All Questions
5,629 questions
0
votes
0
answers
120
views
Mysterious Bound: $\int_{B_{4}}\|D^{2}u\|^{2} \leq 2^{n}$
I am reading through "A GEOMETRIC APPROACH TO THE CALDERON–ZYGMUND ESTIMATES" by Lihe Wang and I am perplexed by an assertion in Lemma 7. The claim is that whenever $\Delta u = f$:
$$\frac{1}...
- 1
3
votes
0
answers
125
views
Extracting moments of $\max(X_1,\ldots,X_k)$ from asymptotic behavior of $\mathbb{E}[(X_1^n+\cdots+X_k^n)^m]$
For fixed $k$ suppose we have $X_1,\ldots,X_k$ non-negative random variables with density functions.
Setting a): We know $\mathbb{E}[(X_1^n+\cdots+X_k^n)^m]$ exactly for any integers $n,m \in \mathbb{...
- 1,295
1
vote
1
answer
185
views
Does $\sum_{n=1}^\infty e^{-n^2 T} \int_0^T e^{n^2 t} \lvert f(t)\rvert \, dt$ converge for $L^1_\text{loc}$ $f : [0,\infty) \to \mathbb{R}$?
Let $f(t) : [0,\infty) \to \mathbb{R}$ be an $L^1_\text{loc}$ function.
Then, I wonder if the following series
\begin{equation}
\sum_{n=1}^\infty e^{-n^2 T} \int_0^T e^{n^2 t} \lvert f(t)\rvert
\, dt
...
- 3,477
11
votes
2
answers
2k
views
L'Hopital rule for upper and lower limit?
I am reading the following paper 1998(H.Hudzik) P.574
It reads using L'Hopital rule$$\liminf_{u\to\infty} \frac{1/\varphi(1/u)}{\psi(u)}=\liminf_{u\to\infty}\frac{\varphi'(u)}{\psi'(u)u^2[\varphi(1/u)]...
15
votes
2
answers
1k
views
If a function $f$ is $(1+\varepsilon)$-times Lebesgue differentiable everywhere, is $f$ a constant function?
Let $f: \mathbb R^n \to \mathbb R$ be a locally integrable function. We say $x \in \mathbb R^n$ is a strong Lebesgue point of $f$ if
$$\lim_{r \to 0} \frac{\int_{B_r (x)} |f(y) - f(x)| \, dy}{r^{n+1+\...
- 6,321
0
votes
0
answers
98
views
Tangent spaces of Lipschitz sub manifolds
Consider $\mathbb{R}^n$, $k<n$, and topological embeddings (homeomorphisms onto image) $f_i : \mathbb{R}^k \supseteq B_1(0) \to \mathbb{R}^n$, $i=1,2$, which are also Lipschitz continuous and ...
- 403
2
votes
1
answer
181
views
Matrix function as gradient
Let $S_n^{++}(\mathbb{R})$ be the space of $n \times n$ symmetric positive definite matrices. For $M \in S_n^{++}(\mathbb{R})$ consider the function $f: X \in S_n^{++}(\mathbb{R}) \mapsto M X^{-1} M$.
...
- 407
4
votes
2
answers
245
views
On the monotonicity of the ratio of two logarithmic expressions
According to this comment and this comment, a positive answer to this recent question (about Bernoulli numbers) would be sufficient to prove the following:
$r:=f/g$ is increasing on $(0,\pi/2)$ from $...
- 128k
4
votes
1
answer
523
views
Is there any strengthened version of Rademacher's Theorem or any counterexample?
The following theorem is well-known in the ordinary analysis textbook:
Theorem: Assume the function $f:U\to\Bbb R^n$ is Lipschitz continuous on an open set $U\subset\Bbb R^m$, then $f$ is almost ...
- 300
3
votes
1
answer
379
views
Convergence of a power series
Consider the numbers $$a_n=\frac{1}{n+1}\sum_{k=0}^{n}\frac{2^{k-1}\binom{n+1}{k}B_k}{2^{s+k-1}-1}, \ n\geq0,$$ where $s\neq1;0;-1;-2;-3;...$ is a fixed real number, and the $B_k$ are the Bernoulli ...
- 463
38
votes
13
answers
5k
views
Continuous relations?
What might it mean for a relation $R\subset X\times Y$ to be continuous, where $X$ and $Y$ are topological spaces? In topology, category theory or in analysis? Is it possible, canonical, useful?
I ...
- 862
2
votes
1
answer
187
views
Local equality of functions implies global equality?
The following question arised in my research, and I was unable to settle it after playing with it for sometime. Let $\{a^k_i\}_{i\geq 1}$ (for $k\in \{1,2,3,4\}$) be four sequences of real numbers. ...
- 1,117
3
votes
1
answer
353
views
Sequential separability on $C_p(X)$
Definition. Let $E$ be a topological space. Suppose that $E$ contains a sequence $\{x_n\}$ such that for every $x\in E$, there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ with $x=\lim x_{n_k}$. ...
- 4,058
54
votes
3
answers
4k
views
Does every real function have this weak continuity property?
In my research I came across the following question :
Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}$, there exists a real sequence $(x_n)_n$, taking infinitely many values, ...
- 4,074
1
vote
0
answers
166
views
Monotone likelihood ratio of convolved power function kernel, $p\ge 3$
It was shown in a previous answer that for: $f(x)=|x|^p$, $\;x\in \mathbb{R}$, $\;p>2$, defining the density:
$$p(\phi;\theta) = \Big(f\big(\hspace{-1pt}\cos(\phi-\theta)\big) - f\big(
\hspace{-1pt}...
- 391
3
votes
1
answer
166
views
A limit arising from Mellin Inversion: How to compute a specific term of an asymptotic series?
So I am wondering if there exists a general procedure for the following problem:
given a monotonically increasing function $f(n)$ which is nonegative on the interval $[0,\infty)$ and grows faster than ...
- 2,689
16
votes
1
answer
3k
views
Did Euler know (unconsciously) to integrate by differentiating?
Considering a method to find the anti-derivative of an (sufficiently smooth) real function by differentiating published some years ago (equation (48) in Kempf et al., New Dirac Delta function based ...
- 4,014
1
vote
2
answers
188
views
Does a measurable $F :[0, T] \to L^p (\mathbb R^d; \mathbb R_{\ge 0})$ have a "flattened" measurable version?
Let $d \in \mathbb N^*,p \in [1, \infty]$ and $T>0$. Let
$$
F :[0, T] \to L^p (\mathbb R^d; \mathbb R_{\ge 0}), t \mapsto F_t
$$
be measurable. I would like to ask if there is a measurable function ...
- 825
4
votes
1
answer
256
views
If a function $f$ is $\varepsilon$-times Lebesgue differentiable, is $f$ continuous?
Let $f: \mathbb R^n \to \mathbb R$ be a locally integrable function. Given an $\varepsilon > 0$, we say $f$ is $\varepsilon$-times Lebesgue differentiable if
$$\lim_{r \to 0} \frac{\int_{B_r (x)} |...
- 6,321
0
votes
1
answer
139
views
A probability distribution, with Fourier transform smaller than $C \exp(-ct^2)$
Is there a probability distribution $\mu$ (with reasonably nice density $f$ on $\mathbb{R}$) such that the Fourier transform (aka. characteristic function) $\psi_\mu(t) = \int_{\mathbb{R}} e^{itx} \, ...
- 1,295
2
votes
1
answer
210
views
What is a subset of $\mathbb{Z}^3$ making $\Bigl( \sin(n \cdot x),\cos(n \cdot x) \Bigr)_{n \in \mathbb{Z}^3}$ linearly independent?
This question was originally posted in ME: https://math.stackexchange.com/questions/4725157/what-is-an-explicit-subset-of-mathbbz3-that-makes-bigl-sinn-cdot-x
but more and more I think about it, this ...
- 3,477
1
vote
1
answer
125
views
Approximation of two densities with a single transformation
Let $p_1$ and $p_2$ be two probability densities and $X_i\sim N(\mu_i,\Sigma_i)$. Write $w(X)\sim p$ if the law of the random variable $w(X)$ has a density equal to $p$. For general densities $p_i$, ...
- 63
15
votes
2
answers
474
views
Generalizations of summation methods of divergence series
If one looks at the "summation proofs" of divergent series such as Grandi's series, one might see a pattern that most of the computation rely on linearity and comparability with the shift ...
31
votes
2
answers
3k
views
A natural construction of real numbers?
Summary
Someone claims $\mathbb{R}$ can be constructed as the following intriguing quotient, which is related to Gromov's bounded cohomology. I want to find out if it is true.
$$\frac{\bigl\{f:\mathbb{...
- 5,230
3
votes
1
answer
219
views
Is there a real/functional analytic proof of Cramér–Lévy theorem?
In the book Gaussian Measures in Finite and Infinite Dimensions by Stroock, there is a theorem with a comment
The following remarkable theorem was discovered by Cramér and Lévy. So far as I know, ...
- 657
6
votes
2
answers
463
views
Spectrum of operator involving ladder operators
The ladder operator in quantum mechanics are the operators
$$a^\dagger \ = \ \frac{1}{\sqrt{2}} \left(-\frac{d}{dq} + q\right)$$
and
$$a \ \ = \ \frac{1}{\sqrt{2}} \left(\ \ \ \!\frac{d}{dq} + q\...
2
votes
0
answers
188
views
Self-adjointness of fractional laplacian
Lets consider the following functional analytic definition of the fractional Laplacian: Consider a (complete, connected, oriented) Riemannian manifold $(M,g)$ with corresponding Laplacian $\Delta_{g}$....
- 1,171
-3
votes
1
answer
167
views
Is there a simple function similar to exp? [closed]
As far as I know exp have such properties:
$f'(x) >0$
$f''(x) >0$
$\lim_{x \to -\infty}f(x)=0$
$\lim_{x \to +\infty}f(x)=\infty$
$f(x)f(-x)=1$
Let's say f(x) comply such rules.
The closest I ...
- 1
11
votes
3
answers
890
views
Structure theorems for compact sets of rationals
Everyone knows the Heine-Borel theorem characterizing compact subsets of Euclidean space. For any $n \in \mathbb N$ a set $A \subseteq \mathbb R^n$ is compact just in case it is closed and bounded (in ...
- 1,882
4
votes
1
answer
288
views
A lower bound for the $L^1$ norm of real trigonometric polynomials
This question is somewhat similar to Minimizing the L1 norm of odd-term trigonometric polynomial. The context of the question is based on the paper Hardy's Inequality and the $L^1$ norm of Exponential ...
- 270
7
votes
0
answers
204
views
Permutations which change the value of a convergent series
I'm interested in the following combinatorial problem: What is a necessary and sufficent condition on a permutation $\sigma : \mathbb{N} \rightarrow \mathbb{N}$, so that there exist a summable ...
- 71
0
votes
0
answers
53
views
Non-linearity of viscosity solutions
I am interested in the following problem.
Let consider the solution of the non-linear PDE on $[0,T]\times\mathbb{R}$ satifying the following Cauchy problem:
$$
\begin{cases}
u_t = F(u_{xx}),\\
u(0,x) =...
- 393
2
votes
1
answer
294
views
Are the jumps of a càdlàg function "summable"?
This question is motivated by the question https://math.stackexchange.com/questions/4644235/ on Math Stack Exchange. First, I need to define a notion of transfinite summability that I have not seen ...
- 708
8
votes
3
answers
1k
views
An infinite series that converges to $\frac{\sqrt{3}\pi}{24}$
Can you prove or disprove the following claim:
Claim:
$$\frac{\sqrt{3} \pi}{24}=\displaystyle\sum_{n=0}^{\infty}\frac{1}{(6n+1)(6n+5)}$$
The SageMath cell that demonstrates this claim can be found ...
- 2,661
7
votes
3
answers
524
views
Rigorous estimates on roots of function
We consider the function
$$f(x) = 1- \frac{1}{N} \sum_{i=1}^N \frac{\sin\left(\tfrac{\pi i}{N}\right)^2}{1+\sin\left(\tfrac{\pi i}{2N}\right)^2-x}.$$
The arguments of the two sines differ by a factor ...
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
0
answers
76
views
Is this extension of n-th derivatives to ordinal-indexed derivatives trivial? [duplicate]
Let $f$ be a function defined everywhere on the real line, which is infinitely differentiable everywhere, in other words, $f$ is everywhere smooth. I define the $\omega$-th derivative, where $\omega$ ...
- 2,023
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
2
votes
1
answer
210
views
Argmax of a function of $n$ variables under linear constraint
(I start by saying that the tags are probably not accurate but I didn't know what to put, so if someone knows what I could tag this question with, let me know in the comments and I'll provide to edit ...
- 739
3
votes
0
answers
245
views
Norm on the space of real analytic functions
The space $C^{\omega}(\Omega)$ of real-valued real analytic functions on the open bounded set $\Omega\subset \mathbb R^n$ does not have any obvious or natural metric which would make it a Fréchet ...
- 203
3
votes
0
answers
154
views
Inequality involving convolution roots
I am struggling with the following problem. Let $f$ be a real smooth function. Let assume that $f$ is:
increasing
strictly convex on $(-\infty,0)$
strictly concave on $(0,+\infty)$
Let $\sigma>0$ ...
- 393
0
votes
1
answer
92
views
If $f(x,t)=\sum_{n \in \mathbb{Z}} a_n(t) e^{in x}$ is $C^\infty$ in $x$ and all $a_n(t)$ continuous, $x$ derivatives of $f$ are continuous in $t$?
This question seem a bit elementary, but I find it more subtle than its looks. So, I post the question here.
Let $f(x,t) : [0,2\pi] \times [0,1] \to \mathbb{C}$ be a function such that $f(0,t)=f(2\pi,...
- 3,477
1
vote
1
answer
151
views
Monotone likelihood ratio of densities based on power function
Given $p,\phi,\theta \in \mathbb{R}$ such that $p>2$ and $0 \le \phi,\theta\le \pi/2$ define the density function:
$$f(\phi;\theta) =
\mbox{$\Large\frac{1}{p B\big(\hspace{-1pt}\frac{3}{2},\frac{p+...
- 391
3
votes
2
answers
203
views
Recovering a set from its projections in varying coordinate systems - a projection hull?
Let me describe the simplest non-trivial case of what I have in mind. Let $V$ be a 2-dimensional $\mathbb{R}$-vector space and fix an isomorphism $V \cong \mathbb{R}^2$, where $\mathbb{R}^2$ is ...
- 7,127
1
vote
0
answers
79
views
Asymptotics of ${2n \choose n+k} {2n \choose n}^{-1}$ when $k$ grows with $n$
The quotient $Q(n,k) := \frac{{2n \choose n+k}}{{2n \choose n}}$ clearly converges to one for $k \in \mathbb{N}$ fixed and $n \rightarrow \infty$. Simultaneously it converges to zero, if $k$ grows ...
- 1,295
3
votes
1
answer
135
views
Recover an $L^1$ integrand by partial differentiation
Denote by $m$ the 2-dimensional Lebesgue measure on $\mathbb{R}^2$. Let $f$ be an element of $L^1(m)$ that takes only nonnegative values. Define $F : \mathbb{R}^2 \rightarrow [0,\infty)$ by
$$F(x,y) = ...
- 41
10
votes
2
answers
1k
views
Proof in constructive mathematics that the principal square root function exists in any Cauchy complete Archimedean ordered field
In classical mathematics, there exists only one Cauchy complete Archimedean ordered field, the Dedekind complete Archimedean ordered field. However, in constructive mathematics, there are multiple ...
- 2,624
7
votes
3
answers
547
views
Maximal Hausdorff dimension of the set on which derivatives do not agree
Let $f, g: [0, 1] \to \mathbb R$ be functions that are differentiable a.e. with $f’ = g’$ almost everywhere. What is the supremal Hausdorff dimension of the set on which $f$ and $g$ are both ...
- 6,321
5
votes
1
answer
534
views
Minimiser of a certain functional
Let $f_i \in L^1 ([0, 1])$ be a sequence of functions equibounded in $L^1$ norm - that is, there exists some $M > 0$ such that $\|f_i\|_{L^1} < M$.
Define the functional $F: L^1([0, 1]) \to \...
- 6,321
2
votes
1
answer
170
views
Log-concavity of the difference of the second anti-derivative of Gaussians
I would like to prove the following but I couldn't manage to do it. Let $a>b>0$ be two real numbers. Let $f$ be the function defined as:
$$\forall \sigma>0, ~\forall x\in\mathbb{R},~f_\sigma(...
- 393