
$$
u_B := \frac1{|B|}\int_B u \, dx.
$$
The standard Sobolev-Poincaré inequality states that if $u \in W^{1,p}(\Omega)$, ...
- 632
2
votes
0
answers
181
views
Is the implication ($f$ is Riemann integrable over $D_1$ and $D_2$) $\Rightarrow $ ($f$ is Riemann integrable over $D=D_1\cup D_2$) true?
Let $D_1,D_2$ be a bounded subset of $\mathbb{R}^n$ and $\partial D_1,\partial D_2$ are both of Lebesgue measure zero (that is to say: $D_1,D_2$ are Jordan measurable).
Also, let $f:D_1\cup D_2=D\...
- 35.5k
1
vote
0
answers
59
views
Analogs of the paralleloram identity in higher degrees
I asked this two months ago in MSE, but nobody answered, so I hope it will be suitable here.
A homogenious polynomial of degree $k\in{\Bbb N}$ on a finite dimensional vector space $X$ over $\Bbb R$ ...
CommunityBot
- 1
4
votes
1
answer
555
views
Construct smooth functions with prescribed derivatives
To be specific, suppose we are given a sequence of smooth functions $\{f_k\}_{k\geq 0}$ on flat torus $\mathbf{T}^2$(you may think of it as doubly periodic functions on $\mathbf{R}^2$ and smooth).
...
- 18.7k
8
votes
1
answer
398
views
A question involving e, floor, and all x > 0
Is $\lfloor(x+1/2)e\rfloor = \lfloor(x+1)(1+1/x)^x\rfloor$ for all $x > 0$?
The question occurred in connection with (nonhomogeneous) Beatty sequences, $\lfloor nr+h\rfloor$, where irrational $r&...
- 54.2k
1
vote
1
answer
261
views
The existence of differential operator of the form $AB=0$
We define $\mathcal A$ is a differential operator of order $n$ with variable coefficients if
$$ \mathcal A:=\sum_{|\alpha|\leq n}A_\alpha (x) D^\alpha $$
where $\alpha$ is an muti-index and $A_\alpha(...
- 21.5k
1
vote
0
answers
111
views
Heat equation inequality
There is an inequality that tells us that for some sufficiently smooth $f$ satisfying $(\partial_t - \Delta )f \le - \delta f^2 +K$ for $\delta,K >0$ that $f$ is bounded by some constant. ...
- 81
9
votes
1
answer
564
views
$L^1$ norm of exponential sum of $n^2 x$
What is the asymptotic order of
$$
\int_0^1 \left| \sum_{n=1}^N e^{2 \pi i n^2 x} \right| ~dx
$$
as $N \to \infty$. This should be known, but I cannot find it in the literature.
- 43.7k
4
votes
1
answer
209
views
Simultaneous approximation of arbitrary functions in Hölder space and in $L^2(\mu)$ by a smooth function and its derivative
Let $\mu$ be a probability measure on the circle $S^1=\mathbb{R}/\mathbb{Z}$ which is singular with respect to the Lebesgue measure $\lambda$. Consider the functions spaces $L^2(\mu)$ on the one hand, ...
- 14.4k
2
votes
1
answer
460
views
Finite trigonometric polynomial
I noticed by numerical and some explicit calculations for a few examples that for real-valued finitely supported functions $\phi \in L^2(\mathbb{R})$ we have that
$T(x):= \sum_{n \in \mathbb{Z}} |\...
- 146
35
votes
19
answers
9k
views
Interesting applications (in pure mathematics) of first-year calculus
What interesting applications are there for theorems or other results studied in first-year calculus courses?
A good example for such an application would be using a calculus theorem to prove a ...
CommunityBot
- 1
0
votes
1
answer
152
views
When can two Cauchy transforms intersect?
Given two polynomials $p$ and $q$ over reals and being guaranteed that both have all roots real I want to know if there is any characterization of the solutions of the equation $\frac{p'}{p} = \frac{q'...
- 54.2k
26
votes
2
answers
2k
views
Analogues of Luzin's theorem
If $X$ is a compact metric space and $\mu$ is a Borel probability measure on $X$, then the space $C(X)$ of continuous real-valued functions on $X$ is a closed nowhere dense subset of $L^\infty(X,\mu)$,...
- 8,903
7
votes
2
answers
437
views
Radial limit does not exist almost everywhere
Problem 4 in Chapter 4 of Stein's book "Real Analysis" says
$\sum_{n\geqslant 0}z^{2^n}$
doesn't have radial limit as $z$ approaches the unit circle from inside almost everywhere. It's fairly easy ...
- 43.7k
11
votes
2
answers
505
views
An inequality for copulas
Suppose that $f$ from $[0,\infty]$ onto $[0,1]$ is completely monotonic on $(0,\infty)$, and let $g$ be the inverse of $f$. For $(u,v)$ in $[0,1]^{2}$, define $C(u,v) = f(g(u)+g(v))$, and let $a = (u+...
- 3,443
5
votes
0
answers
271
views
Is every $C^1$-domain which is homeomorphic to the unit ball in $\mathbb{R}^d$ Lipschitz equivalent to the unit ball?
Suppose we have a domain $\Omega\subset \mathbb{R}^n$ which is homeomorrphic to the unit ball $B(0,1)\subset \mathbb{R}^n$ and such that $\partial \Omega$ is of class $C^1$ (technically, this means ...
- 729
0
votes
1
answer
71
views
Regular curve given implicitly
Let $F:D\subseteq\mathbb{R}^2\to\mathbb{R}$, $D$ open and connected set, be a $C^1 (D)$ application.
What are the minimum requirements for $F$ such that the solutions of the equation $F(x,y)=0$ are ...
- 8,086
4
votes
2
answers
206
views
How to find an ODE with prescribed terminal values?
Let us consider an ODE
$$\frac{dx_t^y}{dt}=g(x_t^y),$$
where y is the initial condition i.e. $x_0^y=y$.
Now, given a function $f$ (increasing and smooth) is it possible to find $g$ (i.e. an ODE) ...
CommunityBot
- 1
7
votes
0
answers
111
views
A monoid-structure on pairs of interlacing polynomials
Let us call a pair of two real polynomials $(P,Q)$ interlacing if $\deg(P)=\deg(Q)+1$, both polynomials have strictly positive leading coefficients and $P,Q$ have only real roots which interlace ...
- 17.9k
2
votes
0
answers
103
views
Writing a function as a sum of functions of bounded diameter
This problem is distilled from one arising in a study of complex random variables, but I've removed as much baggage as I can without (I hope) making it trivial.
Fix $D>0$. A function $f:\mathbb R\...
- 37.7k
8
votes
0
answers
433
views
Heisenberg group: function without vertical derivative
Let $\mathbb H$ be Heisenberg group with vector fields
$$
X=\partial_x - \frac12y\partial_t,\quad Y=\partial_y + \frac12x\partial_t,\quad T=\partial_t
$$
and $U\subset\mathbb H$ is an open set.
I am ...
- 399
4
votes
1
answer
383
views
Horizontal Sobolev space on Carnot group
This question is connected with my previous: Heisenberg group: function without vertical derivative.
Here I am trying to look from another side: what is a difference between Sobolev space and ...
CommunityBot
- 1