All Questions
5,909 questions
2
votes
1
answer
135
views
How to calculate this integral of squared Tricomi hypergeometric function
How to solve this integral
$$
\int_{0}^{\infty}r^2 e^{-\omega r^2}U(-\nu,\frac{3}{2},\omega r^2)^2 \mathrm{d}r
$$
where $\omega>0$ and $\nu \in \mathbb{R} \setminus \left \{ \frac{n-1}{2}\mid n \in ...
29
votes
2
answers
4k
views
Closed formula for a certain infinite series
I came across this problem while doing some simplifications.
So, I like to ask
QUESTION. Is there a closed formula for the evaluation of this series?
$$\sum_{(a,b)=1}\frac{\cos\left(\frac{a}b\right)}{...
- 43.2k
3
votes
1
answer
332
views
Sparse representation for continuous function?
I recently came across the field of "Sparse representation".
A talk is given here : https://www.youtube.com/watch?v=2bW4TkfTk-M.
The goal of sparse representation is taking a signal and ...
- 115
7
votes
1
answer
371
views
Duality of $H^1$ and BMO
While proving that the dual of $H^1$ is $BMO$ in Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, page 143, Stein says that we have $\left\Vert g \right\Vert_{H^1} \...
- 71
-2
votes
1
answer
283
views
Does convergence in probability implies L^1 convergence in probability density function, for bounded random variables?
Let $X_1,X_2,\cdots$ and $Y$ be random variables on $[0,1]$ with smooth density functions $f_1,f_2\cdots$ and $f$. Suppose $X_n\to Y$ in probability. Can we get some convergence of the density ...
0
votes
1
answer
509
views
Possible research directions in analysis? [closed]
I am an undergraduate student who loves basic mathematics in the analysis branch, but I have learned that some directions, for example, harmonic analysis, are already well developed and difficult to ...
- 101
4
votes
1
answer
287
views
Local maxima of the sum of Gaussian functions in *multiple dimensions* are always strict local maxima - prove/disprove/prove conditionally?
This is a follow up of the question in one dimension, that asked to show that the all the maxima of the sum of Gaussian
$$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, x_1 < x_2 < \dots < x_n$$
are ...
- 1,467
3
votes
2
answers
296
views
Domain of spectral fractional Laplacian
Let $(M,g)$ be a complete Riemannian manifold with Laplacian $\Delta:C^{\infty}_{c}(M)\to C^{\infty}_{c}(M)$ (think of $\mathbb{R}^{d}$ if you wish). This operator is essentially self-adjoint in $L^{2}...
- 1,171
0
votes
0
answers
180
views
Proof that the zeroes of certain polynomials are increasing with respect to degree
Choose $k+1$ positive integers $d_j\in\{0,1,2,3,\ldots\}$ and let $d=(d_1,\ldots,d_k)$.
Consider the following polynomial equation over the positive reals:
$$
\sum_{j=1}^{k}\frac1{x^{d_j}} = x^{d_{k+1}...
- 21
17
votes
3
answers
975
views
Evaluating the sum $f(x):=\sum_{n=1}^\infty \frac{1}{n! n^n}(-x^2)^n$ and estimating bounds
For real variable $x$, the function
\begin{equation}
f(x):=\sum_{n=1}^\infty \frac{1}{n! n^n}(-x^2)^n
\end{equation}
clearly has infinite radius of convergence and defines a $C^\infty$ function on $\...
- 3,477
6
votes
3
answers
536
views
A need for analytic continuation of a finite sum function
Let $\varphi(n):=(-1)^{n+1}(n+1)2^{2n}$.
I am able to prove the following identity (${\color{red}{\mathbf{LHS}}}$=infinite series, ${\color{blue}{\mathbf{RHS}}}$=finite sum)
\begin{align*}
{\color{red}...
- 43.2k
0
votes
1
answer
149
views
Equi-coercivity of functionals on a metric space
Definition: A family of functionals $\{F_n: X\to\bar{\mathbb R}\}$ on a metric space $X$ is said to be equi-coercive if, for every $\alpha \in \mathbb{R}$, there is a compact set $K_\alpha$ of $X$ ...
- 1,101
1
vote
0
answers
54
views
Isoperimetric Inequalities in Annular Regions
Let $\Omega$ be an open set in $\mathbb{R}^2$ whose boundary is a rectifiable Jordan curve. Then an old result by Alfred Huber states that
$$
\left(\int_{\partial \Omega} e^u ds\right)^2 \geq 2 \left(...
- 434
5
votes
1
answer
512
views
Norm inequality for the inclusion $L^2(\partial \Omega)\hookrightarrow H^{-1/2}(\partial \Omega)$
Let $\Omega \subset \mathbb{R}^3$ be a lipschitz domain. We then have the trace operator $\tau : H^1(\Omega) \to L^2(\partial \Omega)$ and can define the space $H^{1/2}(\partial \Omega) := \tau(H^1(\...
- 75
1
vote
1
answer
106
views
Upper bound $I (t) := \sup_{x \in \mathbb R^d} \int_{\mathbb R^d} \frac{|x-y|^\alpha}{t^{d/2}} \exp ( - \frac{|\psi(x) - y|^2}{t} ) \, \mathrm d y$
Let $\alpha \in (0, 1)$ and $\psi : \mathbb R^d \to \mathbb R^d$ be a $C^\infty$-diffeomorphism such that $\|\nabla \psi\|_\infty + \|\nabla \psi^{-1}\|_\infty < + \infty$. Let
$$
I (t) := \sup_{x \...
- 825
16
votes
6
answers
2k
views
Alternative proofs sought after for a certain identity
Here is an identity for which I outlined two different arguments. I'm collecting further alternative proofs, so
QUESTION. can you provide another verification for the problem below?
Problem. Prove ...
- 43.2k
2
votes
2
answers
424
views
"Squeezing" the primes?
The logical idea here is to map a curve that encodes the primes into the region $(0,1)^2$ and analyze the distribution there more easily and achieve tight bounds.
To assess the distribution of primes, ...
- 383
8
votes
3
answers
545
views
Approximation of pseudogeometric progression
Let $f_n(x)=1+x+x^{\sqrt{2}}+x^{\sqrt{3}}+x^{\sqrt{4}}+\cdots+x^{\sqrt{n}}$ be a sequence of functions on the interval $[0, 1]$. Is there a good closed form approximation for such a function ( ...
- 728
1
vote
1
answer
179
views
The function $G(x) =(4\pi t)^{-d/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}|y|^k dy$ can be controlled when $|x|\rightarrow \infty$
In this paper, Lemma 6, Pinsky proves that $$H(x) =(4\pi t)^{-d/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}(1+|y|)^m \, dy$$ attains its maximum in $x=0$ for $m<0$. This can also be proven using ...
- 677
10
votes
2
answers
612
views
Proving the simple form of a function from statistical mechanics
I have discovered a pertinent solution to my problem in the article On the Kinetic Theory of Rarefied Gases by Harold Grad and the book Thermodynamik und Statistik by Arnold Sommerfeld, both of which ...
2
votes
1
answer
138
views
Boundedness of an exit time from a campact set
Let $n\geq 1$ and $v\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$. For $x_0\in\mathcal{O}$, let $\big(x(t)\big)_{t\geq 0}$ be the solution of
\begin{align*}
& x(0)=x_0 \\
& \dot{x}=v(x).
\end{...
- 449
23
votes
9
answers
2k
views
Nonseparable counterexamples in analysis
When asking for uncountable counterexamples in algebra I noted that in functional analysis there are many examples of things that “go wrong” in the nonseparable setting. But most of the examples I'm ...
3
votes
1
answer
145
views
Let $\mu : [0, T] \to \mathcal P_2^a (\mathbb R^d), t \mapsto \mu_t$ be absolutely continuous. Is $t \mapsto \mathcal H (\mu_t)$ continuous?
We endow the space $\mathcal P_2^a (\mathbb R^d)$ of absolutely continuous probability measures with finite second moment with the Wasserstein distance $W_2$. Let $\mathcal H (\mu)$ be the relative ...
- 825
0
votes
0
answers
96
views
Hilbert spaces that include algebraic polynomials
This question is motivated by a phrase I found in several books/papers about approximation theory, for example, M.J.D.Powell's Approximation Theory and Methods: ''Let $\mathcal{H}$ be a Hilbert space ...
- 1
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 $$-i \int_{\gamma_\varepsilon} \nabla \log(f(s)) \cdot ds := - i \int_0^1 \nabla (\log f)(\...
2
votes
1
answer
152
views
Growth rate of elementary sequences
We consider three sequences $(x_n),(y_n),(z_n)$, where $(x_n) \in \ell^1$ is positive and the other two sequences are merely assumed to be positive, i.e. $y_n,z_n \ge 0$ where $0<z_n<z_{n+1}$ is ...
0
votes
1
answer
185
views
Can we approximate a Hölder pdf by higher-order Hölder pdf's?
$\newcommand{\RR}{\mathbb R}\newcommand{\NN}{\mathbb N}$
Let $\alpha \in (0, 1)$ and $j \in \NN$. We denote by $H^{j + \alpha} := H^{j + \alpha} ({\RR}^d)$ the space of real-valued functions $f$ on $\...
- 825
29
votes
1
answer
1k
views
About the function $\prod_{k \in \mathbb{N}}(1-\frac{x^3}{k^3})$
I'm wondering if the function $$f(x)=\prod_{k \in \mathbb{N}}\left(1-\frac{x^3}{k^3}\right)$$ has a name, or if there are any properties (especially about derivatives of $f$) have studied so far.
I ...
- 483
0
votes
0
answers
71
views
Nearest integer to fractional power series
Let $k$ be a positive integer. Let
$$\displaystyle f_0(x) = a_n x^{\frac{n}{k}} + \cdots + a_1 x^{\frac{1}{k}} + a_0 + \sum_{h \geq 1} a_{-h} x^{-\frac{h}{k}}$$
be a Laurent series in the variable $x^{...
- 26.9k
-1
votes
1
answer
110
views
Proving that $\max_{w \in B(z)} e^{f(w)} \leq Ce^{f(z)}$
Let $f : \mathbb R^2 \to \mathbb R $ be a smooth function statisfying
$$
0 < \alpha \leq \Delta f(w) \leq \beta < \infty, \ \ \forall w \in \mathbb R^2
$$
where $\Delta$ denotes the Laplace ...
- 437
23
votes
5
answers
2k
views
Axiomatic construction of trigonometric functions
I am able to construct functions $\sin,\cos\colon \mathbb R \to \mathbb R$ satisfying the following properties:
$\sin^2 x + \cos^2 x = 1$,
$\sin(x+y)=\sin x \cos y + \sin y\cos x$, $\cos(x+y)=\cos x \...
- 702
1
vote
1
answer
170
views
fourth-order multivariate Gaussian integral
I am struggling with an integral of form
$$ \int_{\mathbb R^n} y\otimes y~ \langle Ay,y\rangle \, \mathrm d N(0,\Sigma)(y). $$
I assume that it will involve the trace of some product of $R$ and $\...
- 375
4
votes
1
answer
178
views
Compact-open Topology for Partial Maps?
I asked the same question on MathStackExchange a month ago and received no answer. I feel that this would be more suitable for MathOverflow.
Compact open topology is one of the most common ways of ...
- 1,093
1
vote
3
answers
183
views
Evaluating a sinusoidal series
Define the sequence of functions
$$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 ...
- 151
1
vote
1
answer
143
views
Projection of an element of the $n$-simplex onto subset
Let $\mathbb{S}^{n}$ denote the $n$-dimensional probability simplex and let $\{e_1,...,e_{n+1}\}$ be the canonical basis of $\mathbb{R}^{n+1}$. Consider the subset $\mathbb{S}^{n}(K) \subset \mathbb{S}...
1
vote
1
answer
175
views
Analyzable functions and accelero-summation
Is there a complete and rigorous, yet concise, definition of what an analyzable function is, along with the related notion of accelero-summation, both in the sense of Écalle? All of the definitions I ...
- 4,985
25
votes
3
answers
2k
views
Does there exist a continuous function $f(x)$ such that $f(0)=0$ and $0<\lim_{n\to\infty}\prod_{k=1}^n f(k/n)<\infty$?
Does there exist a continuous function $f(x)$ such that $f(0)=0$ and $0<\lim\limits_{n\to\infty}\prod\limits_{k=1}^n f(\frac{k}{n})<\infty$ ?
I do not see any reason why such a function could ...
- 3,577
3
votes
2
answers
293
views
On convergence of convex-concave functions
Let $(f_n)$ be a sequence of twice differentiable functions on $\mathbb R$ such that for each $n$ there exists some $x_n\in\mathbb{R}$ such that:
$f_n$ is strictly convex on $(-\infty,x_n)$,
$f_n$ is ...
- 128k
-1
votes
1
answer
80
views
Regions when a concave function is smaller than another concave function
Let $f_1,f_2:[0,1]\mapsto\mathbb{R}$ be two bounded and concave functions. Assume $f_1(0)<f_2(0)$ and $f_1(1)<f_2(1)$. I want to investigate the set $\mathcal{X}\triangleq\{x\in[0,1]: f_1(x)>...
- 53
1
vote
0
answers
86
views
The intersection of $ n $ cylinders in $ 3D$ space
I posted the question on here, but received no answer
I recently found out about the Steinmetz Solids, obtained as the intersection of two or three cylinders of equal radius at right angles. If we set ...
- 387
2
votes
0
answers
95
views
Can we control the Wasserstein metric between $\mu$ and $\nu$ by their moment difference?
Fix $p \in [1, \infty)$. Let $(\mathcal P_p(\mathbb R^d), W_p)$ be the Wasserstein space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. Let $D_p$ be the collection of ...
- 657
30
votes
1
answer
2k
views
Have any numbers been proven to be normal that weren't constructed to be?
It's easy to construct an example of a number that's normal in a given base, but for most given numbers it's notoriously hard to prove that they're normal.
Has any number ever been proven to be normal ...
- 1,311
6
votes
1
answer
611
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
0
answers
108
views
Archimedean ordered field in which every function is smooth
In constructive mathematics, it is consistent that every function $\mathbb{R} \to \mathbb{R}$ on the Dedekind real numbers is continuous. However, it is not consistent that every function $\mathbb{R} \...
- 2,624
1
vote
1
answer
66
views
Upper bound $I_R := \int_{B_R^c} |x| (P_t \ell_\nu) (x) \, \mathrm d x$ in terms of $R, \nu, t$?
Let $(p_t)_{t >0}$ be the Gaussian heat kernel on $\mathbb R^d$ and $(P_t)_{t >0}$ its induced semi-group, i.e.,
$$
\begin{align}
p_t (x) &:= (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\...
- 825
0
votes
0
answers
619
views
The set of continuous bounded functions $f:X\to Y$ is dense in $L^p(X,Y)$ where $X,Y$ are Polish
It is well known that the set of real-valued continuous functions with compact support is dense in $L^p(\mu)$ where $\mu$ is a Radon measure (see e.g. [Folland, Proposition 7.9]) Clearly, the set of ...
- 305
3
votes
1
answer
198
views
Do radially bounded sets form a bornology?
We call a subset $A$ in a real vector space $E$ radially bounded if it intersects every ray emanating from $0$ via a bounded set. It is easy to see that radially bounded sets in $E$ form a bornology, ...
- 5,529
0
votes
0
answers
71
views
Reference request for equivalent Lipschitz smoothness conditions
For an open set $Z\subseteq\mathbb{R}^n$, let $f: Z\mapsto \mathbb{R}$ be a continuously differentiable function on $Z$, and let $L>0$ be fixed. Also, suppose that (a) $f$ is nonconvex and (b) $f$ ...
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 \...
- 6,321
9
votes
2
answers
793
views
Uniformly Lebesgue differentiable functions
Note: Here $\mu$ denotes Lebesgue measure on $\mathbb R$.
We say a function $f: \mathbb R \to \mathbb R$ is uniformly Lebesgue differentiable if there exists some measurable subset $E$ of $\mathbb R$ ...
- 6,321