All Questions
5,659 questions
6
votes
1
answer
194
views
The most even partition of $\mathbb R$ into measure dense sets
Notation: $\mu$ denotes the Lebesgue measure. Let $\mathcal D$ be the set of Lebesgue measurable subsets of $\mathbb R$ such that itself and its complement have nonzero Lebesgue measure in every ...
- 6,223
6
votes
1
answer
610
views
Total positivity, log-concavity and Pólya frequency
I am not familiar with the definition of total positivity. I am not sure about the link between log-concavity and total positivity.
In a paper On Variation-Diminishing Integral Operators of the ...
- 393
6
votes
2
answers
319
views
Does control on the “magnitude” of the rearrangement give control of the rearranged Cesaro sums?
Let $a_n$ be a nonnegative sequence that Cesaro converges to $K > 0$. We recall this means
$$\frac{1}{N} \sum_{n = 1}^N a_n \to K$$
as $N \to \infty$.
Suppose $a_{\phi_n}$ with $\phi: \mathbb N \to ...
- 6,223
6
votes
3
answers
267
views
Vanishing periodizations $\sum_{k \in \mathbb Z} f(t+ak)$ of a function $f$ for different values of $a$ implies $f=0$?
Consider a continuous function $f : \mathbb R \to \mathbb C$ with rapid decay (e.g. $|f(t)| < e^{-t^2}$). For a constant $a>0$ let
$$
F_a(t) = \sum_{k \in \mathbb Z} f(t+ak)
$$
be the ...
- 127
6
votes
1
answer
222
views
Maximizing $\prod_{i < j} \sin^2(\alpha_i - \alpha_j)$
For an integer $n \geq 2$, define $f_n(\alpha_0, \alpha_1, \ldots, \alpha_{n-1}) = \prod\limits_{0 \leq i < j < n}\sin^2\left(\alpha_i - \alpha_j\right)$ and $$M_n = \max\limits_{(\alpha_0, \...
- 1,625
6
votes
2
answers
399
views
Maximal (minimal) value of $S=x_1^2x_2+x_2^2x_3+\cdots+x_{n-1}^2x_n+x_n^2x_1$ on condition that $x_1^2+x_2^2+\cdots+x_n^2=1$ [closed]
since $x_1^2+x_2^2+\cdots+x_n^2=1$ is sphere,a compact set,so $S$ has a maximal(minimal) value. But when I try to solve it using the Lagrangian multiplier method, I don't know how to solve these ...
- 81
6
votes
1
answer
2k
views
Sobolev functions on $\mathbb{R}^N$ cannot be discontinuous on a $(N-1)$-dimensional submanifold
How can one prove (or where can I find a proof) that if $u \in W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$, then $u$ cannot have a $(N-1)$-manifold of discontinuity points?
- 839
6
votes
1
answer
213
views
A one-dimensional integral minimization problem
Let $\mathscr F$ be the collection of smooth functions $f \colon
\mathbb R \to \mathbb R$ such that
$f \in C^\infty_c(\mathbb R)$, with $\text{supp } f \subset [-1,1]$;
$\int_0^1 x f(x) dx ...
- 391
6
votes
2
answers
2k
views
Bounds on the number of zeros of real analytic functions
Let $F(A)$ be a class of real-analytic function on an interval $A \subset \mathbb{R}$ minus the zero function.
We have the following theorem for $F(A)$.
If $f \in F(A)$ then $f$ has at most ...
- 671
6
votes
3
answers
626
views
Alternating power series $\sum_{k\geq 0}(-1)^k z^{(2k+1)^2}$
Suppose that $f(x):=\sum_{k\geq 0}(-1)^ke^{-(2k+1)^2x}$ has a holomorphic continuation to a neighborhood of $0$, that is, $f(x)=\sum_{n\geq 0}a_n x^n$ for $x> 0$ small. I want to know the value of $...
- 61
6
votes
2
answers
308
views
Recent trends in effective analysis
The references listed at http://en.wikipedia.org/wiki/Computable_analysis have all been published 30-15 years ago. Are the approaches which these references expose still up-to-date and relevant to the ...
user60665
6
votes
1
answer
6k
views
Change of variables formula for Riemann integration and Lebesgue Integration
I've put this question on math.SE for a while without getting any answers. I thought it must be a rather trivial question for MO so that I didn't put it here. But I do want to get some help anyway (...
user14319
6
votes
2
answers
492
views
Does this polynomial have a real zero less than or equal to $1/2$?
Is the smallest root $x$ of
$$
10x^{3}-30x^{2}+\left(30-2\sum_{1\le i<j\le6}\cos^{2}\alpha_{ij}\right)x\\
+2\sum_{1\le i<j\le6}\cos^{2}\alpha_{ij}-\sum_{1\le i<j<k\le6}\cos\alpha_{ij}\cos\...
user538574
6
votes
1
answer
313
views
Convergence of integral averages in $L^1$
Let $f \in L^1 (\mathbb R)$. Suppose $g_n \in L^1 (\mathbb R)$ are a sequence of positive functions.
Define, for each $n$, the function $f_n$ by
$$f_n (x) := \frac{1}{2g_n (x)} \int_{x - g_n (x)}^{x + ...
- 6,223
6
votes
1
answer
376
views
Is there any hope to prove that $g(x)>-4$ if $f(x)<0$?
I have these two functions for $x>0$, $\beta>0$ and $\alpha$ (all reals)
$$ f(x)= \frac{\alpha \; \sin (\beta \; x)}x+4 \cos (\beta\; x) ,\qquad\qquad\qquad\qquad\qquad\qquad\\
g(x)=\frac{\...
user329770
6
votes
1
answer
390
views
Equivariant implicit function theorem
Let $f:\mathbb{R}\times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be a smooth function and $G\subset \operatorname{SO}(n)$ be a $1$-dimensional compact Lie group (diffeomorphic to the circle). ...
- 63
6
votes
1
answer
276
views
How to solve the following ODE with a parameter?
I am considering the following ODE
\begin{equation}
\begin{split}
&\frac{d^2}{dy^2}u + \frac{\alpha}{(1+y^2)^{\frac{r}{2}}}u = \delta(y)\\
&\lim_{|y|\to \infty}u(y) = 0.
\end{split}
\end{...
- 903
6
votes
1
answer
425
views
What is the Borel complexity of this set?
Problem. What is the Borel complexity of the set
$$c(\mathbb Q)=\{(x_n)_{n\in\omega}\in\mathbb R^\omega:\exists\lim_{n\to\infty}x_n\in\mathbb Q\}$$
in the countable product of lines $\mathbb R^\omega$?...
- 42k
6
votes
1
answer
274
views
Limits (growth rates) of power series coefficients
Take two positive integers $m$ and $n$ and consider the rational function
$$G_{m,n}(x,t)=\frac{d}{dx}\left(\frac1{(1-x^m)(1-tx^n)}\right)$$
and the corresponding Taylor expansion as
$$G_{m,n}(x,t)=u_0(...
- 43.2k
6
votes
2
answers
635
views
Does $\int_0^{2\pi} e^{i\theta(t)} (\phi(t))^n dt=0$ $\forall \; n\in\mathbb{N}_0$ imply $\phi$ periodic?
PROBLEM. Let $\theta(t)$ and $\phi(t)$ be two real analytic non-constant functions $[0,2\pi]\rightarrow \mathbb{R}$. I am trying to prove the following claim
If the integral
$$
\int_0^{2\pi} e^{i\...
- 405
6
votes
1
answer
401
views
Why are $\sigma$-algebras preferable to $\sigma$-rings?
The following is said without further explanation in Folland's Real Analysis:
Some authors prefer to take the domains of measures to be $\sigma$-rings rather
than $\sigma$-algebras. The reason is ...
user14319
6
votes
1
answer
2k
views
About weak convergence of probability measure
Suppose $\mu_j$ is a sequence of measures on $\mathbb{R}$. By the definition of weak convergence of measures, $\mu_j$ weak converges to $\mu$ means that for any bounded continuous function $f$, there ...
- 71
6
votes
1
answer
440
views
Are there superexponential Pfaffian functions?
This question is motivated by model theory, but it's really an analysis question (which means it may have an easy analysis answer that I just don't have the background for). Here's the main question, ...
- 1,979
6
votes
1
answer
568
views
Can one show that $|\zeta'(x) / \zeta^2(x)| \leq 1/(x-.5)$ for $x\in\mathbb{R}\cap [1,\infty)$?
I have found that $\left|\frac{\zeta'(x)}{\zeta^2(x)}\right|\leq \frac{1}{x-\frac{1}{2}}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the inequality and got this inequality ...
- 178
6
votes
1
answer
309
views
Is the derivative of a $C^1$ function nonzero almost everywhere on almost every level set?
Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure.
Let $f \in C^1 (\mathbb \Omega)$ for some open, connected, bounded subset $\Omega$ of $\mathbb R^n$. We consider for each $t \...
- 6,223
6
votes
1
answer
289
views
Archimedean ordered fields without maxima and minima in constructive mathematics
In constructive mathematics, let us define an ordered (Heyting) field $F$ to be a commutative ring with a binary relation $<$ which is
irreflexive, where for all $x$, $\neg (x < x)$
asymmetric, ...
- 2,624
6
votes
1
answer
287
views
A characterisation of continuous real functions
Let $f: \mathbb R^n \to \mathbb R$ be a measurable function.
We say $f$ is precise if for every $x \in \mathbb R^n$ and every compact subset $K$ of $\mathbb R^n$ such that for $|K \cap B_\delta (x)|&...
- 6,223
6
votes
1
answer
425
views
Lipschitz property of the symmetric rearrangement
I'm currently reading Talenti's paper "Best constant in Sobolev inequality" and am rather stuck on an argument on pg 363 (or pg 11 if you're reading the pdf). In this section of the paper, ...
- 457
6
votes
1
answer
601
views
Monotonicity of eigenvalues
We consider block matrices
$$\mathcal A = \begin{pmatrix} 0 & A\\A^* & 0 \end{pmatrix}$$ and
$$\mathcal B = \begin{pmatrix} 0 & B\\C & 0 \end{pmatrix}.$$
Then we define the new matrix
$...
- 536
6
votes
1
answer
499
views
Can I find a bump function $\psi$ such that $\nabla\log\psi$ vanishes too?
Consider a bump function supported in the ball of radius $1$, that is $\psi:\mathbb R^n\to\mathbb R$ such that
$\ \psi(x)>0$ for $|x|<1$
$\ \psi(x)=0$ for $|x|\geq 1$
$\ \psi\in C^\infty$.
...
- 311
6
votes
1
answer
314
views
Is $π:\mathcal{C}^∞(M,N)→\mathcal{C}^∞(S,N)$, $π(f)=f|_S$ a quotient map in the $\mathcal{C}^1$ topology?
This question was previously posted on MSE.
Let $M, N$ be smooth connected manifolds (without boundary), where $M$ is a compact manifold, so we can put a topology in the space $\mathcal C^\infty(M, N)$...
- 333
6
votes
1
answer
411
views
Interesting behaviour of binomial coefficients
Let $\binom{n}{k}:=\frac{\Gamma(n+1)}{\Gamma(k+1)\Gamma(1-k+n)}$ be the generalized binomial coefficient then I noticed by playing around with Mathematica that the function $f:[0,n/2] \rightarrow \...
- 63
6
votes
2
answers
633
views
Interpolation space between $L^1\cap L^2$ and $L^1$
In the paper of Bourgain, the way equation (3.78) is deduced from (3.69) and (3.76) seems via the following interpolation result. Let $(X,\mu)$ and $(Y,\nu)$ be two measure spaces and let $T$ be a ...
- 193
6
votes
3
answers
524
views
Reference Request - Recovering a function from its definite integrals (inverse problem)
I'm having a difficult time finding any theory on an inverse problem I've come up against. Let's say I have an unknown function $f:[0,1] \rightarrow \mathbb{R}$, and I know $\int_{a}^{b} f$ for some ...
- 517
6
votes
1
answer
340
views
Inequality for functions on [0,1], continued
Let $0<a<1,\; \psi_a(x)=\displaystyle \prod_{j=0}^\infty (1-a^jx).$ For each $ k\in \mathbb{N},$ set
$$f_k(a;x):=\frac{x^k}{(1-a)(1-a^2)\dots (1-a^k)}\,\psi_a(x).$$
Question. Is it true that, ...
- 783
6
votes
1
answer
288
views
Rademacher type theorem for Alexandrov spaces
The classical Rademacher theorem says that any Lipschitz function on a doman in $\mathbb{R}^n$ has the first derivative almost everywhere.
I am wondering if this result can be generalized as follows. ...
- 21.8k
6
votes
1
answer
352
views
Harmonic maps are light
Assume $f\colon \mathbb{D}\to\mathbb{R}^2$ is a harmonic map
and $x\notin f(\partial\mathbb{D})$. Is it true that $f^{-1}\{x\}$ is totally disconnected?
I hope that the answer is yes.
But actually I ...
- 45k
6
votes
2
answers
231
views
Subsets $X$ such that their Hausdorff outer measure is not finite
Let $H^d:\mathcal{P}(\mathbf{R}^n) \to \mathbf{R}\cup \{\infty\}$ be the $d$-dimensional Hausdorff outer measure on $\mathbf{R}^n$, for some $0<d<n$ with $n$ integer, which is constructed in the ...
- 1,511
6
votes
1
answer
188
views
On continuous perturbations of functions of the first Baire class on the Cantor set
Is it true that for any function of the first Baire class $f:X\to\mathbb R$ on the Cantor cube $X=2^\omega$ there is a continuous function $g:X\to[0,1]$ such that the image $(f+g)(X)$ is disjoint with ...
- 42k
6
votes
3
answers
481
views
Quantum Mechanics and bilinear optimal control theory
I was wondering whether there are any rigorous results about the optimal controllability of Schrödinger operators.
So my question is something like this:
Let $i \partial_t \psi(x,t) = H_0(x)\psi(x,t)...
- 259
6
votes
1
answer
2k
views
Jackson's theorem for partial sum of Fourier series
There is a classical theorem of Jackson stating that the $N$-th partial sum $S_N f$ of the Fourier series of a Lipschitz continuous function $f$ (which is periodic with period 1) satisfies
$$
|f(x) - ...
- 2,207
6
votes
2
answers
3k
views
Multivariable monotonic function
Let $f(x_1, \dots, x_n)$ be a real function on the $n$-dimensional unit cube (that is, mapping $[0,1]^n \mapsto \mathbb{R}$). Assume furthermore that $f$ is monotonic in every coordinate, and that $f$ ...
- 2,207
6
votes
2
answers
2k
views
How to prove the Hahn-Banach constructively
I am just wondering, how to prove the Hahn-Banach theorem constructively for a finite dimensional normed vector space.
Thanks in advance for any helpful answers.
- 71
6
votes
1
answer
379
views
An inequality for a concave function $f(x)=x^{p/2}$
Assume that $p\in(1,2]$, $a,b\ge 1$, $b\le -\frac{1}{2} \left(\cos\frac{\pi }{p}+\sec\frac{\pi }{p}\right)$, and $t\in[0,\pi]$. How to prove this inequality $$\left(\frac{a+\cos t}{b+\cos\frac{\pi }{...
- 333
6
votes
1
answer
300
views
Log-convexity of determinant
Let $f(z):=\langle g(z),g(z)\rangle,$ where $z \mapsto g(z)$ is holomorphic and $\langle \bullet,\bullet\rangle$ is an inner-product on some function space, such as $L^2$, such that $\langle g(z),g(z)\...
6
votes
1
answer
135
views
Small shifts of weakly converging sequences in $L^1$
$\newcommand\R{\mathbb R}$Let $(f_n)$ be a sequence in $L^1(\R)$ converging weakly to some $f\in L^1(\R)$. Let $(a_n)$ be sequence in $\R$ converging to $0$. For each natural $n$, let $g_n$ be the $...
- 128k
6
votes
2
answers
775
views
Upper bounds for Bessel functions
Cosider the K-Bessel function $$K_\nu(x):= \frac\pi 2 \frac{I_{-\nu}(x)-I_\nu(x)}{\sin(\nu\pi)}.$$
See also Watson, G. N., A treatise on the theory of Bessel functions., Cambridge: University Press, ...
- 63
6
votes
1
answer
415
views
Existence of smooth function that characterizes boundary and interior of set
It is well known that every closed set $A \subset \mathbb{R}^{n}$ is the zero level set of some smooth function. It follows that every closed set is also the zero sublevel set of some smooth function, ...
- 351
6
votes
1
answer
128
views
Equivalence of antiderivative in L1 sense and in the usual sense
We say that$\ f$ is differentiable w.r.t to $L_1$ if there exists a$\ g$ such that:
$$
\lim_{h\to 0}\left\Vert\frac{f(x+h)-f(x)}{h} - g(x)\right\Vert_1 = 0
$$
where $\Vert \cdot \Vert_1$ is the $L_1$ ...
- 165
6
votes
1
answer
575
views
Sub-Gaussian decay of convolution of $L^1$ function with Gaussian kernel
I think it might be helpful to put the new statement at the beginning and put the original post at the end. This new statement is more mathematically elegant.
Let $f\geq0$ be in $L^1(\mathbb{R}^d)$ ...
- 1,495