All Questions
5,850 questions
3
votes
0
answers
83
views
Embedding theorems for Dini continuous functions
Are there embedding theorems for the space of Dini continuous functions on a Euclidean domain, or even just on an interval? Ideally, I am looking for something like the classical Morrey inequalities ...
- 3,388
8
votes
1
answer
496
views
A fractional weighted Poincaré inequality
Does there exists a constant $C>0$ such that
$$ \int_{-1}^1 \lvert x\rvert\lvert\partial_x u\rvert^2 \,dx \geq C\, \lVert u\rVert^2_{H^{1/2}((-1,1))},$$
for all $u\in C^{\infty}_0((-1,1))$?
- 6,035
4
votes
1
answer
367
views
Equivalent norms of fractional Sobolev spaces on bounded Lipschitz domain
Let $s>0$, $1<p<\infty$ and let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain. Set $H^{s,p}(\Omega)=\{u\in L^p(\Omega):\exists\tilde u\in L^p(\mathbb R^n),\tilde u|_{\Omega}=u,(I-\...
- 938
0
votes
1
answer
166
views
Matrices and vectors of intervals
I'm working on a project and think that matrices and vectors of intervals will be useful.
I'm aware about interval arithmetic, but there is little information on the internet, regarding matrices and ...
- 128k
74
votes
15
answers
18k
views
$f(f(x))=\exp(x)-1$ and other functions "just in the middle" between linear and exponential
The question is about the function $f(x)$ so that $f(f(x))=\exp (x)-1$.
The question is open ended and it was discussed quite recently in the comment thread in Aaronson's blog here http://...
- 5,342
-2
votes
1
answer
217
views
If a continuous function is differentiable at a point, is it differentiable in some neighborhood around that point? [closed]
This seems like it should be true but I was wondering if anyone could prove it. Thanks!
- 48.8k
4
votes
1
answer
668
views
$f\in C(B_1)\cap W^{1,2}(B_1\setminus \{f=0\})$ implies $f\in W^{1,2}(B_1)$?
In a paper I am writing I need to show that a certain real-valued function $f\in C^0(B_1)$ belongs to the Sobolev space $W^{1,2}(B_1)$ ($B_1$ is the unit ball). So far I have been able to show that ...
- 1,149
2
votes
0
answers
116
views
Behavior at infinity of an $L^2$ function with $L^2$ mixed second derivatives
If $f$, $\nabla_x \cdot \nabla_y f \in L^2(\mathbb{R}^d_x\times \mathbb{R}^d_y)$, what can be said about decay at infinity of $\nabla_x f$, $\nabla_y f$?
It is clear that $(\nabla_x^2 + \nabla_y^2) f \...
- 6,367
0
votes
1
answer
270
views
Nature of $ \sum_{n \geq 1} \frac{ \cos(n) \sin(n+1) }{n} $ [closed]
I'm trying to determine the nature of this series $ \sum_{n \geq 1} \frac{ \cos(n) \sin(n+1) }{n} $, but I'm not getting anywhere. I've tried using the Abel and trigonometric formulas, but I can't ...
- 188k
0
votes
1
answer
117
views
Sufficient conditions for ensuring that a monic polynomial in $\mathbf{Z}[x]$ possesses exclusively simple roots
I am seeking sufficient conditions to ensure that a monic polynomial, denoted as $f$ in $\mathbf{Z}[x]$, possesses exclusively simple roots.
Based on an old paper (this reference), it has been ...
- 91.8k
0
votes
0
answers
98
views
Tangent spaces of Lipschitz sub manifolds
Consider $\mathbb{R}^n$, $k<n$, and topological embeddings (homeomorphisms onto image) $f_i : \mathbb{R}^k \supseteq B_1(0) \to \mathbb{R}^n$, $i=1,2$, which are also Lipschitz continuous and ...
- 403
1
vote
1
answer
126
views
Function orthogonal to $|y-x|$ on $[0,1]$ for every $y \in [0,1]$?
Does there exist an essentially nonzero function $f:[0,1] \mapsto \mathbb{R}$ so that
$$
\int_0^1 |y-x| f(x) \, dx = 0
$$
for every $y \in [0,1]$? I think I see how to show that any such $f$ can't be ...
- 6,486
4
votes
2
answers
485
views
How to get this inequality in Santambrogio's book about optimal transport?
Let $\hat{\varrho}, \tilde{\varrho}$ be probability density functions on $\mathbb R^d$ where $\tilde{\varrho} \in L^{\infty} (\mathbb R^d)$. For $\varepsilon \in [0, 1]$, we define $\varrho_{\...
- 63.9k
0
votes
1
answer
509
views
Possible research directions in analysis? [closed]
I am an undergraduate student who loves basic mathematics in the analysis branch, but I have learned that some directions, for example, harmonic analysis, are already well developed and difficult to ...
- 684
2
votes
0
answers
174
views
Product of marginals absolutely continuous with respect to a Borel probability measure
Let $\mu$ be a Borel probability measure on $\Bbb{R}^{m+n}=\Bbb{R}^m\times\Bbb{R}^n$. Consider its marginal measures $\mu_1(A):=\mu(A\times\Bbb{R}^n)\, (A\in\mathcal{B}(\Bbb{R}^m))$ and $\mu_2(B):=\mu(...
- 3,599
3
votes
1
answer
140
views
Oscillation functions and similar constructs
For given $f$ from reals to reals, the associated oscillation function is defined as follows:
$$\textstyle
osc_f(x):= \lim_{n\rightarrow \infty} [\sup_{y \in B(x, \frac{1}{2^n}) } f(y)-\inf_{z \in B(x,...
- 679
2
votes
0
answers
100
views
Are analytic solutions for the Navier-Stokes equations sufficient?
Generally, we ask for solutions of the Navier-Stokes equations, when the starting conditions are in the Schwartz space.
However, I am wondering, whether it is possible to consider just analytic ...
- 749
9
votes
0
answers
287
views
The approximate mean value theorem / Rolle's theorem in pure constructive mathematics
In the replies of this very similar question, there is a fascinating answer that is beautiful in its simplicity. In particular, it seems to use perhaps the most minimal assumptions one can possibly ...
- 63.9k
3
votes
3
answers
340
views
Extending a $C^1$ function on $\mathbb R^n$ to a set of finite $\mathcal H^{n-2}$ measure
Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure.
Let $n \geq 2$ be an integer, and $E \subset \mathbb R^n$ be a set of finite $\mathcal H^{n-2}$ measure.
Suppose $f: \mathbb R^...
- 16.4k
1
vote
0
answers
70
views
A possible upper bound for a function that satisfies a singular integral inequality
I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality:
$$
|v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left(
|...
- 63.9k
1
vote
1
answer
65
views
Boundedness of maximisers of parametric strictly concave functions
Let $L:[0,1]\times \mathbb R^m\times \mathbb R^n\to \mathbb R$ be defined by
$$L(\lambda, x,y):=\sum_{1\le i\le m}\alpha_i x_i + \sum_{1\le j\le n}\beta_j y_j -\sum_{1\le i\le m, 1\le j\le n} p_{i,j}\...
- 128k
1
vote
1
answer
345
views
Topological degree of differentiable map using line integrals?
Let $f:\mathbb R^2 \to \mathbb C$ be a $C^1$ function that vanishes at a point $x_0.$
I can then define $$-i \int_{\gamma_\varepsilon} \nabla \log(f(s)) \cdot ds := - i \int_0^1 \nabla (\log f)(\...
- 188k
1
vote
1
answer
118
views
A Gaussian measure $\mu$ on $\mathcal{E}'(S^1)$ by Minlos theorem and its value for Sobolev spaces $H^{\alpha}(S^1)$
I posted this question on ME as "A Gaussian measure on $\mathcal{E}'(S^1)$ by Minlos Theorem and its value for $L^2(S^1)$",
but it seems much more nontrivial than I expected... so, I post an ...
- 6,367
1
vote
0
answers
123
views
Dependence of Sobolev embedding theorem constant on smoothness
Assume that $\Omega \subset \mathbb{R}^d$ is "nice" enough and $k$ is a positive real number. Using the Sobolev embedding theorem, we can get that
$$
\|f\|_{W^{0,2d/(d-2k)}\ \ \ \ \ (\Omega)}...
- 63.9k
2
votes
1
answer
189
views
Equivalent characterization of weak derivative in Bochner space
Let $H$ be a hilbert space. A function $v\in L_\text{loc}^1(0,T;H)$ is called the weak derivative of $u \in L_\text{loc}^1(0,T;H)$ iff
$$ \int_0^T u(t) \varphi'(t) \, dt = -\int_0^T v(t) \varphi(t) \, ...
- 2,670
0
votes
1
answer
362
views
Conditions on continuity under Lebesgue measure
Let $h : X \times I \rightarrow \mathbb{R}$ be a continuous function, where $X$ is a compact set of $\mathbb{R}^k$, for some $k$.
Set $\hat{h}(x,t) = 1$ if $h(x,t) \neq 0$, $0$ otherwise.
Define $g : ...
CommunityBot
- 1
23
votes
5
answers
2k
views
PDEs and algebraic varieties
Let $P$ be an order $d$ differential operator with constant coefficients and consider a PDE of the form $Pf = \delta$. Taking the Fourier transform of $P$ we get a degree $d$ polynomial whose zero ...
- 2,689
1
vote
1
answer
139
views
Does there exist a continuous choice of maximizing balls for the Hardy Littlewood maximal function?
Note: Here $\overline B_r (x)$ denotes the closed ball of radius $r$ around $x$.
Let $f \in L^1 (\mathbb R^d)$. We define the averages $A(x, r)$ for $x \in \mathbb R^d$ and $r > 0$ by
$$A(x, r) := \...
- 39.1k
2
votes
0
answers
120
views
On the integer of the form p^a q^b closest to a given integer N
If we give ourselves a number having only one prime factor $p$ and a given natural integer $N$, we know how to give the integer of the form $p^k$ closest (and less than) to this integer $N$ it's ...
- 69
7
votes
0
answers
150
views
The space of analytic associative operations
This question is a follow-up to this old one of mine.
Let $\mathcal{A}$ be the set of functions $\star:\mathbb{R}^2\rightarrow\mathbb{R}$ which are associative and $C^\omega$ (real analytic entire) in ...
- 20.5k
1
vote
0
answers
98
views
Periodicity in one Fourier variable
Let $f:[0,1]\times [0,1] \to \mathbb C$ be a double periodic function (periodic in both variables) that depends real-analytically on its argument.
We can thus write $f$ as $$ f(x) = \sum_{n \in \...
5
votes
0
answers
214
views
Elliptic regularity and Sobolev spaces
Consider a linear partial differential operator $D:C^{\infty}(\mathbb{R}^{d})\to C^{\infty}(\mathbb{R}^{d})$, i.e.
$$D=\sum_{\alpha\in\mathbb{N}^{d}}a^{\alpha}(x)\partial^{\alpha}_{x}$$
where $a$ are ...
- 1,429
4
votes
0
answers
116
views
Lipschitz extension of a flow can still be a flow?
Consider a map $\Phi: [0,T] \times \mathbb{R}^d \to \mathbb{R}^d$, and assume that there exists a set $U \subset \mathbb{R}^d$ such that $\Phi\rvert_{[0,T] \times U}$ is $L$-Lipschitz. It is well ...
- 697
0
votes
1
answer
516
views
A problem of Fourier transform and Hölder condition
Suppose that $f$ is continuous on $[0,1]$. Thus, $f\in L^1(\mathbb{R})$ and its Fourier transform exists, as
$$ \hat{f}(\xi) := \int_\mathbb{R} e^{-2\pi i x \xi} f(x)dx, $$
which can also be written ...
- 5,407
2
votes
0
answers
170
views
finite dimensionality of a subspace of a Banach space
Let $H$ be the space of measurable functions on $(0,1)$ such that
$$ \|u\|_{H}^2 = \int_0^1 x^2\,|\partial_x u|^2\,dx + \int_{0}^1 |u|^2\,dx <\infty.$$
Let $C>0$ be a constant. Suppose that $W \...
- 4,153
6
votes
1
answer
331
views
A combinatorial identity involving increasing functions from $\{1, \dots, n\}$ to itself
This is related to the post An order statistics problem with some interesting geometry. The following identity arose in the context of the problem.
Fix an integer $N \geq 2$. Let $\mathcal S_N^+$ ...
- 50.8k
7
votes
0
answers
254
views
$C^0$-limit of volume-preserving maps on $\mathbb R^n$
Let $f_k:B_1\rightarrow \mathbb R^n$ be a sequence of injective differentiable volume-preserving maps (i.e. $\mu(f_k(A))=\mu(A)$ for any measurable $A\subset B_1$) that converges uniformly to $f:B_1\...
- 435
2
votes
0
answers
120
views
Closure of Laplacian
Let $(M,g)$ be a complete Riemannian manifold and $\Delta$ the (positive) Laplace-Beltrami operator. Now, consider this operator as an operator
$$\Delta:\mathcal{D}(\Delta)\to L^{2}(M)$$
There are two ...
- 1,171
0
votes
0
answers
34
views
It is possible to limit a set of curves in the sense $f(x,y) \leq C f(x_0,y)$?
Suppose you have a continuous function $f:[a,b]\times (-\infty, \infty) \rightarrow \mathbb{R}$. I'm trying to understand if it's possible to conclude that due to the compactness of the interval $[a,b]...
- 677
11
votes
2
answers
1k
views
Twice continuously differentiable implied by existence of limit
I have the following question. Let $f,g:\mathbb{R}\to\mathbb{R}$ be two continuous functions (vanishing at infinity) and assume that
$$
\frac{f(x+t)+f(x-t)-2f(x)}{t^2}\to g(x)
$$
for all $x\in X$ when ...
- 8,992
1
vote
0
answers
102
views
Proving that a quantity is positive (Gaussian density and Gaussian CFD)
$\def\R{\mathbb R}$$\def\aha{{1/2}}$$\def\maha{{1/2}}$
Hi everyone,
I am interested in the following problem:
Let consider the heat equation problem:
$$\forall (t,x) \in \mathbb{R}_+\times\mathbb{R}, ~...
- 393
2
votes
2
answers
615
views
In what sense does the Hermite expansion of a bounded smooth function converge?
Let $f : \mathbb{R} \to \mathbb{C}$ be a smooth and bounded function.
If we denote by $\{ H_n(x) \}$ the sequence of normalized Hermite polynomials, then the Hermite expansion of $f$ is defined as
\...
- 211
2
votes
0
answers
41
views
Blow up for certain classes of distributions
Let $\mathbb D$ be the open unit disc centered at the origin and let $u \in H^{-N}(\mathbb D)$ be a distribution for some natural number $N>0$. Suppose that $$u|_{\mathbb D\setminus \{0\}} \in C^{\...
- 4,153
22
votes
1
answer
4k
views
A challenging (for me) limit calculation
How to calculate the following limit
$$
\lim_{n\to\infty}\sqrt{n}\underbrace{{}\sin(\sin(\sin(\sin(\cdots\sin(\frac{1}{\sqrt{n}})\cdots))))}_{n \text{ sin's}} \text{?}
$$
${}{}$
2
votes
1
answer
159
views
A compact embedding claim
Let $U= (0,1)\times (0,1)$. Consider the weighted Sobolev spaces $H_1$ with the norms
$$ \|u\|_{H_1}^2 = \int_0^1 (\int_0^1 x\,|u(x,y)|^2\,dx) \,dy$$
Let $H_2$ be the weighted Sobolev space with the ...
- 6,035
4
votes
1
answer
339
views
Limit for series of Bessel functions evaluated at zeros
The following series arises in an electrostatics problem for a conducting cylinder:
$$
V=\sum_{n=1}^\infty\frac{J_0(k_n\rho)e^{-k_nz}}{k_nJ_1(k_n)^2}
$$
where $J_i$ is the Bessel function of $i^{th}$ ...
- 232
5
votes
1
answer
272
views
Is the local maximal function bounded from $W^{1, 1}$ to $L^1$?
Let $f \in W^{1, 1} (\mathbb R^d)$. For every $\varepsilon > 0$, we consider the local maximal function $M_\varepsilon f: \mathbb R^d \to \mathbb R$, defined by
$$M f_{\varepsilon} (x) = \sup_{r \...
1
vote
3
answers
181
views
Evaluating a sinusoidal series
Define the sequence of functions
$$f_n(x)=\sum_{m=n}^\infty(-1)^m\frac{x^{2m}}{(2m+1)!} {m \choose n} $$
Is there a closed form expression for arbitrary $n$? It is clear that the result should assume ...
- 17k
1
vote
1
answer
90
views
The number of roots of pseudo-exponential polynomials
Assume that $J$ is the interval $(-\pi,\pi]$. For $k=1,\ldots,2n$, suppose that $\lambda_k$s are real functions on $J$ with $|\lambda_k|=1$, meaning that $\lambda_k(t)$ is either $-1$ or $1$ where $t\...
- 2,060
5
votes
1
answer
512
views
Norm inequality for the inclusion $L^2(\partial \Omega)\hookrightarrow H^{-1/2}(\partial \Omega)$
Let $\Omega \subset \mathbb{R}^3$ be a lipschitz domain. We then have the trace operator $\tau : H^1(\Omega) \to L^2(\partial \Omega)$ and can define the space $H^{1/2}(\partial \Omega) := \tau(H^1(\...
- 2,670