All Questions
5,857 questions
6
votes
2
answers
1k
views
On the uncountability of zero sets
If $f$ is any real-valued function, we define its zero set $Z_f = \{ x : f(x) = 0 \}$. Obviously, the zero set of a nice function can be uncountable. e.g., if $f(x) = 0$ on an uncountable domain.
I ...
- 8,512
4
votes
1
answer
1k
views
General version of Skorokhod representation of random variables
Let $F: \mathbb{R} \to [0,1]$ be cumulative distribution function (cdf). The standard way to build a random variable $\tau$ on $([0,1],\mathcal{B},\text{Leb})$ with $F$ as its cdf is using the ...
- 941
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 ...
- 4,115
4
votes
1
answer
269
views
Real analysis on vector-valued spaces, $L^{p}(\mathbb{R}^N,E)$ ,$H^{s}(\mathbb{R}^N,E)$
I am dealing with some vector-valued Sobolev spaces $H^{s}(\mathbb{R}^N,E)$ where $E$ is a Banach space.
I am looking for references about results for the scalar case $H^{s}(\mathbb{R}^N,\mathbb{C})$...
- 601
0
votes
1
answer
204
views
Are these particular kinds of matrices well known?
Given two positive integers $n$ and $a \leq \frac{n}{2}$ consider a $n \times n$ matrix $A$ such that,
all the diagonal entries are either $a$ or $a+1$
all the non-zero off-diagonal entries are $\pm ...
- 1,893
1
vote
0
answers
88
views
How to show this function is increasing? [closed]
Define function
$$f(\alpha)=\frac{-(4-5\alpha-4\alpha^{2})+\sqrt{32\alpha^{3}+\alpha^{2}-40\alpha+16}}{(1-2\alpha)\alpha}$$
where $\alpha \in [0,\frac{1}{2}]$
I can verify that:
(1) $f(\alpha)\geq 0$...
- 121
3
votes
0
answers
177
views
Interesting stipulation about completely monotone functions
This question relates to a question I asked here. I thought of a well thought out generalization which appears to follow in the situations I've encountered it. I tried to generalize the answer ...
user78249
9
votes
1
answer
948
views
Do partitions of unity exist if we impose additional conditions on the derivatives?
Let $ ~~\cup_{k=-1}^{\infty} U_k = \mathbb{R} $ be an open covering of
$\mathbb{R}$. It is a well known fact that partitions of unity subbordinate to
the cover exists, i.e. there exists smooth
...
- 3,245
0
votes
1
answer
732
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
3
votes
1
answer
301
views
Countable vs. ultra-negligible sets [duplicate]
A subset $A\subset\mathbb{R}$ is negligible if for each $\epsilon>0$ there exists a sequence $(I_n)$ of intervals such that $A\subset\cup_n I_n$ and $\sum_n \vert I_n \vert \leq \epsilon$. Let us ...
- 704
2
votes
0
answers
65
views
Splitting of ordinals of oscillation ranks of a Baire $1$ function
Denny and Tang proved that
Theorem $2.3$ Let $(f_n)$ be a sequence in $\mathfrak{B}_1(K)$ converging pointwise to a function $f.$
Suppose $\sup\{ \beta(f_n):n\in\mathbb{N} \} \leq \beta_0$ and $\...
- 623
2
votes
1
answer
247
views
is $x_{n}\ll \overline{x}_{n}^{2}$?
This question is a cross-post from MSE, cause I didn't get any answer there. I hope it is well suited for MO:
Let $(x_{n})_{n\ge 1}$ be an increasing sequence of positive integers and $\displaystyle{\...
- 6,982
1
vote
0
answers
100
views
singular integral operators
Let $(\Omega,g)$ be a compact domain with smooth boundary and suppose that $g$ is smooth. Let $g_D$ and $g_N$ denote the Dirichlet and Neumann green functions for the Laplace-Beltrami operator.
My ...
- 4,115
-1
votes
1
answer
346
views
An infinite set in a compact space
Let $X$ be a topological space. Is there any characterization for the property that says "for every infinit subset $A$ of $X$ there exists $a\in A$ such that if $f$ be an arbitrary real continuous ...
4
votes
1
answer
143
views
Mean value of a function associated with continued fractions
Suppose that an irrational $x$ in $(0,1)$ has convergents $c(k,x)$, and let
$$d(x) = \sum_{k=0}^{\infty} \mid x - c(k,x)\mid.$$
What is the mean value of $d$?
- 3,443
2
votes
1
answer
257
views
How to characterize singular matrix $X$ that solves det$(X−A)=0$, where $A$ is symmetric positive definite?
Consider real square matrices $X$ and $A$ of same size, where $A$ is known to be symmetric positive definite. I came across the matrix equation $XX^{\top} = AX^{\top}$, which solved for $X$ gives ...
- 1,538
7
votes
0
answers
211
views
Increasing derivatives of recursively defined polynomials
Consider recursively defined polynomials $f_0(x)=x$ and $f_{n+1}(x)=f_n(x)−f_n'(x) x (1−x)$.
These polynomials have some special properties, for example $f_n(0)=0$, $f_n(1)=1$, and all $n+1$ roots of ...
- 225
2
votes
0
answers
279
views
Can a bounded open set in $R^n$ be always approximated from outside with a finite union of dyadic cubes?
Suppose we have a bounded open set $S$ in $R^n$. Consider the collection of closed dyadic cubes $C_k$'s (https://en.wikipedia.org/wiki/Dyadic_cubes). I was wondering if there always exists a finite ...
- 131
3
votes
1
answer
559
views
Determinant of a Certain Positive-Definite Block Matrix
Is there a lower bound for the determinant or minimum eigenvalue of the following $d$ by $d$ matrix in terms of $d$?
$$\Gamma=\left( {\begin{array}{cc}
I & B \\
B^{*} & I \\
\end{array} ...
- 31
0
votes
1
answer
200
views
Number of critical points
Let $f:[0,2\pi]\rightarrow R^2$ be a smooth function such that $f([0,2\pi])$ is a smooth closed simple curve $C$. Suppose $(0,0)$ lies inside the the bounded open region enclosed by $C$ and $f(t)=(x(t)...
3
votes
2
answers
466
views
Question on a Basel-like sum
Hello all,
I have happened upon the following sum:
$ 1^2 + \Big(1 \times \frac{1}{3} + \frac{1}{3} \times 1 \Big)^2 + \Big(1 \times \frac{1}{5} + \frac{1}{3} \times \frac{1}{3} + \frac{1}{5} \times ...
- 237
8
votes
1
answer
597
views
complete metric space
Hallo, I have the following question:
Let $(X,d)$ be a complete metric space. Is then $(X,\operatorname{dist})$ also complete? Here by $\operatorname{dist}$ I mean the metric induced by $d$ by: $\...
- 83
8
votes
2
answers
2k
views
Divergent series expansion in Apéry's proof of the irrationality of $\zeta(2)$ and $\zeta(3)$
UPDATE. I am now making this a CW in the hope someone can improve the content of this question and/or correct the text.
This is a concise version of this math.SE question of mine. I've got an answer ...
2
votes
1
answer
112
views
Enforcing an inequality on series
Let $f:[0,1]\to\mathbb{R}_+$ be a convex, strictly increasing function such that $f(0)=0$ (typically, $f$ is very flat at $0$, i.e. increases very slowly). I would like to prove or disprove the ...
- 14.4k
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
23
votes
0
answers
939
views
A question about small sets of reals
In ZFC, does there exist an uncountable set of reals $A$ such that for every closed measure zero set of reals $B$, we have that $ A + B = \{a+b : a \in A, b \in B\} \neq \mathbb{R}$?
This question is ...
- 9,641
12
votes
1
answer
898
views
Converse to Banach’s fixed point theorem for ordered fields?
Suppose $R$ is an ordered field. Call a continuous map $f: R \rightarrow R$ a contraction if there exists $r < 1$ (in $R$) such that $|f(x)-f(y)| \leq r |x-y|$ for all $x,y \in R$ (where $|x| := \...
- 19.7k
3
votes
1
answer
271
views
Gradient estimate of convex functions
Consider a special type of convex function $g(\cdot):\mathbb{R}^d \to \mathbb{R}_+\cup\{+\infty\}$ such that $g(x)=+\infty$ as $|x|\to \infty$. Then $g$ is differentiable almost everywhere within its ...
- 599
1
vote
1
answer
288
views
A classification of (reasonable) asymptotics
Notations: "eventually" means "for $x$ sufficiently large"; "positive" means "strictly positive".
Let $\exp_1 = \exp$ and $\exp_{n+1} = \exp_n \circ \exp$.
Let $E$ be the set of smooth function $f: ...
2
votes
2
answers
2k
views
Does a bounded real function have an analytic continuation [closed]
Consider the function $f:[0,1]\rightarrow\mathbb{R}$, where
$f$ is real-analytic on the open interval $(0,1)$
$f$ is bounded on the closed interval $[0,1]$ (ie. there is some constant $C$ such that $-...
- 23
0
votes
2
answers
590
views
Bounds on the largest root of a polynomial
Consider the following polynomial: $p(x)=x^{3}-(k-1)x^{2}-(2k-1)x+(k-1)^{2}$, where $k \geq 5$ is a fixed parameter. I am trying to find a strong lower bound on the largest root $x_{\max}$ of the ...
- 6,962
1
vote
1
answer
100
views
Can this equality hold for a nonzero $b$?
Please may you kindly assist me on this integration exercise: For real $a, b$ with $a \neq 0$, consider the equality
$$\int_1^\infty f(x)\sin(a\log \sqrt x)x^b \mathrm{d}x = \int_1^\infty f(x)\sin(a\...
- 19
2
votes
2
answers
421
views
Bi-Lipschitz constant of arc-length parametrisation of convex curve
Assume that $f:[0,2\pi]\to [0,2\pi]$ is a an increasing diffeomorphism, and let $\underline f = \min f'$ and $\overline f = \max f'$ and define $$g(s) = \int_0^s e^{if(t)} dt.$$ Assume that $g(0)=g(2\...
- 89
2
votes
1
answer
889
views
Absolutely continuous functions
it is well known that if a function $f:[0,T]\to\mathbb{R}$ satisfies the inequality
$$\vert f(t)-f(s)\vert\leq \int_s^t{m(r) dr},$$
for $s<t$ and some $m\in L^1([0,T])$ then $f$ is absolutely ...
- 145
2
votes
2
answers
197
views
About preserving real-rootedness of multivariable polynomials
Say $f_i(z_1,z_2,..,z_m)$ are polynomials real rooted in the $z$s for a bunch of polynomials indexed by $i$. When can one say that $\sum_{i} p_i f_i(z_1,z_2,..,z_m)$ is also real rooted?
If ...
- 31
2
votes
0
answers
125
views
Constant periodic Sobolev embedding
Dear mathoverflowers,
I would like to have a reference regarding the optimal constant in the Sobolev embedding
$$
\|u\|_{L^q}\leq C_{s,q}\|u\|_{\dot{H}^s},
$$
($H^s$ denotes the standard L^2 ...
- 843
0
votes
1
answer
297
views
Approximating characteristic functions by cutting the real axis into smaller and smaller pieces
Let $\Lambda_r^*=\frac{1}{2\pi r} \mathbb{Z} \subset\mathbb{R} (r>0)$, let $E\subset\mathbb{R}$ be a Lebesgue measurable set with finite measure $|E|$, define $J_r=(-\frac{1}{4\pi r}, \frac{1}{4\pi ...
- 1,906
1
vote
0
answers
194
views
Cotlar-Stein's Lemma and the Dirichlet kernel
It is well-known that Cotlar-Stein's Lemma can be used to prove the $L^2$ boundedness of the Hilbert transform. See e.g. $L^2$ boundedness of the Hilbert transform via Cotlar-Stein Lemma. Then using ...
- 171
-2
votes
1
answer
880
views
a question regarding the interchange the order of finite summation with finite integration [closed]
Question (1) What are the conditions the complex function $f_n(t)$ and real parameter $B>1$ and positive integer $N>1$ need to satisfy such that the interchange of the finite summation with ...
- 603
1
vote
1
answer
657
views
Local Uniform Convergence
Suppose $f(x)$ is a positive continuous function on $[0,\infty)$ and that $f(x+u)-f(x)\to 0$ as $x\to\infty$ for every given $u\in[0,\infty)$. Prove that, given any $a>0$, $f(x+u)-f(x)\to 0$, as $x\...
- 2,251
2
votes
0
answers
139
views
Existence of solution of a variational inequality
Let $K\subseteq \mathbb{R} ^n$ be closed and convex, and let $F:K \to \mathbb R^n $ be a continuous function. If for every $x,y \in K$ we have $$(x-y)^T(F(x)-F(y))\ge \alpha ||x-y||^2 \, ;\quad \...
- 21
1
vote
0
answers
75
views
Integrability over the closure of a domain [closed]
Let $D$ be a bounded domain in $\mathbb{R}^{N}$ ($N\geq2$) and $E$ a closed subset of $D$ with empty interior. Suppose $f$ is a measurable function defined on $D$ and integrable on $D\setminus E$, i....
- 411
7
votes
1
answer
397
views
Fourier expansion of Takagi-function (everywhere non differentiable function).
Let us consider Takagi-function defined by
$T(x) \colon\!= \sum_{n=0}^{\infty}s(2^nx)/2^n$,
where $s(x) \colon\!\!= \underset{n \in {\Bbb Z}}{\mathrm{min}} \,|x-n|$.
$T(x)$ has its period $1$, so ...
- 181
4
votes
1
answer
251
views
Superadditivity of the lower density
Let $\mu^\star$ be a real-valued function defined on the power set of the positive integers $\mathbf{N}^+$ such that for all $X,Y\subseteq \mathbf{N}^+$ the following axioms hold:
(F1) $\mu^\star(\...
- 1,511
2
votes
1
answer
91
views
Question about optimizing a given function by optimizing an approximation
Let $f$ be a real-valued function. Suppose I want to find a local maximum of $f$, but I decide to work with an ''approximation'' to $f$ --let us call it $g$. What is a suitable notion of ''...
- 170
10
votes
2
answers
766
views
When polynomial f(x^2) can be factored as g(x)·g(-x) ?
In relation to my question Expression for the sum of square roots of zeros of a polynomial
How to characterize polynomials $f(x)$ with rational coefficients such that $f(x^2)=g(x)\cdot g(-x)$ where $...
- 34.3k
3
votes
0
answers
306
views
Metric analogues of bounded variation
A function $f:[a,b]\to\mathbb{R}$ is said to be of bounded variation if
$$ \sup_I \sum_{i=1}^n |f(x_i)-f(x_{i-1})| \le V $$
for some finite $V>0$, where the supremum is over all finite partitions
$...
- 6,541
1
vote
1
answer
195
views
A metric on the set of BV functions, is it mentioned/studied in literature?
I'd like to propose the following metric which operates on the set $M$ of all square integrable functions that are also of bounded variation, of the form $f : (0,1) \to \mathbb{R}$.
Given any $x,y \in ...
- 698
4
votes
2
answers
220
views
existence of a special conformal mapping
Sorry I don't know how to give an appropriate title.
In the complex plane, suppose there is a graph $x+if(x)$ separating the plane into two unbounded components, where $f(x)$ is smooth and bounded, ...
- 41
1
vote
3
answers
188
views
sequences of plane measures converging to a singular one: terminology, etc
We are dealing with very "easy" sequences of uniform measures converging to singular measures (?), as in the following example: let $a$, $b$, and $c$ be vertices of a triangle in $\mathbb{R}^2$, and $...
- 14k