All Questions
5,871 questions
0
votes
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145
views
A question about the duality principle
Suppose $X$ and $Y$ are finite sets and $K:X\times Y\to \mathbb R$ is some function. We get an integral transform from the space of real functions on $X$ to real functions on $Y$ given by $$\Phi_Kf(y)=...
- 133
0
votes
0
answers
182
views
Establishing an upper bound for a dyadic average of a function in $ {L^{p}}([0,1)) $
Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1)) $, where $ p > 1 $. Let
$$
(D_{n})_{n \in \mathbb{N}_{0}} =
\left( \left\{
I^{n}_{j} \stackrel{\text{df}}{=} \left[ \frac{j}{2^{...
- 1
0
votes
0
answers
153
views
extension of function in an abstract metric space
my question is the following.(Maybe my title is not quite proper for this question):
Let $(E,d)$ be a Polish space (or a separable metric space), let $\xi: E\to R_+$ be a Lipschitz function. Now set $...
- 1,835
0
votes
0
answers
145
views
Does there exist this special kind of homeomorphism?
Let $A,B\subset\mathbb{R}^n, n\geq 2,$ are two different shaped spindles. One is thick and one is thin. (Sorry for my unprofessional statements. Unsure about how to say it rigorously.) So there are ...
- 183
0
votes
0
answers
206
views
About approximate eigenvalue
I am in trouble when read the book "D.Henry, Geometric Theory of Semiliner Parabolic Equations". The question is relate to Page 104,proof Lemma 5.1.4.
Suppose $X$ is a real Banach Space, $M$ is a ...
- 103
0
votes
0
answers
428
views
Given an even function how to obtain the most close odd function and vise versa?
Given an even function $f(x)$, how to obtain the most close to it continuous odd function $g(x)$?
By most close I mean that $\int_0^\infty |f(x)-g(x)| dx$ be the minimum possible and the difference $|...
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
$$d(f,g)\rightarrow 0\Longrightarrow f(1)\...
- 1,835
0
votes
0
answers
67
views
Proof that Newton expansion over derivatives has the properties of an integral [duplicate]
Let's consider a Newton expansion over consecutive derivatives of a function:
$$F(x)=\sum_{m=0}^{\infty} \binom {-1}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$
Can it be proven that such ...
- 10.1k
0
votes
2
answers
168
views
Let f:J→R be an absolutely continuous and f'\in...?
Let $f:J\rightarrow \mathbb{R}$ be an absolutely continuous.
Under what kind of extra condition for $f'$, (not $C$) holds the following relation?
$$
\Big | \frac{1}{|I_{1}|}\int_{I_{1}}f'(x)dx- \...
- 111
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
0
votes
0
answers
115
views
Quasi-simmetric function and bi-Lipschitz functions
Assume that $f$ is a homeomorphism of the unit circle onto itself. If $$1/M \le \frac{|f(e^{i(t+s)})-f(e^{i(t)})|}{|f(e^{i(t)})-f(e^{i(t-s)})|}\le M,$$ then we say that $f$ is $M-$quasi-symmetric ...
- 259
0
votes
0
answers
100
views
Two distribution spaces ${\mathcal S}'/{\mathcal P}$ and ${\mathcal S}_\infty'$
Let ${\mathcal S}'$ be the set of all distributions.
Denote by ${\mathcal P}$ the set of all polynomials,
which is embedded into ${\mathcal S}'$ as a closed subspace.
Equip ${\mathcal S'}/{\mathcal P}$...
0
votes
0
answers
60
views
Relative homology of interlevel set
Let us consider a function $f:\mathbb{R}^3→\mathbb{R}$,
$f(x,y,z)=x^3+y^3+z^3-5yz$. Can anybody drop a hint how
to compute relative homology of interlevel sets with coefficients in $\mathbb{R}:
H_{\...
- 181
0
votes
0
answers
127
views
A question of the weights $A_\infty$' equvalent condition in Real &Harmonic analysis
I have a question. The question is to prove:
The weight $w \in A_\infty $if and only if
$\frac{1}{|Q|}\int_Q w(x)dx \cdot \exp\left(\frac{1}{|Q|}\int_Q \log\frac{1}{w(x)}dx\right)\leq C$, for all
...
0
votes
0
answers
94
views
Extending coverings over dense subsets
Let $X$ be a metric space with $D⊆X$ a dense subset.
If there is a covering for $D$, under which conditions on the covering is it possible to guarantee that the covering also covers $X$?
For a ...
- 1
0
votes
0
answers
149
views
Does this sequence of H\"older functions have a limit?
Let $\left\{\alpha_{n}\right\}_{n\in \mathbb{N}}$ a sequence of positive real numbers with
$$\alpha_{n}\in (0,1)\quad \textrm{and}\quad \alpha_{n}>\alpha_{n+1}$$
Moreover suppose
$$\lim_{n\...
- 91
0
votes
0
answers
490
views
Sufficient conditions for continuity of function $y\mapsto\min_{[x_0,y]}\phi$
Let $\phi:\mathbb{R}\to\mathbb{R}$ a continuous function.
Fix $x_0\in\mathbb{R}$ and consider
$$\psi:\mathbb{R}\to\mathbb{R},\ \psi(y)=\min_{\xi\in[x_0,y]}\phi(\xi)\ .$$
Is $\psi$ a continuous ...
- 293
0
votes
0
answers
241
views
Continuity of a function
Let $f\in L^2(\mathbb{R}^3)$ with compact suppport and $z\in\mathbb{C}$. Is the following function continuous for $z\in Q = \{ z : \Re z\in [a,b], \Im \sqrt{z} \in (0,1] \}$:
$$ F(z)=\bigg(\alpha-i\...
- 71
0
votes
1
answer
302
views
An interpolation inequality.
For all $s>0$ define for $\epsilon\in(0,1)$ the function:
\begin{equation}
g(\epsilon)=\sum_{k=0}^{\infty}(1+k)^s(\sqrt{1-\epsilon})^k.
\end{equation}
Prove that $\exists C>0$ and $\phi(s)$ such ...
- 45
0
votes
0
answers
92
views
Class of integrable 0/1-functions "with no null sets."
I am looking for (the name of) a class of functions from $\mathbf{R}^2$ $(\mathbf{R}^n)$ to {0, 1} that are integrable.
Let $f$ be in this class and $E$ be the set of all points where $f$ is equal to ...
- 567
0
votes
0
answers
244
views
Checking whether this would be bounded
It may be better to post this question here. Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric
positive-semi-definite matrix with exactly $m/2$ positive eigenvalues
and every entry of ...
- 1
0
votes
0
answers
382
views
Lambert W-function
I asked this question MSE, but didn't get any answers. Maybe here someone can help.
I need to solve
$$
\theta \rho^{\theta}+r \theta>v
$$
where $\theta \in \mathbb{R}^{+}, -1 < r,v<1, \ 0&...
- 151
0
votes
0
answers
183
views
Continuity of the Shadow of a Nondecreasing Function
So I'm working a lot with monotone nondecreasing functions $f : [0,1] \rightarrow [0,1]$, and I'm defining a certain discrete dynamics on them. Here nondecreasing means $x < y \Rightarrow f(x) \leq ...
- 485
0
votes
1
answer
116
views
Root and sign of a complicated bivariate function
Given two natural numbers $p$ and $i$, such that $0 < i \leqslant 2^p$, let
$$
\Phi(p,i) := \frac{1}{2^p+1}
+ \frac{1}{(i+1)^2} - \frac{1}{2^p}\lg\left(\frac{2^p}{i}+1\right),
$$
where $\lg x$ is ...
0
votes
0
answers
176
views
search for a function satisfying some conditions
Hi everyone, I would like to find a function
$$\Psi\in\mathcal{C}^2: z\in\mathbb{R}\rightarrow\Psi(z)\in\mathbb{R_+}$$
satisfying the following conditions:
$$1-\frac{z\Psi'(z)}{\Psi(z)}+8s\Psi''(z)...
- 69
0
votes
0
answers
193
views
Boundedness of Riemann-like sums on unbounded interval
Hi
I am trying to find suitable conditions (integrability, growth...) on a function $f:\mathbb{R}\to \mathbb{R}$ such that:
\begin{equation}
\sum_{k\in\mathbb{Z}}f(kh)h= \mathcal{O}(1),\qquad h\to 0^+...
0
votes
0
answers
92
views
Lower bound for double sums with power law decay terms.
This question is related to a work in progress about Ballistic-Diffusive phase transition for some random polymers with long range self-repulsion.
The motivation to ask here if the inequality below ...
- 2,044
0
votes
0
answers
345
views
Jacobian of the inversion map
Let $F:Tr(n,\mathbb{R})\cap GL_n(\mathbb{R})\rightarrow Tr(n,\mathbb{R})\cap GL_n(\mathbb{R})$ be the map which sends a matrix $A$ to its inverse $A^{-1}$. If we consider $F$ as a function from $(\...
- 657
0
votes
0
answers
700
views
Sigma algebra generated
Let $\mathcal{L} \subset \mathbb{R}$ the Lebesgue sigma algebra and $\mathcal{B} \subset \mathbb{R}^{n}$ the Borel sigma algebra. I'll denotes by $\mathcal{L} \times \mathcal{B}$ the smallest sigma ...
- 11
0
votes
1
answer
659
views
Under what condition will this set contain a limit point of [0,1)?
Let $T_1,T_2,....T_n$ be numbers such that
$T_k= k$ no. of digits in decimal expansion of an irrational number, say $\alpha$, starting from $(\frac{k(k-1)}{2}+1)^{th}$ digit in the decimal expansion. ...
- 230
0
votes
0
answers
157
views
Matrices satisfying certain pair-wise constraints
Consider given pairs of variables: $a_{ir1},a_{ir2}\in \mathbb{R}^{m \times m}$ and $a_{jr1},a_{jr2}\in \mathbb{R}^{m \times m}$, where $r \in \{1,2,\cdots,t\}$, consider the constraints:
$\sum_{r=1}^...
- 13.9k
0
votes
0
answers
165
views
minimizing the integral of a function over square sets.
Hi!
I'm interested in some problems, but to be honest i'm not sure of the field they belong to.
Let $h(x,y)$ be a bivariate function on $X^2$, where $X$ is some nice topological space (for instance $...
- 1,352
0
votes
1
answer
122
views
Existence of an eigenpair for d-bar operator in the unit disck
Let $\overline{\partial}=\frac{1}{2}(\partial_{x}+\textrm{i} \,\partial_y)$ and let $D$ be the unit disc in the complex plane. For each $\lambda \in \mathbb C$, consider the problem:
$$ \overline{\...
- 4,135
-1
votes
1
answer
354
views
The grail of functional analysis?
Does $g\in C([0,1],[0,1])=A$ exist such that $\{g^n ,n\in\mathbb N\}$ is dense in $A$ provided with the uniform norm?
with $g^2=g \circ g $
If we can find $g$ then $F$ a closed of $A$, $id \in F$ ...
- 4,074
-1
votes
1
answer
369
views
Would this go to 0 [closed]
Let $t_{m}$ be the sup of the sum of the pairwise distances
between any $2m$ points in the unit disk. Does $t_{m}/m^{2}$ go to
$0$ as $m\rightarrow\infty$?
- 1
-1
votes
3
answers
589
views
on compact support distributions [closed]
If $f$ a distribution with compact support then they exist $m$ and measures $f_\beta$,$|\beta|\leq m$ such that
$$f=\sum_{|\beta|\leq m}\frac{\partial^\beta f_\beta}{\partial x^\beta}$$
how to ...
- 57
-1
votes
1
answer
113
views
Interpolation inequality $\int_{\mathbb R} u^3 dx \le \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$ [closed]
Let $u \in C^\infty(\mathbb R)$. Is it true that the following interpolation inequality holds?
$$\int_{\mathbb R} u^3 dx \lesssim \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$$
- 217
-1
votes
1
answer
227
views
Solving the integral identity $ \int_{a}^{b} f(x)dx = \int_{a}^{b} f(x)g(x)dx. $ [closed]
We know that 0 is the additive identity and 1 is the multiplicative identity. In the same spirit let us define the integral identity as follows.
Definition: Let $f(x)$ be integrable in $(a,b)$. If ...
- 5,470
-1
votes
1
answer
139
views
$L^1$ convergence
Setting
For $i \in \mathbb{N}$, consider two sequences $f_i,g_i \in L^1(\mathbb{R})$ such that $$ f_i \rightarrow_{L^1} f \in L^1(\mathbb{R}) $$ and also $$ g_i \rightarrow_{L^1} g \in L^1(\mathbb{R})...
- 125
-1
votes
1
answer
152
views
Topological characterization of invertible real matrices [closed]
Let $n\geq 2$ be an integer. Consider the topological space $M_n$ of $n$-by-$n$ matrices with real entries.
Can you give a short non-constructive proof of the existence of a continuous function $M_n\...
- 23
-1
votes
1
answer
185
views
eigenvalues of $I\otimes B\otimes C + A\otimes I \otimes C + A\otimes B \otimes I $
Let $A$, $B$ and $C$ be symmetric matrices.
What can we say about eigenvalues of $I\otimes B\otimes C + A\otimes I \otimes C + A\otimes B \otimes I $?
- 461
-1
votes
1
answer
122
views
Divergent summation [closed]
Let $(x_i)_{i=0}^\infty$ be a sequence such that $0<x_i<1\ \forall i \in \mathbb{N} \cup {0}$.Consider the following series:
$$\sum_{i=1}^\infty \frac{x_i}{\left(\sum_{k=0}^{i-1} x_k \right)^2}.$...
- 21
-1
votes
1
answer
223
views
Centroid of $\Omega$ and $\partial\Omega$ concides then $\Omega$ must be a ball
Hi I just happened to have a small question. If we have
$$\frac{\int_\Omega x}{|\Omega|}=\frac{\int_{\partial\Omega} x}{|\partial\Omega|}$$
for a simply connected set $\Omega$ with analytic boundary. ...
- 421
-1
votes
1
answer
110
views
Proving that $\max_{w \in B(z)} e^{f(w)} \leq Ce^{f(z)}$
Let $f : \mathbb R^2 \to \mathbb R $ be a smooth function statisfying
$$
0 < \alpha \leq \Delta f(w) \leq \beta < \infty, \ \ \forall w \in \mathbb R^2
$$
where $\Delta$ denotes the Laplace ...
- 437
-1
votes
1
answer
96
views
Limiting points of elementary set
I consider the following set
$$A:=\left\{ \frac{3mn}{2(m^2+mn+n^2)}; m,n \in \mathbb Z; \text{ and }m,n \text{ are not both zero}\right\}$$
Is it possible to identify the closure of $A$ in the reals?
- 124
-1
votes
1
answer
246
views
Maximal commutative subrings of the endomorphism ring of a vector space
Let $\mathbb{F}$ be a field, and $\mathbf{V}$ a possibly uncountably generated $\mathbb{F}$-vector space. Let $\mbox{End}_\mathbb{F}(\mathbf{V})$ be the endomorphism ring of $\mathbf{V}$. That the ...
- 1,959
-1
votes
1
answer
1k
views
A question about pointwise convergence of Fourier transform in $N$-dimensions
I am retreating back on this statement, after some explorations and calculation
Bow to Willie and others who were skeptical on this. Main difficulty can be seen in this reference. But I must mention ...
- 698
-1
votes
1
answer
4k
views
Lipschitz condition on the first derivative of a function? [closed]
If the derivative of a function is lipschitz,,,does it mean that the function itself is also lipschitz? Any proof for that?
- 101
-1
votes
1
answer
167
views
Space of distributions on $[0,1]^2$: weakly compact or not?
Let $X_1,X_2$ be distributions on $[0,1]$ and let $X=(X_1,X_2)$ be the joint distribution of $X_1,X_2$. Let $\mathcal{X}$ be the set of all such joint distribution $X$.
Question 1: Does $\mathcal{X}$ ...
- 349
-1
votes
1
answer
80
views
Regions when a concave function is smaller than another concave function
Let $f_1,f_2:[0,1]\mapsto\mathbb{R}$ be two bounded and concave functions. Assume $f_1(0)<f_2(0)$ and $f_1(1)<f_2(1)$. I want to investigate the set $\mathcal{X}\triangleq\{x\in[0,1]: f_1(x)>...
- 53