All Questions
731 questions
3
votes
1
answer
308
views
$f: [0,1]\rightarrow L^1(\Omega)$ as a (measurable?) function from $[0,1]\times \Omega\rightarrow \mathbb{R}$
Given a map from $\big([0,1], \mathcal{B}[0,1], m\big)$ to a Banach space $(X, \|\cdot \|)$. There are strong measurable functions (they are the point wise a.e. limit of simple functions) and weak ...
- 485
3
votes
1
answer
681
views
measure zero in R but not in R^2
I want to find some subset of R^2 which its intersection with every vertical line is measure zero if we see it as a subset of R and it is not measure zero in R^2?
- 33
3
votes
0
answers
646
views
On properties on a certain functional
Consider the following function:
$$F(z) = \omega(z)\sin^2\left(\frac{c\Gamma(z)}{z}\right)$$
Here, $\omega(z)$ is a weight we have to construct and $c$ is a constant.
The following three conditions ...
- 375
3
votes
2
answers
210
views
Bounding integral expression with total variation of integrand
Consider the following integral expression:
$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{(g(x)-g(y))(x-y)}{|x-y|^{3}} d x d y $$
for $\epsilon>0$, $f \in L^\infty(\mathbb R)$,...
user139844
2
votes
0
answers
228
views
Integrating an n-fold Cauchy product of a Fourier series
I posted this on Math Stack Exchange one month ago, but did not receive any responses. The original question (in a simplified form) can be found here.
Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be ...
- 161
2
votes
0
answers
117
views
Bounding integral expression with BV norm of integrand
Consider the following integral expression:
$$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$
for $\epsilon>0$, $f \in L^\...
user139844
2
votes
0
answers
159
views
Are there hereditarily square-boxed plane continua?
A plane continuum is a bounded, closed and connected subset of the plane.
A bounding box $B$ for a plane continuum $C$ is
a rectangle $B=[a,b]\times[c,d]$ (including sides and interior)
such that $C$ ...
- 1,375
2
votes
1
answer
203
views
Does this maximisation problem admit a finite upper bound?
Let $\mathcal M_2$ be the space of real $2\times 2$ matrices and $\mathcal S_2\subset \mathcal M_2$ be its subset consisting of positive semidefinite elements, i.e. $A\in \mathcal S_2$ iff $A$ is ...
- 1,389
2
votes
2
answers
197
views
$L^p$ domination of mixed partial derivatives by the unmixed ones?
Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has
$$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...
- 128k
2
votes
0
answers
250
views
Dense property of intersection of Sobolev space
I'm using Muscalu and Schlag's textbook (online notes) to study Littlewood-Paley theory in harmonic analysis, where I encounter the following claim:
Pick an arbitrary real number $s$, we have that the ...
- 197
2
votes
1
answer
382
views
Continuous real function on germs
Let $C_0^{m,n}$ be the space of germs of continuous maps from $\mathbb{R}^m$ to $\mathbb{R}^n$, located at $0\in\mathbb{R}^m$, with the usual inductive limit topology. One can also consider $C_0^{m,n}$...
- 21.5k
2
votes
2
answers
243
views
Given a specific function $f$, how to compute the left-inverse of $f$ in the sense of $\approx$?
For a non-negative function $\varphi$ defined on $[0,\infty)$, the left-inverse $\varphi^{-1}$ of $\varphi$ is defined by setting, $\forall t\geq 0$,
$$\varphi^{-1}(t):=\inf\{u\geq0:\varphi(u)\geq t\}....
- 51
2
votes
0
answers
946
views
On a deceptively tricky calculus problem
Motivation for this question: If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dots B_n'$ also satisfies the same inequality
Let $A$ be a non-constant operator acting on $C^...
- 90
2
votes
0
answers
216
views
Is $f$ defined by $f(x) = t\mapsto G(t , x(t))$ differentiable?
Let us consider $X = AC([0 , 1] , \mathbb{R}^n)$, and $Y=L^{1} ([0,1] , \mathbb{R}^n )$ as Banach spaces with their usual norms. Let $G: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ be a ...
- 369
2
votes
2
answers
485
views
Dual space of the completion of the space of Lipschitz functions
This question is a continuation of this post : Metrization of a topological vector space
Let $C_{lip}(\mathbb R^d)$ be the space of Lipschitz functions on $\mathbb R^d$. We endow $C_{lip}(\mathbb R^...
user128095
2
votes
2
answers
336
views
Metrization of a topological vector space
Let $C(\mathbb R^d)$ be the space of continuous functions on $\mathbb R^d$, and $C_{lip}(\mathbb R^d)\subset C(\mathbb R^d)$ be the subspace of Lipschitz functions. We endow $C_{lip}(\mathbb R^d)$ ...
user128095
2
votes
1
answer
404
views
Sturm Liouville problems for non-classical orthogonal polynomials
It is known that for the classical orthogonal-polynomials there exist a set of Sturm Liouville problems. E.g. , the Hermite polynomial of order $n$ is a solution of $$y''(x) -xy'(x)+ny(x)=0 \, .$$
My ...
- 3,574
2
votes
1
answer
996
views
Derivative and Jacobian determinant of solution of ODE [closed]
Let $\Phi$ be the unique solution of
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\Phi(x,0) = x \quad x \in \mathbb{R}^N
\end{cases}$$
where we have assumed $f$ smooth.
...
user124345
1
vote
1
answer
487
views
New differintegral formula: how is it related to other differintegral formulas?
Lets define new differintegral formula as
$$\mathbb{D}^s_xf(x)= \sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$
or, equivalently,
$$\mathbb{D}^s_xf(x)= \lim_{t\to s} \...
- 10.1k
1
vote
1
answer
154
views
BV function with absolutely continuous divergence
Let $f:\Omega \subset \mathbb{R}^N \to \mathbb{R}^N$ be a vector field such that $f \in BV(\Omega)$.
Suppose that $\mathrm{div} f$ is absolutely continuous with respect to the Lebesgue measure and ...
- 839
1
vote
1
answer
263
views
Does global boundedness ruin Stone-Weierstrass denseness?
Let $X$ be any topological space and denote by $\tau_X$ the topology on $C_b(X;\mathbb{R})$ that is induced by the family of seminorms $(\|\cdot\|_\psi\mid\psi\in B_0(X))$ with $\|f\|_\psi:=\sup_{x\in ...
- 463
1
vote
1
answer
758
views
meromorphic extension of a function
Let $\Lambda\in \mathbf{C}$ be a discrete subset. We assume that $\mathrm{Re}(\lambda)<0$ for all the $\lambda\in \Lambda$. For $i\in \mathbf{N}$, $\lambda\in \Lambda$, let $m_{i,\lambda}\in \...
- 1,111
1
vote
2
answers
194
views
Continuity of the densities of a stochastic process
Let $X=(X_t)_{t\in I}$ ($I\subset\mathbb{R}$ an interval) be a stochastic process with continuous sample paths and such that $X_t$ admits a continuous Lebesgue density $\chi_t\in C(\mathbb{R}^d)$ for ...
- 463
0
votes
1
answer
176
views
Symmetry of fractional laplacian
Let $\Omega\subset\mathbb{R}^n$, let $s\in [1/2,1)$, let $u\in C^{1,2s-1+\epsilon}(\Omega)$ such that: $u=0$ on $\mathbb{R}^n\setminus\Omega$, and: $u\in C^{0,s}(\mathbb{R}^n)$, is true that:
$$\int_{\...
- 339
0
votes
2
answers
178
views
"Find a representation [using Mellin transform] of 𝑓(𝜇,𝛽) as Gauss hypergeometric function in variable 𝜇"
This is a follow-up to the first comment (by Nemo) to the posting Compute the two-fold partial integral, where the three-fold full integral is known . (I also just asked this as a comment to that ...
- 723
0
votes
1
answer
557
views
Is the limsup or liminf of n-wise independent events independent?
Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space.
Consider events indexed by $m, n \in \mathbb N$:
$ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent.
$A_{m,1}...
- 247
0
votes
2
answers
125
views
Is there a modification of $f$ on a null set such that $F: [0, T] \to L^p ({\mathbb R}^d), t \mapsto f(t,\cdot)$ is Bochner measurable?
Let $T>0$ and $p \in [1, \infty)$. Let $f \in L^p ([0, T] \times {\mathbb R}^d)$. By a theorem in this thread, there is a Lebesgue null subset $N$ of $[0, T]$ such that $f(t, \cdot)$ is Lebesgue ...
- 835
0
votes
1
answer
186
views
Poisson kernel, $E^{(x, y)}\text{exp}\{i\theta X_t - \theta Y_t\} = e^{i\theta x - \theta y}$
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. How do I see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\text{exp}\{...
user76167
0
votes
2
answers
403
views
Application of uniform boundedness principle
$\DeclareMathOperator\Lip{Lip}$Let $\Lip_0(\mathbb R^d)$ be the space of Lipschitz functions $f:\mathbb R^d\to\mathbb R$ vanishing at zero, i.e., $f(0)=0$, and equipped with the norm $\|f\|:=\|\nabla ...
user128095
15
votes
2
answers
473
views
Generalizations of summation methods of divergence series
If one looks at the "summation proofs" of divergent series such as Grandi's series, one might see a pattern that most of the computation rely on linearity and comparability with the shift ...
114
votes
34
answers
86k
views
Why do we teach calculus students the derivative as a limit?
I'm not teaching calculus right now, but I talk to someone who does, and the question that came up is why emphasize the $h \to 0$ definition of a derivative to calculus students?
Something a teacher ...
87
votes
8
answers
16k
views
Why is Lebesgue integration taught using positive and negative parts of functions?
Background: When I first took measure theory/integration, I was bothered by the idea that the integral of a real-valued function w.r.t. a measure was defined first for nonnegative functions and only ...
- 50.6k
78
votes
5
answers
8k
views
Does pointwise convergence imply uniform convergence on a large subset?
Suppose $f_n$ is a sequence of real valued functions on $[0,1]$ which converges pointwise to zero.
Is there an uncountable subset $A$ of $[0,1]$ so that $f_n$ converges uniformly on $A$?
Is there a ...
- 31.5k
67
votes
9
answers
7k
views
Taking "Zooming in on a point of a graph" seriously
In calculus classes it is sometimes said that the tangent line to a curve at a point is the line that we get by "zooming in" on that point with an infinitely powerful microscope. This explanation ...
- 12.1k
64
votes
8
answers
6k
views
Two (probably) equal real numbers which are not proved to be equal?
Can someone give me a nice example of two computable real numbers which are believed but not proved to be equal?
I never really understood the assertion that "the reals do not have decidable equality"...
63
votes
6
answers
12k
views
Why isn't integral defined as the area under the graph of function?
In order to define Lebesgue integral, we have to develop some measure theory. This takes some effort in the classroom, after which we need additional effort of defining Lebesgue integral (which also ...
- 1,229
46
votes
2
answers
7k
views
Is $\sum_{k=1}^{n} \sin(k^2)$ bounded by a constant $M$?
I know $\sum_{k=1}^{n} \sin(k)$ is bounded by a constant. How about $\sum_{k=1}^{n} \sin(k^2)$?
- 573
43
votes
0
answers
819
views
A kaleidoscopic coloring of the plane
Problem. Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap D)=\...
- 41.8k
37
votes
3
answers
3k
views
An entropy inequality
Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$, and set $K=\sum_i\sqrt{X(i)Y(i)}$ so that $Z:=\frac{1}{K}\sqrt{XY}$ is also a probability measure on $\{1,2,\dots,n\}$. How can we prove the ...
- 11.4k
35
votes
3
answers
4k
views
A curious determinantal inequality
In my study, I come across the following curious inequality, which I do not know a proof yet (so I am asking it here).
Let $A, B$ be $n\times n$ (Hermitian) positive definite matrices. It is very ...
- 1,748
34
votes
2
answers
2k
views
Is it always possible to calculate the limit of an elementary function?
I already asked this question on https://math.stackexchange.com/questions/2691331/is-it-always-possible-to-calculate-the-limit-of-an-elementary-function but as I received no answer; maybe it is not as ...
- 496
33
votes
1
answer
2k
views
How quickly can the derivative of an everywhere differentiable function change sign?
Let $f : [a,b] \to \Bbb R$ be everywhere differentiable with $f'(a) = 1$ and $f'(b) =-1$.
By Darboux theorem, we know that $f'([a,b])$ is an interval containing $[-1,1]$. In particular, the set $\{x \...
- 635
33
votes
1
answer
2k
views
For which maps $S^1\to S^1$ is the winding number defined?
There are two classes of maps $S^1\to S^1$ for which I know how to define the winding number:
• Continuous maps:
Using the unique path lifting property of the universal covering map $\mathbb R\to S^...
- 43.2k
32
votes
4
answers
4k
views
Is a random subset of the real numbers non-measurable? Is the set of measurable sets measurable?
One might say, "a random subset of $\mathbb{R}$ is not Lebesgue measurable" without really thinking about it. But if we unpack the standard definitions of all those terms (and work in ZFC), it's not ...
- 2,200
31
votes
13
answers
6k
views
Classic applications of Baire category theorem
I've seen Baire category theorem used to prove existence of objects with certain properties. But it seems there is another class of interesting applications of Baire category theorem that I have yet ...
31
votes
1
answer
2k
views
Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$
Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ such that $f(x)=0$ on $\mathbb{R}\setminus (-1,1)$. Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least $n+1$ zeros on $(-1,1)$
I ...
user46896
29
votes
3
answers
2k
views
Wanted: Positivity certificate for the AM-GM inequality in low dimension
I'm seeking for a Certificate of Positivity for the AM-GM inequality in five variables
$$a^5+b^5+c^5+d^5+e^5-5abcde\;\ge 0\qquad\forall\,a,b,c,d,e\ge 0\,.$$
Can one write the LHS as a sum
$\,\...
- 489
28
votes
4
answers
3k
views
Prove that $\left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n\geq x^n+1$
Let $x>0$ and $n$ be a natural number. Prove that:
$$\left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n\geq x^n+1.$$
This question is very similar to many contests problems, but ...
- 1,054
28
votes
4
answers
2k
views
For a continuous function $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ does $(f(x)-f(y)) (f(\frac{x+y}{2}) - f(\sqrt{xy}))=0$ imply that $f$ is constant?
Suppose that $f: \mathbb{R}^+ \to \mathbb{R}^+$ is a continuous function such that for all positive real numbers $x,y$ the following is true :
$$(f(x)-f(y)) \left ( f \left ( \frac{x+y}{2} \right ) - ...
- 850
26
votes
2
answers
12k
views
About the definition of Borel and Radon measures
I am trying to understand the notion of Radon measure, but I am a little bit lost with the different conventions used in the litterature.
More precisely, I have a doubt about the very definition of ...
- 281