All Questions
5,909 questions
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 ...
- 3,487
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.
$$\frac{\bigl\{f:\mathbb{...
- 5,230
1
vote
1
answer
125
views
Approximation of two densities with a single transformation
Let $p_1$ and $p_2$ be two probability densities and $X_i\sim N(\mu_i,\Sigma_i)$. Write $w(X)\sim p$ if the law of the random variable $w(X)$ has a density equal to $p$. For general densities $p_i$, ...
- 63
15
votes
2
answers
474
views
Generalizations of summation methods of divergence series
If one looks at the "summation proofs" of divergent series such as Grandi's series, one might see a pattern that most of the computation rely on linearity and comparability with the shift ...
3
votes
1
answer
220
views
Is there a real/functional analytic proof of Cramér–Lévy theorem?
In the book Gaussian Measures in Finite and Infinite Dimensions by Stroock, there is a theorem with a comment
The following remarkable theorem was discovered by Cramér and Lévy. So far as I know, ...
- 657
6
votes
2
answers
463
views
Spectrum of operator involving ladder operators
The ladder operator in quantum mechanics are the operators
$$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
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
-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
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
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
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:
$$
\begin{cases}
u_t = F(u_{xx}),\\
u(0,x) =...
- 393
13
votes
2
answers
915
views
Topological vector spaces (reference request)
In his book Topological Function Spaces Arhangel'skii says that "it is well known that every nontrivial locally convex linear topological space $X$ is homeomorphic to a space of the form $Y \...
- 674
8
votes
3
answers
1k
views
An infinite series that converges to $\frac{\sqrt{3}\pi}{24}$
Can you prove or disprove the following claim:
Claim:
$$\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)
$$
be the ...
- 127
2
votes
1
answer
210
views
Argmax of a function of $n$ variables under linear constraint
(I start by saying that the tags are probably not accurate but I didn't know what to put, so if someone knows what I could tag this question with, let me know in the comments and I'll provide to edit ...
- 749
3
votes
0
answers
245
views
Norm on the space of real analytic functions
The space $C^{\omega}(\Omega)$ of real-valued real analytic functions on the open bounded set $\Omega\subset \mathbb R^n$ does not have any obvious or natural metric which would make it a Fréchet ...
- 203
3
votes
0
answers
154
views
Inequality involving convolution roots
I am struggling with the following problem. Let $f$ be a real smooth function. Let assume that $f$ is:
increasing
strictly convex on $(-\infty,0)$
strictly concave on $(0,+\infty)$
Let $\sigma>0$ ...
- 393
0
votes
1
answer
92
views
If $f(x,t)=\sum_{n \in \mathbb{Z}} a_n(t) e^{in x}$ is $C^\infty$ in $x$ and all $a_n(t)$ continuous, $x$ derivatives of $f$ are continuous in $t$?
This question seem a bit elementary, but I find it more subtle than its looks. So, I post the question here.
Let $f(x,t) : [0,2\pi] \times [0,1] \to \mathbb{C}$ be a function such that $f(0,t)=f(2\pi,...
- 3,487
3
votes
2
answers
203
views
Recovering a set from its projections in varying coordinate systems - a projection hull?
Let me describe the simplest non-trivial case of what I have in mind. Let $V$ be a 2-dimensional $\mathbb{R}$-vector space and fix an isomorphism $V \cong \mathbb{R}^2$, where $\mathbb{R}^2$ is ...
- 7,127
1
vote
1
answer
151
views
Monotone likelihood ratio of densities based on power function
Given $p,\phi,\theta \in \mathbb{R}$ such that $p>2$ and $0 \le \phi,\theta\le \pi/2$ define the density function:
$$f(\phi;\theta) =
\mbox{$\Large\frac{1}{p B\big(\hspace{-1pt}\frac{3}{2},\frac{p+...
- 391
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
$$F(x,y) = ...
- 41
10
votes
2
answers
1k
views
Proof in constructive mathematics that the principal square root function exists in any Cauchy complete Archimedean ordered field
In classical mathematics, there exists only one Cauchy complete Archimedean ordered field, the Dedekind complete Archimedean ordered field. However, in constructive mathematics, there are multiple ...
- 2,624
7
votes
3
answers
547
views
Maximal Hausdorff dimension of the set on which derivatives do not agree
Let $f, g: [0, 1] \to \mathbb R$ be functions that are differentiable a.e. with $f’ = g’$ almost everywhere. What is the supremal Hausdorff dimension of the set on which $f$ and $g$ are both ...
- 6,313
5
votes
1
answer
534
views
Minimiser of a certain functional
Let $f_i \in L^1 ([0, 1])$ be a sequence of functions equibounded in $L^1$ norm - that is, there exists some $M > 0$ such that $\|f_i\|_{L^1} < M$.
Define the functional $F: L^1([0, 1]) \to \...
- 6,313
2
votes
1
answer
170
views
Log-concavity of the difference of the second anti-derivative of Gaussians
I would like to prove the following but I couldn't manage to do it. Let $a>b>0$ be two real numbers. Let $f$ be the function defined as:
$$\forall \sigma>0, ~\forall x\in\mathbb{R},~f_\sigma(...
- 393
1
vote
1
answer
191
views
Will this "tree" cover all rational numbers in a range?
Question
I am making a tree using the following two functions:
$$f(x)=\frac{x}{r},\quad g(x)=\frac{x+b}{r}$$
where $1<r<2$ and $0<b$ are rationals. Everything is a real number here.
The ...
- 433
1
vote
0
answers
122
views
Implicit function theorem / Implicit selections when Jacobian not invertible
I saw the attached result in the book by Dontchev and Rockafellar.
It requires the Jacobian to be of full rank m. I suspect this condition can be further relaxed. Assume that we know that the columns ...
- 393
4
votes
2
answers
305
views
Generalization of van der Corput's estimate on oscillatory integrals
Question: Given exponents $0<\alpha<\beta$ and an interval
$[a,b]\subset(0,\infty)$ are there constants $C,d>0$ such that for any
$\lambda_1,\lambda_2\in\mathbb{R}$,
$$\left|\int_a^be(\...
- 1,701
3
votes
1
answer
557
views
Is there real or complex analytic function whose positive real zeros are the primes?
Related to this question
Q1 Is there real or complex analytic function $f(x)$ such
that its positive real zeros are the primes and it is
given in closed form of compositions of already named ...
- 25.4k
3
votes
2
answers
2k
views
Expected gradient vs. gradient of expectation
Suppose a function $f(x): \mathbb R^d \mapsto \mathbb R^D$, and its stochastic approximator, $g(x; W): \mathbb R^d \mapsto \mathbb R^D$. Here $W$ is some random variable. Then $g(x; W)$ is unbiased in ...
- 205
16
votes
1
answer
888
views
Kakeya crossed-needles problem
The Kakeya needle problem asks for the
minimum area planar region in which one can completely turn around a line segment through
a series of translations and rotations. There is no minimum: There are &...
- 151k
4
votes
1
answer
209
views
Why is there a $\mathcal{H}^d$-null set in the definition of d-rectifiable set?
Given a set $A \subset \mathbb{R}^n$, this is called d-rectifiable if it can be covered by a countable union of images of lipshitz functions from $\mathbb{R}^d $ to $ \mathbb{R}^n $ and a $\mathcal{H}^...
- 749
3
votes
1
answer
242
views
Closed subset of unit ball with peculiar connected components
Let $n\geq 2$ and denote by $B\subset \mathbb{R}^n$ the closed unit ball.
Does there exist a closed subset $A\subset B$ containing $0\in \mathbb{R}^n$ with the following properties i,ii,iii?
i) $\{0\}$...
- 722
1
vote
0
answers
59
views
Study of the properties of a non-local ODE
I am studying the following non-local ODE
$$\dot p(x) \nu_{\varepsilon, \alpha}(x) + \int_{x}^{2x_0}\frac{\dot p(s)}{s + \varepsilon} ds = c \quad \text{for } x \in [0,2x_0].$$
The number $x_0$ can ...
- 452
9
votes
0
answers
522
views
Does the intersection of middle third and middle half Cantor sets contain an irrational number?
Let $C_\frac{1}{3}$ be the middle third Cantor set, that is, the set of real numbers in the interval $[0,1]$ which can be written in base $3$ using only digits $0$ and $2$.
Likewise let $C_\frac{1}{2}$...
- 2,934
27
votes
4
answers
8k
views
Proofs of Young's inequality for convolution
For $1\leq p,q \leq \infty$ such that $\frac1p +\frac1q\geq 1$, Young's inequality states $\|f\star g\|_r\leq \|f\|_p\|g\|_q$ (we work on $\mathbf{R}^d$ here), where $1+\frac1r = \frac1p+\frac1q$. ...
- 3,425
4
votes
1
answer
643
views
Explicit and fast error bounds for approximating continuous functions
Main Question
This question is about finding explicit, calculable, and fast error bounds (no hidden constants) when approximating continuous functions with polynomials or simpler functions to a user-...
- 697
1
vote
0
answers
70
views
A possible upper bound for a function that satisfies a singular integral inequality
I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality:
$$
|v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left(
|...
6
votes
2
answers
622
views
Forcing the uniqueness of a solution of an ODE
For $n\geq 1$, $f_n\in\mathcal{C}^1([0,1],\mathbb{R})$ such that $f_n(x)\geq\sqrt{x}$ for $x\in[0,1]$, and
$$\lim\limits_{n\to+\infty}\sup_{x\in[0,1]}\big|f_n(x)-\sqrt{x}\big|= 0.$$
Let $y_n$ be the ...
- 449
3
votes
1
answer
96
views
Let $g$ be the heat kernel. Are there constants $C_1, C_2>0$ such that $\frac{g(t_1, \cdot)}{t_1} \le C_1 \frac{g(C_2 t_2, \cdot)}{\sqrt{t_2}}$?
We consider the heat kernel
$$
g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R,\quad (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$
Let $0 < t_1 < t_2 &...
- 657
1
vote
2
answers
163
views
Transcendental functions with two prescribed values
Let $\alpha$ and $\beta$ two algebraic numbers lying in unit ball. Let $T:=(t_k)_k$ be an increasing sequence of positive integers such that $t_{k+1}/t_k$ tends to $1$ as $k\to \infty$.
I would like ...
- 515
7
votes
2
answers
538
views
How should I understand the "$C^\infty$ functions" whose domain is the dual of $C^\infty(\mathbb{R}^n)$?
I am reading Colombeau's book "New Generalized Functions and Multiplication of Distributions" and he uses the notation $C^\infty({C^\infty}'(\Omega))$ out of nowhere.
Here $\Omega$ is any ...
- 3,487
5
votes
2
answers
788
views
Do non-zero derivatives imply tangent lines (and vice versa)?
Let $\gamma : \mathbb{R} \rightarrow \mathbb{R}^2$ be any continuous function, with image given by $C_\gamma$.
We can say that $\gamma$ has an image tangent at $t \in \mathbb{R}$ if there exists $\...
- 700