All Questions
757 questions
10
votes
2
answers
1k
views
On equibounded sequences in $L^\infty$
Let $f_n: [0, 1] \to \mathbb R$ be a sequence of positive functions in $L^\infty$ (hence a fortiori in $L^1$) that are equibounded in $L^\infty$ norm - that is $\sup_{n \in \mathbb N} \|f_n\|_{L_\...
- 6,155
10
votes
2
answers
666
views
Reference request: Extensions of Wiener's Tauberian Theorem
Wiener's Tauberian Theorem says that linear combinations of translations of a function $f$ are dense in $L^1(\mathbb{R})$ if and only if the zero set of the Fourier transform of $f$ is empty. This is ...
- 710
10
votes
2
answers
766
views
When polynomial f(x^2) can be factored as g(x)·g(-x) ?
In relation to my question Expression for the sum of square roots of zeros of a polynomial
How to characterize polynomials $f(x)$ with rational coefficients such that $f(x^2)=g(x)\cdot g(-x)$ where $...
- 34.3k
10
votes
1
answer
1k
views
Within ZFC, is $2^{\aleph_0}<2^{\aleph_1}$ provable/independent?
So, I ask whether from the ZFC axioms one can prove X that every uncountable set has strictly more than continuum many subsets, or whether X is independent of the ZFC axioms. Note that (within ZFC) ...
- 3,584
10
votes
3
answers
830
views
Find the inverse of a matrix that is very similar to the Hilbert matrix
The standard Hilbert matrix $H$ is given by
$$H_{ij}=\frac{1}{i+j-1},$$
and it has an inverse given for example in this MO question.
Now I have encountered a matrix $M$ of similar form, namely,
$$...
- 153
10
votes
2
answers
1k
views
Can the integration of integrable sections of a measurable function of two variables ever result in a non-measurable function?
I spent some time searching MathOverflow for a problem that would resemble the one given below, but it turned out to be a rather futile endeavor. I was led to this problem in my attempts to construct ...
- 1,396
9
votes
1
answer
358
views
Relaxation of notion of positive definite function
A function $f:\mathbb{R}\to\mathbb{R}$ is called positive definite (in the semigroup sense) if for all $n\geq 1$ and $x_1,\ldots,x_n\in\mathbb{R}$ pairwise different the matrix $(f(x_i+x_j))_{i,j=1}^n$...
- 3,031
9
votes
1
answer
652
views
Scaling in Mehta's integral
The following expression is known as Mehta's integral and deeply connected to random matrix theory:
$$\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \prod_{i=1}^n e^{-...
- 124
9
votes
1
answer
10k
views
Can the supremum of continuous functions be discontinuous on a set of positive measure? [closed]
Given a sequence of continuous functions $f_n(x)$, all defined on a compact set $D$ and assuming $f_n(x)$ is uniformly bounded. Let $f(x) = sup_n f_n(x)$.
It is clear that $f(x)$ is not necessarily ...
- 93
9
votes
2
answers
652
views
Is $\mathbb{Q}$ the orbit of a rational function under iteration?
In this previous post I asked for the smallest set of continuous real functions that could generate $\mathbb Q$ by iteration starting from $0$. Surprisingly one continuous function suffices.
In the ...
- 4,862
9
votes
2
answers
2k
views
Does the Weierstrass function have a point of increase?
Problem
The Weierstrass function $W(x)$ is given by
$W(x)=\sum_{n\geq 0} a^n \cos(b^n \pi x)$
where $0< a <1$ and $b$ is an odd integer such that $ab > 1+3\pi/2$.
A function $f:\mathbb{R}\...
- 491
9
votes
1
answer
621
views
Uniqueness of solutions of Young differential equations
Consider the following one dimensional Young differential equation:
\begin{align*}
&Y_t=\int_0^t Y_s dX_s,\quad t\in[0,1];\\
&Y_0=0.
\end{align*}
Here the driving process $X$ is a bounded ...
- 931
9
votes
1
answer
917
views
A Besicovitch-type Covering Theorem
In the book The Geometry of Domains in Spaces by Krantz and Parks, the authors proved the weak $(1,1)$-type estimate of the maximal function $M_\mu f$, where $\mu$ is a Radon measure, using their ...
- 1,245
9
votes
1
answer
3k
views
Is every finite Borel measure on a locally compact Hausdorff, $\sigma$-compact and separable space automatically regular?
The conditions stated in the question seem mouthful and a bit arbitrary, so let me provide some backgrounds.
Definition
Let $\mu$ be a Borel measure on a topological space. We say:
$\...
- 83
9
votes
4
answers
2k
views
How may I find all continuous and bounded functions g with the following property?
Find all continuous and bounded functions $g$
with :
$$\forall x \in \mathbb R, 4g(x)=g(x+1)+g(x-1)+g(x+\pi)+g(x-\pi).$$
I have posted this question here, but received no answer.
- 4,074
9
votes
1
answer
499
views
Subspaces of $L^2(0,1)$ dense on every truncation $L^2(c,1)$
It may be better to move this to a separate question.
Let me call a linear subspace $V \subset L^2(0,1)$ to be tame if, for every linear subspace $W \subset V$, either $W$ is dense in $L^2(0,1)$, or ...
- 13.8k
9
votes
2
answers
1k
views
Is there a differentiable but nonsmooth version of the continuous Implicit Function Theorem?
From the result discussed in Does the inverse function theorem hold for everywhere differentiable maps? (which I'll call the differentiable nonsmooth Inverse Function Theorem) one can obtain a ...
- 2,049
9
votes
3
answers
2k
views
Smallest root of a degree 3 polynomial
Is it true that the smallest root $t$ of the polynomial
$$
20 t^3 - 30 t^2 + (12 - 4 \cos^2 \alpha - 4 \cos^2 \beta - 4 \cos^2 \gamma) t + \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma - 2 \cos \alpha \...
- 171
9
votes
2
answers
2k
views
Stokes theorem with corners
I've found the following version of Stokes' theorem in G. Stolzenberg's lecture notes 19:
Notation:
for $1 \le n \le m$
$\Lambda(m, n) = \{ \lambda: \{1,...,n\} \rightarrow \{1,...,m\} \ | \ \...
- 103
9
votes
2
answers
552
views
Asymptotic behavior of Sturm-Liouville eigenvalues
I have two questions.
Consider the operator $Av = -v'' + a(x)v$ on $I = (0, L)$, with zero Dirichlet condition and $a \in C([0, L])$.
Let $(\lambda_n)$ denote the sequence of eigenvalues of $A$....
- 369
9
votes
1
answer
845
views
Convergence of sequences formed by orthocenters, incenters, and centroids in repeated triangle constructions
I asked this question on MSE here.
Given a scalene triangle $A_1B_1C_1$ , construct a triangle $A_{n+1}B_{n+1}C_{n+1}$ from the triangle $A_nB_nC_n$ where $A_{n+1}$ is the orthocenter of $A_nB_nC_n$, ...
- 541
9
votes
3
answers
934
views
local behavior of a finite Borel measure
Let $\mu$ be a finite Borel measure on $\mathbb{R}^n$. I am interested in how does $\mu(B(x,r))$ behave, where $B(x,r)$ is the open ball of radius $r$ centered at $x$. For instance, as far as I recall,...
- 1,839
9
votes
1
answer
950
views
Sort-of converse of Kolmogorov zero-one theorem
Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. The Kolmogorov zero-one theorem states that
Suppose we have independent random variables $X_1, X_2, ...$. Then $\forall \ A \in \bigcap_n ...
- 247
9
votes
2
answers
758
views
Number of critical points of smooth functions on $S^1$
Let $u$ be a smooth function on the unit circle $S^1$ such that $\int_{S^1}ux_j=0$, for $j=1,2$. Is the number of critical points of $u$ strictly bigger than 2?
9
votes
2
answers
791
views
Asymptotic difference between a function and its "binomial average"
(I posted this question on Math.SE a few weeks ago. I got a few comments, but nothing definite, and so I thought I would try MO.)
The origin of this question is the identity
$$\sum_{k=0}^n \binom{n}{...
- 3,283
9
votes
1
answer
451
views
Improper integral $\int_0^1 \frac{\exp(ctx)}{\sqrt{(\exp(bt)-1)(1-\exp(atx))-(1-\exp(at))(\exp(btx)-1)}} dx$ with $-a$ and $b$ positive
Is the following function real analytic in $t>0$:
$$F(t)=\int_0^1\frac{\exp(ctx)}{\sqrt{(\exp(bt)-1)(1-\exp(atx))-(1-\exp(at))(\exp(btx)-1)}} dx,$$
where $-a$ and $b$ are positive, and $c\not=a$?
...
- 353
9
votes
3
answers
657
views
Degree necessary of a polynomial?
Given $-1<a<b<0$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[a,b]$ at every $x\in[b^2,a^2]$ and $f(0)=0$. What is minimum degree that is needed and maximum degree that ...
- 13.9k
9
votes
0
answers
569
views
A standard name for a function satisfying the intermediate value theorem?
Do you know any (standard) name for a function $f:\mathbb R\to\mathbb R$ having the following weak intermediate value property:
$(*)$ for any connected subset $C\subset \mathbb R$ and points $a,b\...
- 41.8k
8
votes
3
answers
747
views
How does the parity of $n$ affect the properties of $\mathbb{R}^n$? [closed]
Does the parity of the dimension of $\mathbb{R}^n$ affect its structure/properties? As in, does it make a difference if $n$ is even or odd?
8
votes
1
answer
783
views
Real-rooted polynomials
I proposed this question at MO which was resolved neatly by Gerald Edgar in the form
$$
u_n(x) =
{2}^{n-1}\prod _{k=0}^{n-1}(2x+2k+1)
-{2\,n-1\choose n-1}\prod _{k=0}^{n-1}(x+k).$$
Now that we ...
- 43.2k
8
votes
1
answer
2k
views
Does integrating with respect to a finitely additive measure respect addition?
Let $X$ be a set and $\mathcal{A} \subseteq P(X)$ a $\sigma$-algebra. Assume $\nu : \mathcal{A} \to [0,\infty]$ is a finitely additive measure. If $f : X \to [0,\infty]$ is a measurable function, we ...
- 3,830
8
votes
1
answer
734
views
Almost Arzela Ascoli
Definitions:
We say a sequence of continuous functions $f_n: [0, 1] \to \mathbb R$ is equicontinuous on average if for every $x \in [0, 1]$ and $\varepsilon > 0$ there exists some $\delta > 0$ ...
- 6,155
8
votes
2
answers
784
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 ...
- 29.7k
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
8
votes
3
answers
1k
views
An infinite series that converges to $\frac{\sqrt{3}\pi}{24}$
Can you prove or disprove the following claim:
Claim:
$$\frac{\sqrt{3} \pi}{24}=\displaystyle\sum_{n=0}^{\infty}\frac{1}{(6n+1)(6n+5)}$$
The SageMath cell that demonstrates this claim can be found ...
- 2,661
8
votes
1
answer
997
views
A seemingly trivial property of differentiable functions
NOTE. This is not really the question I wanted to ask. Somehow I forgot to mention that I am assuming $f$ is continuous. However, since Iosif's answer has been well-received I have left this question ...
- 1,149
8
votes
1
answer
1k
views
Calculating a curvature tensor by polarization
I'm reading some articles by Siu and Nannicini on the Weil-Petersson metric associated to families of compact Kahler-Einstein manifolds. In each article Siu and Nannicini construct a Weil-Petersson ...
- 4,343
8
votes
1
answer
384
views
Are $f\sqrt{1+g^2}$ and $fg\sqrt{1+g^2}$ smooth if $f,fg,fg^2$ are smooth?
Suppose that $f$ and $g$ are functions from $\mathbb R$ to $\mathbb R$ such that the functions $f,fg,fg^2$ are smooth, that is, are in $C^\infty(\mathbb R)$. Does it then necessarily follow that the
...
- 128k
8
votes
2
answers
2k
views
Do proper Zariski closed sets of algebraic sets have measure zero
This is a question related to another question I asked: here.
Say we induce a probability measure that is absolutely continuous with respect to to Lebesgue measure onto an irreducible real algebraic ...
- 81
8
votes
1
answer
485
views
An inequality related to Riesz–Thorin theorem, determinants and $L_p$ norm
Let $a, b, c \in \mathbb{R}^n$ , $p \in [1, +\infty)$, prove that
$$\left( \sum_{1\leq i < j <k \leq n} \left| \det\left(\begin{matrix} a_i & b_i & c_i \\
a_j & b_j & c_j \\
...
- 563
8
votes
1
answer
838
views
Density of prime pairs whose gap is less than the average gap
By the prime number theorem we know that the "average gap" between the first $n$ primes is $\ln p_n$. I would like to know the density of consecutive prime pairs whose gap is less than the average gap ...
- 5,470
8
votes
2
answers
644
views
Given any sequence of interpolating nodes, can we find a continuous function $f$ whose interpolating polynomials doesn't converge to $f$ point-wise
Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
- 1,209
8
votes
0
answers
314
views
How to prove that $ \sum_{m=0}^{\infty} { \Gamma\{(1+2m)/\alpha\}\over \Gamma(1/2+m)} { (-t^2/4)^{m}\over m !} \ge (\alpha/2)^{3}\exp(-t^{2}/4) $
I would love to prove the following inequality
$$
{1\over \sqrt{\pi} } \sum_{m=0}^{\infty}
\Gamma\{(1+2m)/\alpha\}
{ (-t^2)^{m}\over (2m) !}=$$
$$
\sum_{m=0}^{\infty}
{ \Gamma\{(1+2m)/\alpha\}\over \...
- 178
8
votes
3
answers
296
views
Shrinking subset and product
Given a segment and a value $c$ less than the segment length, let $A_1,\dots,A_n$ be finite unions of intervals on the segment. We choose a finite union of intervals $B$ with $|B|=c$ that maximizes $|...
- 1,209
8
votes
0
answers
334
views
Criterion for smooth functions [duplicate]
Let $f:\mathbb{R}→\mathbb{R}$ a real-valued function and $m,n∈\mathbb{N}^∗$ coprime, i.e. greatest common divisor of m and n is 1, and define $f^m:=f\cdot f\cdot\ldots\cdot f.$
Show that
$$f^m,f^n\in ...
- 81
8
votes
3
answers
429
views
A density claim
Suppose that $g_k\in C([1,2])$, $k\in \mathbb N$ are continuous functions such that $\|g_k\|_{C([1,2])} \leq \epsilon^k$ for some sufficiently small $\epsilon>0$. Is the following claim true:
If $f\...
- 4,135
8
votes
2
answers
559
views
How can we show that if $f$ is convex, then $\liminf_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0$?
Let $d\in\mathbb N$ and $f:\mathbb R^d\to\mathbb R$ be convex with $$\int e^{-f(x)}\:{\rm d}x<\infty\tag1.$$ How can we show that $$\liminf_{|x|\to\infty}\frac{x\cdot\nabla f(x)}{|x|}>0?$$ $f$ ...
- 167
8
votes
2
answers
478
views
Whitney extension theorem preserving monotonicity
This question is related to Monotone version of one-dimensional Whitney extension theorem.
Let $m$ be a positive integer or $m=\infty$.
Suppose that $E\subset\mathbb{R}$ is a closed set and $f:E\to\...
- 28k
7
votes
4
answers
3k
views
Upper bound of the expectation of sum of the absolute value pairs
We have two arrays $A,B$ of length $n$. All values are i.i.d drawn from same distribution on $[0,1]$. If we sort $A,B$ in non-decreasing order and let $A_{(i)},B_{(i)}$ denote the i-th value in the ...
user124861
7
votes
3
answers
390
views
Bounds on polynomial values
Assume $f(x)\in\Bbb{R}[x]$ is a polynomial of degree $n$.
Question. If $\int_{-1}^1f^2(x)\,dx=1$, is it true that
$$\vert f(x)\vert\leq \frac1{\sqrt2}(n+1), \qquad \text{for $\vert x\vert\leq1$}\,\,\...
- 43.2k