All Questions
712 questions
7
votes
1
answer
449
views
Stronger version of Besicovitch covering theorem
I'm wondering if the following strengthening of the Besicovitch covering theorem holds: Suppose $A\subset\mathbb R^n$ is a bounded subset and suppose $x\mapsto r_x$ is a function $A\to(0,\infty)$. Is ...
- 1,761
7
votes
2
answers
1k
views
Two different kinds of definitions of $C^k(\overline{\Omega})$ — extension and restriction
This is cross-posted in MSE.
I have seen two different kinds of definitions of the notation $C^k(\overline{\Omega})$ — by "extension" of functions on $\Omega$ or by "restriction" of functions on $\...
user14319
6
votes
1
answer
729
views
An $L^1$ function but (really) no better?
Question: For a smooth, bounded domain $\Omega\subset \mathbb R^d$, does there exist a function $u\in L^1(\Omega)$ such that
$u\not\in L^\Phi(\Omega)$ for any Orlicz space $\Phi$?
For the definition ...
- 5,380
6
votes
0
answers
405
views
Using the Lorentz operators to build polynomials that converge to a continuous function
Questions
Let $f(\lambda):[0,1]\to (0,1)$ have a $\beta-\lfloor\beta\rfloor$)-Hölder continuous $\lfloor\beta\rfloor$-th derivative, where $\beta>0$.
Find explicit bounds, with no hidden constants,...
- 697
6
votes
3
answers
748
views
Clarification and Proof of Inequality (8.11) in Analytic Number Theory by Iwaniec and Kowalski
I am studying inequality (8.11) from Analytic Number Theory by Iwaniec and Kowalski. It is found on top of page 200. In bottom of page 199, the authors prove that
$$
|S_f(N)|^2 \leq N + \frac{2N^2}{q} ...
- 111
6
votes
1
answer
2k
views
Sobolev functions on $\mathbb{R}^N$ cannot be discontinuous on a $(N-1)$-dimensional submanifold
How can one prove (or where can I find a proof) that if $u \in W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$, then $u$ cannot have a $(N-1)$-manifold of discontinuity points?
- 839
6
votes
2
answers
225
views
On a trigonometric inequality by Huygens
The following inequality, ascribed to Huygens, appeared in this post:
\begin{equation*}
1-\frac43\,\frac{\sin^3\theta/2}{\theta-\sin\theta}
>(1-\cos\theta/2)\Big(\frac35-\frac3{1400}\frac{\...
- 128k
6
votes
3
answers
1k
views
Dependence of error on mesh for Riemann sums
Suppose $f$ is continuous on $[a,b]$ with $I = \int_a^b f(x)\: dx$,
and for every $\epsilon > 0$ let $\delta(\epsilon)$ be the largest
$\delta > 0$ such that every Riemann sum arising from a ...
- 19.7k
6
votes
2
answers
409
views
Existence and uniqueness of an Euler-type ODE with varying parameters
Consider this ODE on $[1, \infty)$
$(r^2 - 2ar)f''(r) + 2(r-a) f'(r) - ({4a} + m(m+1))f(r) = -4af(1) $
with initial conditions
$\frac{a}{1-2a} f(1) + f'(1) = C, \qquad \lim_{r\to \infty} f(r) = 0$
...
- 969
6
votes
1
answer
423
views
What is the Borel complexity of this set?
Problem. What is the Borel complexity of the set
$$c(\mathbb Q)=\{(x_n)_{n\in\omega}\in\mathbb R^\omega:\exists\lim_{n\to\infty}x_n\in\mathbb Q\}$$
in the countable product of lines $\mathbb R^\omega$?...
- 41.8k
6
votes
1
answer
2k
views
Analysis of solutions to a nonlinear ODE
Consider the following ODEs:
$\phi^2=\phi''\sqrt{1-\phi'^2}$, or $\phi^2=-\phi''\sqrt{1-\phi'^2}$.
Is there any theory (e.g. comparison theorems) which analyzes solutions of the above ODEs? I am only ...
- 591
6
votes
1
answer
601
views
Monotonicity of eigenvalues
We consider block matrices
$$\mathcal A = \begin{pmatrix} 0 & A\\A^* & 0 \end{pmatrix}$$ and
$$\mathcal B = \begin{pmatrix} 0 & B\\C & 0 \end{pmatrix}.$$
Then we define the new matrix
$...
- 536
6
votes
2
answers
303
views
Is there a set $S\subseteq [0,1]$ with $|S|=2^{\aleph_0}$ and distinct pairwise distances?
Short version of question. Is there a set $S\subseteq [0,1]$ with $|S|=2^{\aleph_0}$ such that all points of $S$ have distinct pairwise distances?
Formal version of question. If $X$ is a set, let $[X]...
- 48.9k
6
votes
1
answer
313
views
Convergence of integral averages in $L^1$
Let $f \in L^1 (\mathbb R)$. Suppose $g_n \in L^1 (\mathbb R)$ are a sequence of positive functions.
Define, for each $n$, the function $f_n$ by
$$f_n (x) := \frac{1}{2g_n (x)} \int_{x - g_n (x)}^{x + ...
- 6,155
6
votes
2
answers
336
views
On frequency decay of an integral transform of a function
Suppose $f \in C^{\infty}_c((-1,1))$ and assume that there exists constants $a,b>0$ such that
$$
\bigg|\int_{\mathbb R} f(t) \,e^{\tau t^2+i\tau t}\,dt\bigg| \leq a\,e^{-b|\tau|},$$
for all $\tau \...
- 4,135
6
votes
1
answer
901
views
Fundamental Theorem of Algebra, via algebra
I know there are already a couple of questions on this on the site, but I haven't seen an answer to this particular form...
We know, from the Fundamental Theorem of Algebra, that the complex ...
- 1,687
6
votes
2
answers
401
views
Intuition and analogue of Wraith axiom from synthetic differential geometry
In synthetic differential geometry, an object $M$ verifies the Wraith axiom if for all functions $\tau:D\times D\to M$ which are constant on the axes $\tau(d,0)=\tau(0,d)=\tau(0,0)$ for all $d\in D$, ...
- 10.5k
6
votes
1
answer
816
views
Is the $L^\infty$ norm of the derivative the same under the Hausdorff and Lebesgue measure?
Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure, and $\|f\|_{L^\infty (\mathcal H^k)}$ denotes the $L^\infty$ norm of a function $f$ with respect to $\mathcal H^k$.
Let $\Omega$...
- 6,155
6
votes
1
answer
3k
views
Proving the interior of a dual cone is the set of vectors whose inner product is strictly positive on the cone
Apologies for posting such a simple question to mathoverflow. I've have been stuck trying to solve this problem for some time and have posted this same query to math.stackexchange (but have received ...
- 283
6
votes
2
answers
635
views
Does $\int_0^{2\pi} e^{i\theta(t)} (\phi(t))^n dt=0$ $\forall \; n\in\mathbb{N}_0$ imply $\phi$ periodic?
PROBLEM. Let $\theta(t)$ and $\phi(t)$ be two real analytic non-constant functions $[0,2\pi]\rightarrow \mathbb{R}$. I am trying to prove the following claim
If the integral
$$
\int_0^{2\pi} e^{i\...
- 405
6
votes
4
answers
614
views
Number of intervals needed to cross, Brownian motion
Let $B_t$ be a standard Brownian motion. Let $E_{j, n}$ denote the event$$\left\{B_t = 0 \text{ for some }{{j-1}\over{2^n}} \le t \le {j\over{2^n}}\right\},$$and let$$K_n = \sum_{j = 2^n + 1}^{2^{2n}} ...
user80013
6
votes
1
answer
791
views
Is there a continuous function $f$ satisfying the following Zygmund condition but not differentiable.
Suppose that a continuous function $f$ on the line and satisfies
$$
|f(x+2h)−2f(x+h)+f(x)|\leq const \frac{|h|}{(\log\frac{1}{|h|})^{\beta}}\,\,\,\,\,\,\text{where}\,\,\,\, \beta \in(0, 1]
$$
...
- 111
6
votes
1
answer
188
views
On continuous perturbations of functions of the first Baire class on the Cantor set
Is it true that for any function of the first Baire class $f:X\to\mathbb R$ on the Cantor cube $X=2^\omega$ there is a continuous function $g:X\to[0,1]$ such that the image $(f+g)(X)$ is disjoint with ...
- 41.8k
6
votes
1
answer
1k
views
About the generating structure of Borel field
This is a graduate-level measure theory problem. I have thought throught it and asked on math.SE but received no satisfying answer.
On P.32 of [P.Billingsley] Probability and Measure, 3ed, 1993, the ...
- 8,071
6
votes
3
answers
1k
views
Orthonormal basis in $W^{1,2}([0,1])$
Consider the Hilbertspace $W^{1,2}([0,1])$ (i.e. Sobolev space) with the standard inner product which is defined by: $(f,g) = (f,g)_{L^{2}([0,1])} + (f',g')_{L^{2}([0,1])}$. Here $[0,1]$ is not ...
- 63
6
votes
1
answer
289
views
Archimedean ordered fields without maxima and minima in constructive mathematics
In constructive mathematics, let us define an ordered (Heyting) field $F$ to be a commutative ring with a binary relation $<$ which is
irreflexive, where for all $x$, $\neg (x < x)$
asymmetric, ...
- 2,624
6
votes
0
answers
632
views
Generating functions in countable commutative monoids
Let $f: \mathbb{N}_0 \rightarrow \mathbb{C}$ be a function. The power series of $f$ can be viewed as the function $\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$ where $q \in \mathbb{...
- 405
6
votes
0
answers
309
views
Have we discovered constructions for natural fractional dimensional spheres?
I have been thinking about a couple different problems in fractal geometry (including I one deleted because it was ill posed) and realize they all depend in a fundamental way on the problem of: Can we ...
- 2,689
6
votes
1
answer
808
views
Must the Minkowski sum of a Borel set and a *closed* ball be Borel?
Let A be a Borel set in R^n. Must then A + B(0,1) be Borel?
Here B(0,1) is the closed ball centered at 0 of radius 1.
I know that Erdos and Stone gave an example of a compact set (it is Cantor) and a ...
- 63
6
votes
2
answers
231
views
Subsets $X$ such that their Hausdorff outer measure is not finite
Let $H^d:\mathcal{P}(\mathbf{R}^n) \to \mathbf{R}\cup \{\infty\}$ be the $d$-dimensional Hausdorff outer measure on $\mathbf{R}^n$, for some $0<d<n$ with $n$ integer, which is constructed in the ...
- 1,511
5
votes
2
answers
483
views
Are there any known approaches of generalizing functions that do not have a limit at infinity to values at infinity?
Let's consider the affinely extended real line. The functions that have a limit on positive or negative infinity $\lim_{x\to+\infty} f(x)$ or $\lim_{x\to-\infty} f(x)$ can be generalized to the values ...
- 10.1k
5
votes
0
answers
195
views
What are the possible $L^{\infty}$ closures of an integration-invariant linear subspace of $C([0,1],\mathbb{R})$?
Let $S \subset C([0,1],\mathbb{R})$ be an $\mathbb{R}$-linear subspace that is invariant under the $T := \int_0^x$ integration operation: if $g \in S$ then the function $f = Tg$ defined pointwise by $...
- 13.8k
5
votes
1
answer
410
views
Is there always a real $x$ such that $\cos n_1 x + \cos n_2 x + \cos n_3 x < -2$?
Problem: Given three positive integers $0 < n_1 < n_2 < n_3$. Is there always a real number $x$ such that
$$\cos n_1 x + \cos n_2 x + \cos n_3 x < -2?$$
- 1,053
5
votes
2
answers
297
views
Is the $W^{1, \infty}$ limit of differentiable a.e. functions also differentiable a.e.?
Let $f_n$ be a sequence of continuous, differentiable a.e. functions on $[0, 1]$ with
$f_n \to f$ uniformly for some continuous $f$.
$f'_n - g \to 0$ in $L^\infty$ for some measurable $g$,
where we ...
- 6,155
5
votes
1
answer
712
views
Does this condition imply absolute continuity?
Let $f: [0, 1] \to \mathbb R$ be a measurable function. Define the (possibly infinite valued) upper and lower Dini derivative $D^+ f, D^- f: [0, 1] \to [-\infty, \infty]$ by
$$D^+ f (x) := \limsup_{y \...
- 6,155
5
votes
0
answers
143
views
Error of midpoint method for differentiable functions
Is it the case that for every differentiable function $f$ on $[0,1]$ (with finite one-sided derivatives at the endpoints), the midpoint method of estimating $\int_0^1 f(x) \: dx$ has error $o(1/n)$?
...
- 19.7k
5
votes
2
answers
301
views
Euler–Maclaurin formula in $\mathbb{Z}^d$
I was wondering whether there is a Euler–Maclaurin formula of sorts for expressions such as
$$
\sum_{x \in [a,b]^d\cap \mathbb{Z}^d} f(x) - \int_{[a,b]^d}f(x)
$$
where $d\ge 2$ is an integer, $a,b \...
- 446
5
votes
1
answer
243
views
How much time does a function spend above or below its average value around a point?
Given a locally integrable function $f: \mathbb R \to \mathbb R$, define $
K: \mathbb R \times \mathbb R+ \to \mathbb R$ by
$$
K(x, r) :=
\begin{cases}
1, & \text{if }f(x) > \dfrac{1}{2r}\...
- 2,069
5
votes
2
answers
565
views
Geometry of Level sets of elliptic polynomials in two real variables
Updated:
A polynomial $P(x,y)\in \mathbb{R}[x,y]$ is called an elliptic polynomial if its last homogeneous part does not vanish on $\mathbb{R}^2\setminus\{0\}$.The two answers to this post provide a ...
- 356
5
votes
0
answers
1k
views
Boundary of an open, bounded and convex set in $\mathbb{R} ^n$
Let $U$ be an open, bounded and convex set in $\mathbb{R} ^n$. Since $\partial U$ is a rectifiable set it follows that up to a set of $H^{n-1}$-measure zero $\partial U$ is contained in a countable ...
5
votes
2
answers
922
views
What is the status of the extreme value theorem in forms of constructive mathematics, such as Smooth Infinitesimal Analysis?
In certain intuitionistic frameworks the extreme value theorem cannot be proved. Depending on the exact framework, counterexamples can be constructed as well; see for example pp. 294-295 in
Troelstra,...
- 16.6k
5
votes
0
answers
221
views
Can we construct a computable sequence of trigonometric polynomials that converges pointwise to a given continuous function defined on the torus?
Consider any continuous function $f$ on an $m$-dimensional torus $\mathbb{T}^m$. Can we construct a sequence of band limited functions (trigonometric plynomials), with the band width (degree of the ...
- 698
5
votes
1
answer
167
views
Upper bound for the $n$-th derivative of a rational function $\frac{f}{f+g}$
Let $f$ and $g$ be real polynomials with nonnegative coefficients. Let
$$
h = \frac{f}{f+g}.
$$
I want to prove that the $n$-th derivative of $h$ satisfies:
There exists $C > 0$ such that
$$
|h^{(...
- 187
5
votes
2
answers
503
views
Minimizing $x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1$
Look at the expression
$$
f(x_1,x_2,x_3) = x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1.
$$
The numbers $x_1,x_2,x_3$ are non-negative, and I assume that $x_1+x_2+x_3=3$. This is a sum of squares and "...
- 2,207
5
votes
1
answer
564
views
Convergence of discrete Laplacian to continuous one
I make the following observation:
Let $\Delta^{(n)}$ be the discrete Laplacian on $\mathbb{C}^n$ (ie the $n\times n $ matrix with diagonal $-2$ and upper/lower diagonal $1$.)
This one has eigenvalues ...
- 536
5
votes
3
answers
620
views
Poisson equation on manifolds
Let $(\mathcal{M},g)$ be a compact Riemannian manifold with Levi-Civita connection $\nabla$. It is well-known that the Poisson equation
$$\Delta u=f$$
does have a solution on $C^{\infty}(\mathcal{M})$ ...
- 1,171
5
votes
0
answers
270
views
Differential operators that preserve real-rootedness
Is there some description of polynomial differential operators, $\mathcal{D}=\sum f_i(x) D_x^i$ such that, if $h$ is a polynomial all of whose roots are in $[0,1]$, then so are all the roots of $\...
- 156k
5
votes
1
answer
425
views
"Interlacing property" of certain polynomials
I posted this question on MO which was quickly and decidedly answered by Noam D. Elkies.
Once more referring to the same set of polynomials
$$u_n(x) =
{2}^{n-1}\prod _{k=0}^{n-1}(2x+2k+1)
-{2\,n-1\...
- 43.2k
5
votes
2
answers
594
views
Taylor $k$-differentiability of a real function at a point
I am interested in the standard name for the following weak form of $k$-differentiability.
Definition. A function $f:\mathbb R\to\mathbb R$ is called Taylor $k$-differentiable at a point $x_0$ if ...
- 41.8k
5
votes
2
answers
358
views
Linear transport equation with unbounded coefficients
Consider the PDE
$$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$
with Schwartz initial data $f(0,x) = f_0(x) \in \mathscr S(\mathbb R^n).$
I am wondering then if $q$ and all its ...
- 124