All Questions
5,908 questions
2
votes
0
answers
161
views
Improving a bound from Taylor's Theorem
For this problem, suppose $g:\mathbb{R}\rightarrow\mathbb{R}$ is such that $g\in\mathcal{C}^{k}(\mathbb{R})$, and there exists $\epsilon>0$ such that
\begin{align*} \epsilon<|g^{(k)}(x)|<\...
- 21
3
votes
1
answer
97
views
Number of small projections
Suppose $X$ is a finite subset of the plane and for $0\leq \theta<\pi$, let $l_\theta$ denote the line through the origin having angle $\theta$ with the positive $x$-axis. For how many values of $\...
- 79.9k
3
votes
1
answer
2k
views
Is the space of test functions separable? [closed]
Consider the space $\mathcal D(\mathbb{R}^n)$ of smooth functions (in the sense of having continuous derivatives of all orders) which are compactly supported. Endow it with its usual topology, i.e., ...
- 3,968
10
votes
3
answers
2k
views
The intersection of $n$ cylinders in $3$-dimensional space
A standard question in vector calculus is to calculate the volume of the shape carved out by the intersection of $2$ or $3$ perpendicular cylinders of radius $1$ in three dimensional space. Such ...
- 148k
5
votes
1
answer
279
views
A problem on the boundedness of maximal operator by using linearization method
We know that the maximal operator is bounded on $L^{p}(\mathbb{R}^{n})$ where $n\geq 1$ and $1<p<\infty$ and the proof would be contained in many classical harmonic analysis books. Here I find a ...
- 333
4
votes
1
answer
465
views
Julia sets without Montel's theorem
Let $J(c)$ be the Julia set of $f(z)=z^2 +c$ defined as the closure of repelling periodic orbits. Is there a way to prove that $J(c)$ is the boundary of the basin of attraction of attractive fix ...
- 3,709
1
vote
1
answer
223
views
A linear algebraic q-difference equation [SOLVED]
I would like to solve the following algebraic linear q-difference equation:
\begin{equation}
a\left(x\right)f\left(x\right)=f\left(qx\right)
\end{equation}
The parameter $q$ is real, positive and ...
- 133
2
votes
0
answers
88
views
System of 2 linear q-difference equations with singular matrix
I would like to solve the following algebraic linear system of q-difference functional equations:
\begin{cases}
a_{11}\left(x\right)f\left(x\right)+a_{12}\left(x\right)g\left(x\right)=f\left(qx\right)...
- 133
19
votes
5
answers
1k
views
Floors of powers of reals, how much do the first few determine the next?
Call an integer sequence $\mathbf{x}=\left( x_1,x_2,\cdots \right)$ feasible if it is $f(r)=\left(\lfloor r \rfloor, \lfloor r^2 \rfloor, \lfloor r^3 \rfloor,
\ldots, \lfloor r^n \rfloor, \ldots \...
- 30.1k
9
votes
1
answer
224
views
Is it always possible to "encircle" exactly $n$ points in an infinite subset of $\mathbb{R}^d$ without limit points?
Let $d$ be a positive integer, and let $\mathbb{R}^d$ be endowed with the
Euclidean metric. Given an infinite set $S \subset \mathbb{R}^d$ without limit points
and a positive integer $n$, is there ...
CommunityBot
- 1
4
votes
1
answer
189
views
Weak ergodicity of nonhomogenous products of 0-1 matrices
Here is a question which probably has a negative answer, but I couldn't find any literature directly on it.
Let $(A_n)$ be a sequence of rectangular 0-1 matrices (that is, the entries are restricted ...
- 23.2k
2
votes
2
answers
509
views
Banach algebra of BV functions
I would like to find a reference for the proof that functions of bounded variation make a Banach algebra. Same question for $BV\cap L^\infty$.
CommunityBot
- 1
6
votes
1
answer
218
views
Approximating an iteratively defined function
Let $f_0,f_1,\ldots$ be a sequence of functions $f_n : [0,1] \rightarrow R$ defined as follows:
$$f_0(x) =1+2x$$
$$f_{n}(x) := \left\{\frac{5+t}{2} : \text{ where t solves } f_{n-1}\left(\frac{x}{t}...
- 60.5k
1
vote
1
answer
100
views
Estimating a quantity from an estimate in its integral
I am reading a paper in which the following argument is made. We have two positive real valued functions $f(x)$ and $g(x)$. We know that $$\int_0^x \int_0^y f(z) \ dz \ dy \leq g(x).$$
It is then ...
- 91.8k
1
vote
1
answer
289
views
Compactly supported smooth function with Laplace transform bounded on a cone
My question is if it is possible to find a compactly supported smooth function $\varphi:\mathbf{R}\to \mathbf{R}$ s.t. the following integration $\int_{\mathbf{R}}\varphi(t)e^{itx}e^{tx}dt$ stays ...
- 91.8k
1
vote
0
answers
184
views
A bound for a product in BMO
The question: Let's consider $f\in L^\infty(\mathbb{T})$ and $g\in BMO(\mathbb{T})$. I'm trying to figure out if the following inequality is true
$$
\|fg\|_{BMO}\leq C\|f\|_{L^\infty}\|g\|_{BMO}.
$$
...
- 843
1
vote
1
answer
714
views
Quantitative version of of Riemann Lebesgue Lemma
I'm wondering if there exists a "Quantitative version of of Riemann Lebesgue Lemma" at least for the following case
$
\int_{1}^{\infty}F(t)e^{-2\pi i wt}dt
$
where $F(t)$ is a Piecewise cont. ...
- 16.2k
2
votes
1
answer
90
views
Expressions in "continued" monotone functions
Recall continued fractions: http://en.wikipedia.org/wiki/Continued_fraction
Now take a look at this question: https://math.stackexchange.com/questions/601846/the-limit-of-displaystyle-lim-n-to-infty-...
CommunityBot
- 1
2
votes
1
answer
951
views
A special case of the Divergence theorem
I am interested in the following statement:
Let $F$ be a vector field in $\mathbb{R}^n$ that is $C^1$-smooth in a
domain $U$, continuous up to the boundary $\partial U$, and vanishing on $\...
- 2,834
27
votes
2
answers
1k
views
Continuous functions $f$ with $f(A)$ linearly independent when $A$ is independent
Is there any characterization of continuous functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ such that for any linearly independent set $A$ (over the rationals) $f(A)$ is also linearly independent ?
- 82.7k
14
votes
0
answers
633
views
Classes of (non-continuous) functions with the fixed point property
Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$.
...
CommunityBot
- 1
0
votes
0
answers
45
views
compactness related to some distance defined on the space of increasing functions2
Let $I=[0,1]$ and denote by $C^{+}(I)$ the space of continuous increasing functions. Can we find a distance $d$ for $C^+(I)$ such that the set of the form
$$d(f,g)\rightarrow 0\Longrightarrow f(1)\...
- 1,835
3
votes
1
answer
494
views
A question on Grassmannian
Let $V$ be the space of $4$ by $4$ Hermitian matrices, that
is a vector space of dimension $16$ over $\mathbb{R}$. Is the uniform
measure of
$$
\left\{ W\in Gr\left(5,V\right):W \text{contains no ...
CommunityBot
- 1
6
votes
0
answers
2k
views
Are planar Lipschitz curves countable unions of graphs?
More precisely:
Question:
Let $\gamma \colon [0,1] \to \mathbb{R}^2$ be Lipschitz. Do there exist Borel (or Suslin) sets $A_i \subset \mathbb{R}^2$ and directions $v_i \in \mathbb{R}^2$, for ...
- 3,270
5
votes
1
answer
893
views
Isolated critical points
Is the following statement true or false?
Let $f:U\subset{\bf R}^n\to{\bf R}$ be a $C^2$-function (or $C^k$, with $k>2$; or real analytic) defined in a neighborhood $U$ of $0$. Assume that $0$ is ...
CommunityBot
- 1
5
votes
1
answer
857
views
Hausdorff metric on C[0,1]
Let us consider $C[0,1]$, the space of continuous functions $f\colon [0,1] \to \mathbb{R}$. It comes usually with the metric of the maximum, or of the supremum, $d_{L^{\infty}}$. Each element $f$ in $...
- 45k
23
votes
3
answers
3k
views
Is there a function defined on real numbers which is continuous from the left, but not from the right, everywhere
I am teaching Mathematical analysis. A student asked this question. I think this is a good question, but don't know the answer.
- 31.5k
0
votes
2
answers
590
views
Bounds on the largest root of a polynomial
Consider the following polynomial: $p(x)=x^{3}-(k-1)x^{2}-(2k-1)x+(k-1)^{2}$, where $k \geq 5$ is a fixed parameter. I am trying to find a strong lower bound on the largest root $x_{\max}$ of the ...
- 11.8k
2
votes
0
answers
227
views
Is there an absolutely continuous function $f$
Is there an absolutely continuous function $f$ satisfying
$$
|f(x+\delta)+f(x-\delta)-2f(x)|\leq \mbox{const}\frac{|\delta|}{\log \frac{1}{|\delta|}},\,\,\, |\delta|<1,
$$
which is not $C^{1}$?
- 2,494
0
votes
0
answers
67
views
Proof that Newton expansion over derivatives has the properties of an integral [duplicate]
Let's consider a Newton expansion over consecutive derivatives of a function:
$$F(x)=\sum_{m=0}^{\infty} \binom {-1}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$
Can it be proven that such ...
- 10.1k
0
votes
1
answer
169
views
Are all discrete-analytic funtions as defined here also natural?
Let's define a discrete-analytic function as a function that is equal to its Newton expansion:
$$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)=\sum_{m=0}^{\infty} \binom {x}m \sum_{k=...
- 30.1k
2
votes
0
answers
814
views
Quantifying the “flatness” of functions which are the Fourier transforms of positive functions
Short version of question: I'm trying to understand the extent to which a function is prevented from being "flat" as a result of being the Fourier transform of a positive function. That is, the extent ...
- 21
2
votes
0
answers
245
views
Is $f$ an absolutely continuous function? [closed]
Let
$$
f(x)=\sum_{n=1}^{\infty}\frac{\sin(2^{n}\pi x)}{n\cdot2^{n}}, \,\,\,\,\,\, x\in [-1, 1].
$$
Is $f$ an absolutely continuous function? If yes how can I show it? If not how about on total ...
- 111
11
votes
2
answers
1k
views
Is sigma-additivity of Lebesgue measure deducible from ZF?
Is sigma-additivity (countable additivity) of Lebesgue measure (say on measurable subsets of the real line) deducible from the Zermelo-Fraenkel set theory (without the axiom of choice)?
Note 1. ...
- 39.8k
8
votes
2
answers
3k
views
Finite measure on the power set
Let $X$ be an uncountable set, and let $\Omega$ be the power set of $X$, viewed as a $\sigma$-algebra. Does there exist a positive $\sigma$-additive measure of finite total mass on $(X, \Omega)$ such ...
- 14k
0
votes
2
answers
168
views
Let f:J→R be an absolutely continuous and f'\in...?
Let $f:J\rightarrow \mathbb{R}$ be an absolutely continuous.
Under what kind of extra condition for $f'$, (not $C$) holds the following relation?
$$
\Big | \frac{1}{|I_{1}|}\int_{I_{1}}f'(x)dx- \...
- 3,699
2
votes
2
answers
283
views
A general inequality about spherical mean of a function
suppose $\overline u(r)=\frac{1}{\omega_{n-1}}\int_{S^{n-1}}u(r,w)dw,0<r<1,$ is the average of $u(r,w)$ on sphere $S^{n-1}$,where $(r,w)$ are the polar coordinates in $R^n$.
My question is ...
- 2,494
2
votes
0
answers
131
views
Representing quasianalytic functions in several variables
For functions in a quasianalytic Denjoy-Carleman class we have the property that their Taylor expansions at a point (the origin) determines the function. For classes that don't only contain analytic ...
- 807
7
votes
0
answers
174
views
On derivatives of polynomials majorized by $\max(1,|x|^d)$
In the course of generalizing the Bernstein-Markov theorem to normed space, Harris came up with the following question.
Suppose that $p$ is a real polynomial satisfying $|p(x)| \leq (1+|x|)^d$. How ...
- 1,906
2
votes
0
answers
428
views
Weak relative compactness in $L^1_{loc}$.
In my work I stumbled upon a proposition (without proof, alas), which I can't really prove.
Suppose we have a family of functions $\left\{\phi_\epsilon (t,x,v)\right\}_{\epsilon\in(0,1]}$, and $M(v)$ ...
- 173
3
votes
2
answers
295
views
Finding a simpler "local" lower bound for a rational function
I have obtained as the expression for some quantity the following gargantuan formula:
$$ \frac{k^8 + 3k^7 + 8k^6 + 3k^5 - 16k^4 - 32k^3 + 63k^2 - 34k + 6}{k^6 + 3k^5 + 6k^4 - 24k^2 + 21k - 5}$$.
...
- 188k
2
votes
2
answers
1k
views
Is there a Calderon-Zygmund decomposition for $L^p$ function
The Calderon-Zygmund decomposition for a $L^1$ function is well known, which says for any $f\in L^1$, then we can decompose $f$ into a good term $f$ and a bad term $\sum b_k$, such that for any $\...
- 679
0
votes
0
answers
405
views
Dual of the space of vector valued Borel measures
What is the dual of the space of all vector valued Borel measures?
- 21
7
votes
1
answer
507
views
Is the mapping $f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}$ surjective?
Is the mapping
$$
f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}
$$
surjective?
If not, what is its image?
If yes, what can be said about ...
- 1,875
1
vote
1
answer
527
views
An Integral Functional Equation
Let $f$ be a non-negative function supported and integrable on the positive real axis, such that
$$\int_0^\infty f(x+y)p(y) dy = c[p] f(x), $$
where $c[p]$ a number (functional) dependent on function $...
- 2,239
8
votes
3
answers
800
views
Continuous functions as uniformly continuous function
Three question concerninng metrics on the real line:
Is there a metric $d$ on $\Bbb{R}$ such that a function $f : (\Bbb{R},d) \longrightarrow (\Bbb{R},d)$ ( or $f : \Bbb{R} \longrightarrow (\Bbb{R},...
CommunityBot
- 1
1
vote
1
answer
426
views
Roots of the derivative as symmetric functions of the roots of the polynomial
Let $p(t)=(t^2-a_1^2)\ldots(t^2-a_n^2)$ be an even polynomial with distinct real non-zero roots. Can the roots of its derivative $p'(t)$ be expressed nicely (e.g. as rational symmetric functions) in ...
- 37.7k
1
vote
1
answer
215
views
What's the asymptotic behavior of this function at large distance? [closed]
This question is based on some Physics motivation. Define a distance function $f(\mathbf{r})=\int_{\Omega }d^2k\int_{\Omega }d^2q \cos[(\mathbf{k}-\mathbf{q})\cdot\mathbf{r}]$, where $\mathbf{r},\...
- 113
1
vote
0
answers
122
views
Does the difference quotient of an absolut cont. funct. converge in L^1?
Assume that $\mu$ is a finite Radon measure on the real line and $f$ is integrable wrt. $\mu$. Define
$F(x)=\int_{]\infty;t]}f(y)d\mu(y) $
Is the following statement true?
The functions $d_h:x\...
- 11
4
votes
1
answer
766
views
Preimage of a smooth function
Suppose we are given a smooth function $f\colon \mathbb{R}^n \rightarrow \mathbb{R}$ and some number $c$. What can be said about the preimage $f^{-1}(c)$.
There's the theorem on regular preimages, ...
- 60.5k