All Questions
5,848 questions
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
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:
$$f_0(x) =1+2x$$
$$f_{n}(x) := \left\{\frac{5+t}{2} : \text{ where t solves } f_{n-1}\left(\frac{x}{t}...
- 13k
12
votes
1
answer
991
views
The geometric-mean factorial
Think of the factorial as $f(n) = n \odot (n-1) \odot \cdots \odot 2 \odot 1$,
where $\odot$ is the binary operator for multiplication, $\cdot$. This suggests exploring replacing
$\odot$ with other ...
- 151k
12
votes
4
answers
831
views
Relating the roots of polynomials to the solution sets of certain functional equations
Consider a functional equation of the following form:
$$\sum_{k=0}^n a_k\,\underbrace{f(f(\cdots f}_{k}(x)\cdots )=0\quad \big(f:\,\mathbb{R}\to\mathbb{R},\;a_i\in \mathbb{R},\;\text{and}\;f^0=\text{...
- 453
49
votes
2
answers
3k
views
Is a function with nowhere vanishing derivatives analytic?
My question is the following: Let $f\in C^\infty(a,b)$, such that $f^{(n)}(x)\ne 0$, for every $n\in\mathbb N$, and every $x\in (a,b)$. Does that imply that $f$ is real analytic?
EDIT. According to a ...
- 2,933
18
votes
2
answers
2k
views
Generalization of Darboux's Theorem
Darboux's Theorem. If $f:[a,b]\to\mathbb R$ is differentiable and $f'(a)<\xi<f'(b)$, then there exists a $c\in (a,b)$, such that $\,f'(c)=\xi$.
Does any of the following generalizations
Let $U\...
- 2,933
7
votes
2
answers
2k
views
Does anyone know what is the right reference for the following simple lemma from harmonic analysis?
The lemma says that given $\lambda\geq 1$, $p\geq 1$, $a_j\geq 0$, for a collection of balls $\{B_j\}_{j\in\mathbb{N}}$ in $\mathbb{R}^n$, it holds
$$\bigg\|\sum_j a_j\chi_{\lambda B_j}\bigg\|_p\leq C(...
- 1,881
1
vote
1
answer
100
views
Estimating a quantity from an estimate in its integral
I am reading a paper in which the following argument is made. We have two positive real valued functions $f(x)$ and $g(x)$. We know that $$\int_0^x \int_0^y f(z) \ dz \ dy \leq g(x).$$
It is then ...
- 13
13
votes
2
answers
2k
views
An alternative proof of the Łojasiewicz inequality
Is there a "brute force proof" of the Łojasiewicz inequality? By "brute force" I mean a proof without introducing the machinery of semianalytic sets and so on but only using elementary results (i.e., ...
- 1,727
33
votes
5
answers
12k
views
Differentiable functions with discontinuous derivatives
For years I've taught my honors calculus students about functions like (the continuous extension of) $x^2 \sin 1/x$, and for just as many years I've told them that they won't encounter functions like ...
- 19.7k
1
vote
1
answer
289
views
Compactly supported smooth function with Laplace transform bounded on a cone
My question is if it is possible to find a compactly supported smooth function $\varphi:\mathbf{R}\to \mathbf{R}$ s.t. the following integration $\int_{\mathbf{R}}\varphi(t)e^{itx}e^{tx}dt$ stays ...
- 11
1
vote
0
answers
184
views
A bound for a product in BMO
The question: Let's consider $f\in L^\infty(\mathbb{T})$ and $g\in BMO(\mathbb{T})$. I'm trying to figure out if the following inequality is true
$$
\|fg\|_{BMO}\leq C\|f\|_{L^\infty}\|g\|_{BMO}.
$$
...
- 843
1
vote
1
answer
713
views
Quantitative version of of Riemann Lebesgue Lemma
I'm wondering if there exists a "Quantitative version of of Riemann Lebesgue Lemma" at least for the following case
$
\int_{1}^{\infty}F(t)e^{-2\pi i wt}dt
$
where $F(t)$ is a Piecewise cont. ...
- 167
2
votes
1
answer
90
views
Expressions in "continued" monotone functions
Recall continued fractions: http://en.wikipedia.org/wiki/Continued_fraction
Now take a look at this question: https://math.stackexchange.com/questions/601846/the-limit-of-displaystyle-lim-n-to-infty-...
- 2,205
3
votes
0
answers
314
views
Is a particular set of polynomials dense in a set of functions?
Let us consider the set $\mathcal{F}$ of strictly increasing continuous functions from $[0;1]$ on $[0,1]$ that cancel in $0$ and are equal to $1$ in $1$. So, if $f\in \mathcal{F}$ one has $f(0)=0$ and ...
- 31
9
votes
2
answers
1k
views
Is there a differentiable but nonsmooth version of the continuous Implicit Function Theorem?
From the result discussed in Does the inverse function theorem hold for everywhere differentiable maps? (which I'll call the differentiable nonsmooth Inverse Function Theorem) one can obtain a ...
- 2,049
2
votes
1
answer
950
views
A special case of the Divergence theorem
I am interested in the following statement:
Let $F$ be a vector field in $\mathbb{R}^n$ that is $C^1$-smooth in a
domain $U$, continuous up to the boundary $\partial U$, and vanishing on $\...
0
votes
0
answers
45
views
compactness related to some distance defined on the space of increasing functions2
Let $I=[0,1]$ and denote by $C^{+}(I)$ the space of continuous increasing functions. Can we find a distance $d$ for $C^+(I)$ such that the set of the form
$$d(f,g)\rightarrow 0\Longrightarrow f(1)\...
- 1,835
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
993
views
Full-rank rectangular matrices over GF(2)
Given positive integers $k$, $m$, $n$, let $A$ be an $m \times n$ matrix over $GF(2)$ constructed as follows. Let $X_1, \ldots, X_m$ be independent random subsets of $\{1,\ldots,n\}$ with cardinality ...
- 54.2k
6
votes
1
answer
380
views
Asymptotic value of a multivariate integral
The following question is a simple case of a type of problem that occurs in combinatorial enumeration problems.
Define
$$F(x_1,\ldots,x_n) = \frac{1}{(2\pi)^{n/2}}\exp\biggl( -\frac12\sum_{j=1}^n ...
- 37.7k
5
votes
1
answer
893
views
Isolated critical points
Is the following statement true or false?
Let $f:U\subset{\bf R}^n\to{\bf R}$ be a $C^2$-function (or $C^k$, with $k>2$; or real analytic) defined in a neighborhood $U$ of $0$. Assume that $0$ is ...
5
votes
1
answer
857
views
Hausdorff metric on C[0,1]
Let us consider $C[0,1]$, the space of continuous functions $f\colon [0,1] \to \mathbb{R}$. It comes usually with the metric of the maximum, or of the supremum, $d_{L^{\infty}}$. Each element $f$ in $...
- 283
27
votes
2
answers
1k
views
Continuous functions $f$ with $f(A)$ linearly independent when $A$ is independent
Is there any characterization of continuous functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ such that for any linearly independent set $A$ (over the rationals) $f(A)$ is also linearly independent ?
- 447
23
votes
3
answers
3k
views
Is there a function defined on real numbers which is continuous from the left, but not from the right, everywhere
I am teaching Mathematical analysis. A student asked this question. I think this is a good question, but don't know the answer.
- 527
0
votes
2
answers
590
views
Bounds on the largest root of a polynomial
Consider the following polynomial: $p(x)=x^{3}-(k-1)x^{2}-(2k-1)x+(k-1)^{2}$, where $k \geq 5$ is a fixed parameter. I am trying to find a strong lower bound on the largest root $x_{\max}$ of the ...
- 6,962
0
votes
0
answers
67
views
Proof that Newton expansion over derivatives has the properties of an integral [duplicate]
Let's consider a Newton expansion over consecutive derivatives of a function:
$$F(x)=\sum_{m=0}^{\infty} \binom {-1}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$
Can it be proven that such ...
- 10.1k
2
votes
0
answers
814
views
Quantifying the “flatness” of functions which are the Fourier transforms of positive functions
Short version of question: I'm trying to understand the extent to which a function is prevented from being "flat" as a result of being the Fourier transform of a positive function. That is, the extent ...
- 21
2
votes
0
answers
227
views
Is there an absolutely continuous function $f$
Is there an absolutely continuous function $f$ satisfying
$$
|f(x+\delta)+f(x-\delta)-2f(x)|\leq \mbox{const}\frac{|\delta|}{\log \frac{1}{|\delta|}},\,\,\, |\delta|<1,
$$
which is not $C^{1}$?
- 111
2
votes
0
answers
245
views
Is $f$ an absolutely continuous function? [closed]
Let
$$
f(x)=\sum_{n=1}^{\infty}\frac{\sin(2^{n}\pi x)}{n\cdot2^{n}}, \,\,\,\,\,\, x\in [-1, 1].
$$
Is $f$ an absolutely continuous function? If yes how can I show it? If not how about on total ...
- 111
0
votes
1
answer
169
views
Are all discrete-analytic funtions as defined here also natural?
Let's define a discrete-analytic function as a function that is equal to its Newton expansion:
$$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)=\sum_{m=0}^{\infty} \binom {x}m \sum_{k=...
- 10.1k
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. ...
- 16.6k
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?
$$
\Big | \frac{1}{|I_{1}|}\int_{I_{1}}f'(x)dx- \...
- 111
3
votes
0
answers
770
views
Kullback-Leibler Divergence of Stationary Distributions of Markov chains
Consider two finite Markov chains on the same state space, both assumed to be irreducible, with transition matrices $P$ and $Q$ and associated stationary distributions $\pi$ and $\tilde \pi$. Is it ...
- 418
4
votes
0
answers
988
views
Probability distribution function for singular value sum of Gaussian random matrix
Let $\mathbf{X}$ be an $N \times N$ random matrix with IID Gaussian entries. They can be standard normal, but $N$ is not large: that is $N$ $<$ 6, typically. Call its singular value decomposition (...
2
votes
2
answers
283
views
A general inequality about spherical mean of a function
suppose $\overline u(r)=\frac{1}{\omega_{n-1}}\int_{S^{n-1}}u(r,w)dw,0<r<1,$ is the average of $u(r,w)$ on sphere $S^{n-1}$,where $(r,w)$ are the polar coordinates in $R^n$.
My question is ...
- 21
7
votes
0
answers
174
views
On derivatives of polynomials majorized by $\max(1,|x|^d)$
In the course of generalizing the Bernstein-Markov theorem to normed space, Harris came up with the following question.
Suppose that $p$ is a real polynomial satisfying $|p(x)| \leq (1+|x|)^d$. How ...
- 1,906
2
votes
0
answers
131
views
Representing quasianalytic functions in several variables
For functions in a quasianalytic Denjoy-Carleman class we have the property that their Taylor expansions at a point (the origin) determines the function. For classes that don't only contain analytic ...
- 807
3
votes
2
answers
295
views
Finding a simpler "local" lower bound for a rational function
I have obtained as the expression for some quantity the following gargantuan formula:
$$ \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}$$.
...
- 6,962
3
votes
1
answer
935
views
Beurling density and interpolation
Let $\Lambda=\{\lambda_n\}_1^\infty$ a set of points on the real line. We denote by $\bar{n}(r)$ the largest number of points in any interval $[x,x+r]$, $r>0$. Define the upper uniform density (...
- 859
0
votes
0
answers
405
views
Dual of the space of vector valued Borel measures
What is the dual of the space of all vector valued Borel measures?
- 21
2
votes
2
answers
1k
views
Is there a Calderon-Zygmund decomposition for $L^p$ function
The Calderon-Zygmund decomposition for a $L^1$ function is well known, which says for any $f\in L^1$, then we can decompose $f$ into a good term $f$ and a bad term $\sum b_k$, such that for any $\...
- 879
2
votes
0
answers
428
views
Weak relative compactness in $L^1_{loc}$.
In my work I stumbled upon a proposition (without proof, alas), which I can't really prove.
Suppose we have a family of functions $\left\{\phi_\epsilon (t,x,v)\right\}_{\epsilon\in(0,1]}$, and $M(v)$ ...
- 173
7
votes
1
answer
507
views
Is the mapping $f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}$ surjective?
Is the mapping
$$
f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}
$$
surjective?
If not, what is its image?
If yes, what can be said about ...
- 19.6k
1
vote
1
answer
527
views
An Integral Functional Equation
Let $f$ be a non-negative function supported and integrable on the positive real axis, such that
$$\int_0^\infty f(x+y)p(y) dy = c[p] f(x), $$
where $c[p]$ a number (functional) dependent on function $...
- 2,239
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 ...
- 131
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, ...
- 355
0
votes
1
answer
322
views
Regularity for the Mean Value Theorem
Consider the most classical form of the Mean Value Theorem: given a positive continuous function $f\in C([0,2])$ and a continuous function $g\in C([0,2])$, there exist $c\in(\frac{1}{2},\frac{3}{2})$...
- 2,494